DEFINITION OF LOGIC and TYPES OF LOGIC

Logic is the study of valid reasoning.

Logic is a word that comes from the Greek word λογική pronounced as logikē that means, the study of reasoning.

Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science.

Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking.

In philosophy, the study of logic falls in the area of epistemology, which asks: "How do we know what we know?"

In mathematics, it is the study of valid inferences within some formal language.

Logic has origins in several ancient civilizations, including ancient India, China and Greece.

Logic was established as a discipline by Aristotle, who established its fundamental place in philosophy.

The study of logic was part of the classical trivium.

Averroes defined logic as "the tool for distinguishing between the true and the false"

Richard Whately, defined logic as "the Science, as well as the Art, of reasoning"

Frege, defined logic as "the science of the most general laws of truth".

Logic is often divided into two parts, inductive reasoning and deductive reasoning.

The Inductive reasoning involves in drawing general conclusions from specific examples. We can also say that Inductive Reasoning involves in deriving at unknown conclusions from known facts. This is the reason why the conclusions of Inductive reasoning are probable and not certain.

The Deductive Reasoning involves in drawing logical conclusions from definitions and axioms. We can also say that deductive Reasoning involves in deriving known conclusions from known facts. As a result, the conclusions of deductive reasoning are certain.

Traditional classification of Propositions

Traditional logicians have divided propositions into singular and general.

Singular propositions have a single individual as a subject. This means, in a singular proposition, the subject is a singular individual thing and predicate is a class of individuals.

General propositions have a group of individuals as a subject. This means, in a General proposition, we have a group of individuals as a subject as well as a group of individuals as a predicate.

The general propositions are of two types, universal and general.

When the general proposition says something about the entire group indicated in the subject, it is known as a universal proposition.

When the general proposition says something about a part of the group indicated in the subject, it is known as a particular proposition.

Both singular and general propositions are either affirmative or negative. When we are told that the subject has the quality indicated in the predicate, the proposition is said to be affirmative. When we are told that the subject does not have the quality indicated in the predicate, the proposition is said to be negative.

In case of affirmative propositions, in singular proposition, the quality indicated in the group stated in the predicate is applicable to the individual indicated in the subject, while in general proposition, it either is applicable to the entire group indicated by the subject, as in universal propositions, or to a part of the group indicated by the subject, as in particular propositions.

In case of negative propositions, in singular proposition, the quality indicated in the group stated in the predicate is not applicable to the individual indicated in the subject, while in general proposition, it is either not applicable to the entire group indicated by the subject, as in universal propositions, or not applicable to a part of the group indicated by the subject, as in particular propositions.

According to this, the general propositions are classified into four categories.

These are:

A = Universal affirmative

E = Universal negative

I = Particular affirmative

O = Particular negative

In the next post, we shall see the relationship between these four types of general proposition types.

OPPOSITION OF PROPOSITIONS AND VENN DIAGRAMS

Traditional logicians classified propositions into two types, namely, singular and general.

When the subject of a proposition represents one single individual, it is a singular proposition.

When the subject of a proposition represents a group of individuals, it is a general proposition.

General propositions are further classified into Universal and Particular.

When the subject tells something about the whole group represented by it, the proposition is known to be universal.

When the subject tells something about some members of the group represented by it, the proposition is known to be particular.

The propositions are also classified using another criteria of quality and this makes them affirmative or negative.

So, both the singular as well as general propositions are either affirmative or negative.

As a result, we have four types of general propositions, as the general propositions have both the quality as well as quantity.

The four types of general propositions are:

A, E, I, & O.

The quantity and quality of these are as follows:

A = Universal affirmative
E = Universal negative
I = Particular affirmative
O = Particular negative

Relation of opposition between these propositions is as follows:

When two universal propositions differ in quality, they are known as CONTRARY.

When two particular propositions differ in quality, they are known as SUB-CONTRARY.

When two propositions with same quality, differ in quantity, they are known as SUB-ALTERN.

When two propositions differ both in quality and quantity, they are known as CONTRADICTORY.

The relation of truth values between these opposite propositions is as follows:

If a universal proposition is true, its contrary is false, its sub-altern is true and its contradictory is false.

If a universal proposition is false, its contrary is uncertain, its sub-altern is uncertain and its contradictory is true.

If a particular proposition is true, its sub-contrary is uncertain, its sub-altern is uncertain and its contradictory is false.

If a particular proposition is false, its sub-contrary is true, its sub-altern is false and its contradictory is true.

The general propositions represent the relation of two groups indicated by the subject and predicate, and so, they can be represented symbolically using the venn diagram method used in mathematics.

To do this, we use two intersecting circles.

The circle on the left represents the subject, and the one on the right, represents the predicate.

To represent "A" proposition, we shade the part of the circle of subject, that is outside that of predicate. This shows that the set of subject outside the predicate is empty.

To represent "E" proposition, we shade the part of the circle of subject, that is inside predicate. This shows that the set of subject inside the predicate is empty.

To represent "I" proposition, we put a cross in the part of the circle of subject, that is inside that of predicate. This shows that the set of subject inside the predicate is not empty.

To represent "O" proposition, we put a cross in the part of the circle of subject, that is outside that of predicate. This shows that the set of subject outside the predicate is not empty.

Pl check the images for venn diagrams and opposition of proposition.

When these propositions are symbolized, we change them in a specific format so that we can show the class membership of the subject and the predicate terms. This method is known as the method of Quantification and the symbols used to indicate the quantity of the subject are known as quantifiers.

Let us see how this is done:

Singular propositions:

Affirmative:

Ramu is a boy.

is symbolized as:
Br

Negative:

Sita is not a boy.

is symbolized as:
~Bs

General propositions:

These are of four kinds as we have seen earlier. They are symbolized as follows:

"A" proposition:

Subject-less:

Everything perishes.

will be written as:
Given any x, x is Perishable.

This is symbolized as follows:
(x)(Px)

With subject:

All S is P.

will be written as:
Given any x, if x is S, then x is P

This is symbolized as follows:
(x)(Sx>Px)

[Since the implication sign cannot be put due to font limits of the portal, so, a similar sign is put here]

"E" proposition:

Subject-less:

Nothing is Permanent.

will be written as:
Given any x, x is not Permanent.

This is symbolized as follows:
(x)(~Px)

With subject:

No S is P.

will be written as:
Given any x, if x is S, then x is not P

This is symbolized as follows:
(x)(Sx>~Px)

[Since the implication sign cannot be put due to font limits of the portal, so, a similar sign is put here]

"I" proposition:

Subject-less:

Lions exist

will be written as:
There is an x, such that, x is a Lion.

This is symbolized as follows:
(Ex)(Lx)

[Since the Existential quantifier sign that is actually reverse as a mirror image as actual E, cannot be put due to font limits of the portal, so, E is put here]

With subject:

Some S is P.

will be written as:
There is an x, such that, x is S and x is P.

This is symbolized as follows:
(Ex)(Sx.Px)

[Since the Existential quantifier sign that is actually reverse as a mirror image as actual E, cannot be put due to font limits of the portal, so, E is put here]

"O" proposition:

Subject-less:

Ghosts do not exist

will be written as:
There is an x, such that, x is not a Ghost.

This is symbolized as follows:
(Ex)(~Gx)

[Since the Existential quantifier sign that is actually reverse as a mirror image as actual E, cannot be put due to font limits of the portal, so, E is put here]

With subject:
Some S is not P.

will be written as:
There is an x, such that, x is S and x is not P.

This is symbolized as follows:
(Ex)(Sx.~Px)

[Since the Existential quantifier sign that is actually reverse as a mirror image as actual E, cannot be put due to font limits of the portal, so, E is put here]

When we symbolize, the first letter of the subject term is taken as a capital letter and small x is written after it to indicate the singular variable that is quantified in the beginning.

This is how we symbolize the general propositions in traditional classification.

ARISTOTELIAN SYLLOGISTIC DIVISION
Kinds of Division

• Logical division divides a class into its subclasses
– E.g., mammals into monotremes, marsupials & placentals
– Division is useful for
• determination of exact relationships among related things
• formulation of definitions

• Other kinds of division

– Physical division divides a whole into its parts
• E.g., a complex machine into its simple mechanical parts

– Metaphysical division divides an entity into its qualities,

• e.g.,a species into its genus & difference
– man into animality & rationality
• a substance into its attributes
– sugar into color, texture, solubility, taste, &c.
• a quality into its dimensions
– sound into pitch, timbre, volume

How to Divide

• Logical Division

– begins with a summum genus
– proceeds through intermediate genera
– ends at the infimae species
– NB: It does not continue to individuals

• The results of division should meet these criteria:

1. The subclasses of each class should be coextensive with the
original class.
2. The subclasses of each class should be mutually exclusive.
3. The subclasses of each class should be jointly exhaustive.
4. Each stage of a division should be based on a single principle.

Kinds of Classification

• Classification is the technique of inquiry in which similar individuals and classes are grouped into larger classes.
– E.g., how are steam, diesel, & gasoline engines related to one another?

Natural Classification
• Natural classification is a scheme that provides theoretical understanding of its subject matter
– E.g., classification of living things into monerans, protistans, plants, fungi and animals
• The concept “monerans” is now obsolescent because it does not provide sufficient theoretical clarity.

Artificial Classification
• Artificial classification is a scheme established merely to serve some particular human purpose
– E.g., classification of plants as crops, ornamentals, and weed

Classification and Division Compared

• The result of a classification will look like the result of a division.
• Classification begins with a individuals or small classes and works
towards a summum genus.
– i.e., it works in the direction opposite to that of division
• Classification begins with a set of apparently related things found in
the world (i.e., it is based on experience) and builds from there.
– Hence, it is well-suited to natural objects.
– But it will work with any kind of object.

Two Overly Ambitious Ideals

• Pure division
– begins with the summum genus and
– divides on the basis of a priori considerations
• i.e., it is based on logical possibility, not experience

• Dichotomous division
– divides on the basis of the presence or absence of a particular feature
• (NB: Classification can also be dichotomous.)
• Striving for these ideals
– works well with mathematical objects, &c.
– does not work well with natural objects (e.g., kinds of animals)
– guarantees a division that meets criteria (2) – (3)
– sometimes provides more insight than alternative divisions.
• But “ dichotomous division is often difficult and often impracticable”—Aristotle, Parts of Animals I.2-3
• Sometimes, class Rules notification (a bottom-up approach) is more practical.
RULES OF DIVISION:

When we are using logical division, we need to follow certain rules. thesde are as follows:

One division must follow only one criteria. It must be either physical or metaphysical.
The division criteria must be mutually exclusive and collectively exhaustive.
All the parts of an entity being explained must be covered by the division.
No extra members must be suggested as parts of the entity explained during the process of division.

FALLACIES OF DIVISION:

When we fail to follow the above rules, we end up in committing the following fallacies:

Division by cross criteria: When we divide something by using two or more criteria at the same time, we commit this fallacy. e.g. when we divide Indians into "Hindus, Muslims, Christians, Sikh, Rich, poor, Tall, short, Fair, Dark, introverts and extroverts"; we are committing this fallacy as we are using many criteria, both of physical as well as metaphysical divisions at the same time. at the same time.
Too narrow division: when we exclude some of the members from the group or some qualities of the entity being explained, we commit this fallacy. e.g. Quadrilateral into, square and rectangle. Here we exclude many other types of quadrilaterals and so the division becomes too narrow as it leaves out many other members that actually belong to this group.
Too wide division: when we include some members that actually do not belong to the group as we are dividing, our division becomes too wide. e.g. birds into single coloured & multi-coloured. Here, many other single coloured and multi-coloured things and beings get indicated as part of the group of bired, so it is a too wide division.

DEDUCTION VERSUS INDUCTION

We cannot fully understand the nature or role of inductive syllogism in Aristotle without situating it with respect to ordinary, “deductive” syllogism.

Aristotle’s distinction between deductive and inductive argument is not precisely equivalent to the modern distinction.

Contemporary authors differentiate between deduction and induction in terms of validity. (A small group of informal logicians called “Deductivists” dispute this account.)

According to a well-worn formula, deductive arguments are valid; inductive arguments are invalid.

The premises in a deductive argument guarantee the truth of the conclusion: if the premises are true, the conclusion must be true. The premises in an inductive argument provide some degree of support for the conclusion, but it is possible to have true premises and a false conclusion.

Although some commentators attribute such views to Aristotle, this distinction between strict logical necessity and merely probable or plausible reasoning more easily maps onto the distinction Aristotle makes between scientific and rhetorical reasoning (both of which we discuss below).

Aristotle views inductive syllogism as scientific (as opposed to rhetorical) induction and therefore as a more rigorous form of inductive argument.

We can best understand what this amounts to by a careful comparison of a deductive and an inductive syllogism on the same topic.

If we reconstruct, along Aristotelian lines, a deduction on the longevity of bileless animals, the argument would presumably run: All bileless animals are long-lived; all men, horses, mules, and so forth, are bileless animals; therefore, all men, horses, mules, and so forth, are long-lived.

Defining the terms in this syllogism as: Subject Term, S=men, horses, mules, and so forth; Predicate Term, P=long-lived animals; Middle Term, M=bileless animals, we can represent this metaphysically correct inference as: Major Premise: All M are P. Minor Premise: All S are M. Conclusion: Therefore all S are P. (Barbara.)

As we already have seen, the corresponding induction runs: All men, horses, mules, and so forth, are long-lived; all men, horses, mules, and so forth, are bileless animals; therefore, all bileless animals are long-lived. Using the same definition of terms, we are left with: Major Premise: All S are P. Minor Premise: All S are M (convertible to All M are S). Conclusion: Therefore, all M are P. (Converted to Barbara.)

The difference between these two inferences is the difference between deductive and inductive argument in Aristotle.

Clearly, Aristotelian and modern treatments of these issues diverge. As we have already indicated, in the modern formalism, one automatically defines subject, predicate, and middle terms of a syllogism according to their placement in the argument.

For Aristotle, the terms in a rigorous syllogism have a metaphysical significance as well. In our correctly formulated deductive-inductive pair, S represents individual species and/or the individuals that make up those species (men, horses, mules, and so forth); M represents the deep nature of these things (bilelessness), and P represents the property that necessarily attaches to that nature (longevity). Here then is the fundamental difference between Aristotelian deduction and induction in a nutshell.

In deduction, we prove that a property (P) belongs to individual species (S) because it possesses a certain nature (M); in induction, we prove that a property (P) belongs to a nature (M) because it belongs to individual species (S). Expressed formally, deduction proves that the subject term (S) is associated with a predicate term (P) by means of the middle term (M); induction proves that the middle term (M) is associated with the predicate term (P) by means of the subject term (S).

Aristotle does not claim that inductive syllogism is invalid but that the terms in an induction have been rearranged. In deduction, the middle term joins the two extremes (the subject and predicate terms); in induction, one extreme, the subject term, acts as the middle term, joining the true middle term with the other extreme. This is what Aristotle means when he maintains that in induction one uses a subject term to argue to a middle term.

Formally, with respect to the arrangement of terms, the subject term becomes the “middle term” in the argument.

Aristotle distinguishes then between induction and deduction in three different ways. First, induction moves from particulars to a universal, whereas deduction moves from a universal to particulars. The bileless induction moves from particular species to a universal nature; the bileless deduction moves from a universal nature to particular species. Second, induction moves from observation to language (that is, from sense perception to propositions), whereas deduction moves from language to language (from propositions to a new proposition).

The bileless induction is really a way of demonstrating how observations of bileless animals lead to (propositional) knowledge about longevity; the bileless deduction demonstrates how (propositional) knowledge of a universal nature leads (propositional) knowledge about particular species.

Third, induction identifies or explains a nature, whereas deduction applies or demonstrates a nature. The bileless induction provides an explanation of the nature of particular species: it is of the nature of bileless organisms to possess a long life. The bileless deduction applies that finding to particular species; once we know that it is of the nature of bileless organisms to possess a long life, we can demonstrate or put on display the property of longevity as it pertains to particular species.

One final point needs clarification. The logical form of the inductive syllogism, after the convertibility maneuver, is the same as the deductive syllogism. In this sense, induction and deduction possess the same (final) logical form.

But, of course, in order to successfully perform an induction, one has to know that convertibility is possible, and this requires an act of intelligence which is able to discern the metaphysical realities between things out in the world. We discuss this issue under non-discursive reasoning below.

LAWS OF THOUGHT

During the 18th, 19th, and early 20th Century, scholars who saw themselves as carrying on the Aristotelian and Medieval tradition in logic, often pointed to the “laws of thought” as the basis of all logic. One still encounters this approach in textbook accounts of informal logic. The usual list of logical laws (or logical first principles) includes three axioms: the law of identity, the law of non-contradiction, and the law of excluded middle. (Some authors include a law of sufficient reason, that every event or claim must have a sufficient reason or explanation, and so forth.) It would be a gross simplification to argue that these ideas derive exclusively from Aristotle or to suggest (as some authors seem to imply) that he self-consciously presented a theory uniquely derived from these three laws. The idea is rather that Aristotle’s theory presupposes these principles and/or that he discusses or alludes to them somewhere in his work. Traditional logicians did not regard them as abstruse or esoteric doctrines but as manifestly obvious principles that require assent for logical discourse to be possible.

The law of identity could be summarized as the patently unremarkable but seemingly inescapable notion that things must be, of course, identical with themselves. Expressed symbolically: “A is A,” where A is an individual, a species, or a genus. Although Aristotle never explicitly enunciates this law, he does observe, in the Metaphysics, that “the fact that a thing is itself is [the only] answer to all such questions as why the man is man, or the musician musical.” This suggests that he does accept, unsurprisingly, the perfectly obvious idea that things are themselves. If, however, identical things must possess identical attributes, this opens the door to various logical maneuvers. One can, for example, substitute equivalent terms for one another and, even more portentously, one can arrive at some conception of analogy and induction. Aristotle writes, “all water is said to be . . . the same as all water . . . because of a certain likeness.” If water is water, then by the law of identity, anything we discover to be water must possess the same water-properties.

Aristotle provides several formulations of the law of non-contradiction, the idea that logically correct propositions cannot affirm and deny the same thing:

“It is impossible for anyone to believe the same thing to be and not be.”

“The same attribute cannot at the same time belong and not belong to the same subject in the same respect.”

“The most indisputable of all beliefs is that contradictory statements are not at the same time true.”

Symbolically, the law of non-contradiction is sometimes represented as “not (A and not A).”

The law of excluded middle can be summarized as the idea that every proposition must be either true or false, not both and not neither. In Aristotle’s words, “It is necessary for the affirmation or the negation to be true or false.” Symbolically, we can represent the law of excluded middle as an exclusive disjunction: “A is true or A is false,” where only one alternative holds. Because every proposition must be true or false, it does not follow, of course, that we can know if a particular proposition is true or false.

Despite perennial challenges to these so-called laws (by intuitionists, dialetheists, and others), Aristotelians inevitably claim that such counterarguments hinge on some unresolved ambiguity (equivocation), on a conflation of what we know with what is actually the case, on a false or static account of identity, or on some other failure to fully grasp the implications of what one is saying.

INDUCTION:

Induction is a type of inference where we go from known to unknown or from less general to more general. Here, from the things that are known, we say something about things that are not known. This is the reason why in induction we always say something more than what we already know of. So, Induction, a form of argument in which the premises give grounds for the conclusion but do not make it certain. Induction is contrasted with deduction, in which true premises imply a definite conclusion, the conclusion of Induction is always probable. The probability rate changes as per strength of evidence. Unlike deductive arguments, inductive reasoning allows for the possibility that the conclusion is false, even if all of the premises are true.

Induction is of two types, perfect and imperfect. Perfect induction takes support of deduction in later stages to establish a certain conclusion, while imperfect induction does not do this.

There are two main types of imperfect induction. they are, Simple enumeration and Analogy.

Simple enumeration is a method of arriving at a generalization on the basis of uniform uncontradicted observation of something. This conclusion can be disproved by observing just one single contrary instance. Yet, the conclusion by simple enumeration is highly probable when the number of observed instances is really high. But if one is arriving at a conclusion on the basis of very limited observation, the conclusion is less probable and hence, it is termed as hasty generalization or illicit generalization.

Analogy is a type of imperfect induction where we are comparing two things, persons, groups or classes. while doing so, we observe some similarities and on the basis of these, we infer some further similarity, as we find an additional quality in one of the two compared things, persons, groups or classes. Here, if the observed similarities are relevant to the additional quality, then our conclusion is likely to be true and we may say that Analogy is good Analogy. But if the observed qualities are not relevant to the additional quality, then our conclusion about predicting the additional similarity is not likely to be true, so, we say that such an analogy is Bad Analogy.

In law, we need to use simple enumeration and Analogy to infer things from circumstantial evidence. Of them analogy is more useful in legal matters. Also, while using precedent law, we use analogy to indicate the support of past decided cases in our matter.

When we see a person following some pattern of behavior or thinking or actions, while talking of the Modus Operandi of that person, we are using simple enumeration as we talk of the generalized pattern of behavior of that person. This is the method followed by criminal investigators quite often. They determine the Modus Operandi of a criminal to find out the criminal and / or to track the criminals. This is a very common practice used by the police in registering the crime record of certain criminals while maintaining their files.

While contesting any matter, the lawyers use analogy in arguing about similar matters, or actions done by an individual in similar situations, to infer about the truth of the statement given by any witness. For example, if it is shown that the witness had reacted in a particular way in the past in similar situations, or has reacted in a particular way in similar situation created in court, then, one can infer that he must have reacted exactly in same way when the actual event had happened that the witness was witnessing. This type of inference adds to the weight-age in argument in court.

Similarly, when we are arguing any matter, we may come across previously decided matters of same type in the same court, or higher court or another court. We use the citation of these matters as case law or precedent law to lead the judge to the conclusion we want, and the procedure of inductive argument that we use in this type of matter is of analogy. This is why is is said that Analogy is of great use in legal arguments.

Proposition Relationships in EDUCTION @ glance

Relationship	Changed	Type	Original
Original = S-P			All S is P	No S is P	Some S is P	Some S is not P
			A	E	I	O
Obverse =	All S is non P	A		All S is non P
S-P	No S is non P	E	No S is non P
	Some S is non P	I				Some S is non P
	Some S is not non P	O			Some S is not non P

Converse =	All P is S	A				X
P-S	No P is S	E		No P is S		X
	Some P is S	I	Some P is S		Some P is S	X
	Some P is not S	O				X

Obv Conv=	All S is non P	A		All S is non P		X
P-S	No S is non P	E				X
	Some S is non P	I				X
	Some S is not non P	O	Some S is not non P		Some S is not non P	X

Part Inv =	All non S is P	A			X	X
S-P	No non S is P	E			X	X
	Some non S is P	I		Some non S is P	X	X
	Some non S is not P	O	Some non S is not P		X	X

Full Inverse =	All non S is non P	A			X	X
S-P	No non S is non P	E			X	X
	Some non S is non P	I	Some non S is non P		X	X
	Some non S is not non P	O		Some non S is not non P	X	X

Part Con +ve	All non-P is S	A			X
P-S	No non P is S	E	No non P is S		X
	Some non P is S	I		Some non P is S	X	Some non P is S
	Some non P is not S	O			X

Full Con +ve	All non P is non S	A	All non P is non S		X
P-S	No non P is non S	E			X
	Some non P is non S	I			X
	Some non P is not non S	O		Some non P is not non S	X	Some non P is not non S

CONVERTING A STATEMENT INTO LOGICAL FROM

There are four standard forms of categorical propositions such as A, E, I and O-propositions having the structure of the form, 'All S is P' 'No S is P’, 'Some S is P' and 'Some S is not P' respectively. Thus, we know that the logical structure of any categorical proposition exhibits the following four items in the order as given below.
Quantifier (Subject term) copula (predicate term)
Here the first item is the 'quantifier' (or more precisely the words expressing the quantity of the proposition). It is attached to the subject term only. The second item in any logical proposition is the subject term. The predicate term, that expresses something about the subject, comes after the copula. The copula is placed in between the subject and predicate term.
Further, the quality of the proposition is expressed in and through the copula. The copula and the predicate term are respectively the third and fourth logical elements of a categorical proposition. Thus, a categorical proposition which is in standard form must exhibit explicitly the subject, the predicate, the copula, its quality and quantity. Let us call a categorical proposition regular if it is in its standard form, otherwise it is called irregular.
In our ordinary language most of the categorical propositions are irregular in nature. Even though there are irregular categorical propositions they can be put in their regular form. It should be noted that while reducing an irregular categorical proposition into its standard form, we should pay enough attention to the meaning of the proposition so that the reduced proposition is equivalent in meaning to its irregular counterpart.
Before describing the method of reduction of irregular propositions into their regular forms, it is profitable to understand the reasons for irregularity of a categorical proposition: The irregularity of any categorical proposition may be due to one or more of these following factors.
(i) The copula is not explicitly stated; rather it is mixed with the main verb which forms the part of the predicate
(ii) Though the logical ingredients of a categorical proposition are present in the sentence yet are not arranged in their proper logical order.
(iii) The quantity of a categorical proposition is not expressed by a proper word like 'all', 'no' (or none), 'some' or it does not contain any word to indicate the quantity of the proposition.
(iv) All exclusive, exceptive and interrogative propositions are clearly irregular.
(v) The quality of the proposition is not specified by attaching the sign of negation to the copula.
Keeping these factors in mind, let us describe systematically the method of reduction of an irregular categorical proposition into its standard form (or into a regular proposition). Below we describe the method of reduction.

I. Reduction of categorical propositions whose copula is not stated explicitly
In our ordinary use of language, very often the copula is not explicitly or separately expressed but is mixed with the main verb. The main verb in such a case forms the part of the predicate. The moment copula is identified; the other items of a logical proposition are brought out in a usual manner. We know that the copula of any logical proposition must be in present tense of the verb "to be" with or without the sign of negation.
Now let us consider an example of an irregular proposition, where the copula is not explicitly stated. "All sincere students deserve success". This is an irregular proposition, as the copula is clearly mixed with the main verb of the proposition. The method of reducing such irregular sentences into regular ones is as follows. The subject and the quantifier of the irregular proposition should remain as they are, while the rest of the proposition may be converted to a class forming property (i.e. term) which would be our logical predicate.
In our above example 'All' is the quantifier attached to the subject 'sincere students'. We should not touch the quantifier nor the subject term of the proposition, they should remain where they are. On the other hand, the rest of the proposition 'deserve success' should be converted into a class forming property 'success deserving'. This should be our logical predicate. Then we link the subject term with the predicate term with a standard copula. Thus,
"All sincere students deserve success." Irregular proposition.
"All sincere students are success deserving." A - Proposition.
"All people seek power." Irregular proposition.
"All people are power seekers." A - Proposition.
"Some people drink Coca Cola." Irregular proposition.
"Some people are Coca Cola drinkers." I – proposition

II. Irregular propositions where the usual logical ingredients are all present but are not arranged in their logical order.
Consider the following examples of irregular propositions. "All is well that ends well" and "Ladies are all affectionate." In these cases, first we have to locate the subject term and then rearrange the words occurring in the proposition to obtain the regular categorical proposition. Such reductions are usually quite straight forward. Thus we reduce the above two examples as given below.
"All is well that ends well." Irregular proposition
"All things that end well are things that are well." A - Proposition
"Ladies are all affectionate." Irregular proposition
"All ladies are affectionate." A – Proposition

III. Statements in which the quantity is not expressed by proper quantity words. Some propositions do not contain word like 'All', 'No', 'some' or contain no words to indicate the quantity. We reduce such a type of irregular proposition into its logical form as explained below.
Here we have to consider two sub-cases : sub-case (i) where there is indication of quantity but no proper quantity words like 'All', 'No', on 'Some' are used and Sub case (ii) where the irregular proposition contains no word to indicate its quantity.
Sub-case (i): Affirmative sentences that begin with words like 'every', 'any', 'each' are to be treated as A-propositions, where such words are to be replaced by the word "all" and rest of the proposition remains as it is or may be modified as necessary. The followings are some of the examples of this type.
"Every man is liable to commit mistakes." Irregular proposition.
"All men are persons who liable to commit mistakes." A - Proposition.
"Each student took part in the competition." Irregular proposition.
"All students are persons who took part in the competition." A - Proposition.
"Any one of my students is laborious." Irregular proposition.
"All my students are laborious." A - Proposition.
A negative sentence that begins with a word like 'every', 'any', 'each', or 'all' is to be treated as an O-proposition. Any such proposition may be reduced to its logical form as shown below.
"Every man is not honest". Irregular proposition
"Some men are not honest." O - Proposition
"Any student cannot get first class." Irregular proposition.
"Some students are not persons who can get first class." O - Proposition.
"All is not gold that glitters." Irregular proposition.
"Some things that glitter are not gold." O - Proposition.
Sub- Case (ii):
"Sentences with singular term or definite singular term without the sign of negation are to be treated as A-proposition. For example, "Ram is mortal.", "The oldest university of Orissa is in Bhubaneswar." are A-propositions.
Here the predicate is affirmed of the whole of the subject term. On the other hand, sentences with singular term or definite singular term with the sign of negation are to be treated as E-propositions. For example, "Ram is not a student" and "The tallest student of the class is not a singer" are to be treated as E-propositions. These are cases where the predicate is denied of the whole of the subject term.

IV. “Sentences beginning with the words like 'no', 'never', 'none' are to be treated as E-propositions. The following sentence is an example of this type.
"Never men are perfect." Irregular proposition
"No man is perfect." E – Proposition

V. Affirmative sentences with words, like 'a few', 'certain', 'most', 'many' are to be treated as I-propositions, while negative sentences with these words are to be treated as
O-propositions. Since the word 'few' has a negative sense, an affirmative sentence beginning with the word 'few' is negative in quality. A negative sentence beginning with the word 'few' is affirmative in quality because it involves a double negation that amount to affirmation. The following are examples of above type.
"A few men are present." Irregular proposition.
"Some men are present." I - proposition.
"Certain books are good." Irregular proposition.
"Some books are good." I - proposition.
"Most of the students are laborious." Irregular proposition.
"Some students are laborious." I - proposition.
Here we may note that 'most' means less then 'all' and hence it is equivalent to 'some'.
"Many Indians are religious." Irregular proposition.
"Some Indians are religious." I - proposition.
"Certain books are not readable." Irregular proposition
"Some books are not readable." O - Proposition
"Most of the students are not rich." Irregular proposition.
"Some students are not rich." O - Proposition
"Few men are above temptation." Irregular proposition
"Some men are not above temptation." O - Proposition
"Few men are not selfish." Irregular proposition
"Some men are selfish.' I

VI. Any statement whose subject is qualified with words like 'only', 'alone', 'none but', or 'no one else but' is called an exclusive proposition. This is so called because the term qualified by any such word applies exclusively to the other term. In such cases the quantity of the proposition is not explicitly stated.
The propositions beginning with words like 'only', 'alone', 'none but' etc are to be reduced to their logical form by the following procedure. First interchange the subject and the predicate, and then replace the words like 'only', 'alone' etc with 'all'. For example,
"Only Oriyas are students of this college." Irregular proposition.
"All students of this college are oriyas." A - Proposition.
"The honest alone wins the confidence of people." Irregular Proposition.
"All persons who win the confidence of people are honest." A-proposition.

VII. Propositions in which the predicate is affirmed or denied of the whole subject with some exception is called an exceptive proposition. An exceptive proposition may be definite or indefinite. If the exception is definitely specified as in case of "All metals except mercury are solid" then the proposition is to be treated as universal and if the exception is indefinite, as in case of "All metals except one is solid", the proposition is to be treated as particular.
"All metals except mercury are solid." is a universal proposition which means
"All non-mercury metals are solid."
Now let us consider an example where the exception is indefinite. For example, "All students of my class except a few are well prepared", it is to be reduced to an I-proposition as given below.
"All students of my class except a few are well prepared." Irregular proposition.
"Some students of my class are well prepared." I - proposition.

VIII. There are impersonal propositions where the quantity is not specified. Consider for example, "It is cold", "It is ten O'clock". In such cases propositions in question are to be reduced to A-proposition because the subject in each of these cases is a definite description.
"It is cold". Irregular proposition
"The whether is cold." A - Proposition.
"It is ten O'clock." Irregular proposition.
"The time is ten O'clock." A - Proposition.
There are some propositions where the quantity is not specified. In such cases we have to examine the context of its use to decide the quantity. For example, consider following sentences (1) "Dogs are carnivorous", (2) "Men are mortal", (3) "Students are present." In first two examples, the quantity has to be universal but in the third case, it is particular. Thus, their reductions into logical form are as follows.
"Dogs are carnivorous." Irregular proposition.
"All dogs are carnivorous." A - Proposition.
This is so because we know that "being carnivorous' is true of all dogs.
"Men are mortal." Irregular proposition.
"All men are mortal." A - Proposition
Here 'being mortal' is generally true of men. But in the proposition "Students are present", we mean to assert that some students are present". So the proposition "Men are mortal" is reduced to "All men are mortal" But in the example "Students are present", 'being present' is not generally true of all students.
So the proposition "Students are present" is reduced to "Some dents are present" which is an I-proposition. Thus the context of use of a proposition determines the nature of the proposition.

IX. Problematic propositions are particular in meaning. For example "The poor may be happy" should be treated as a particular proposition, because what such a proposition asserts is that it is sometimes true and sometimes false.
Thus, "The poor may be happy" is reduced to "Some poor people are happy", which is an I-proposition

X. Similarly, there are propositions where the quantity is not specified but their predicates are qualified by the words like 'hardly', 'scarcely', 'seldom'. Such propositions should be treated as particular negative. For example, "Businessmen are seldom honest", is an irregular proposition. It is reduced to "Some businessmen are not honest". If such a proposition contains the sign of negation that these proposition is to be treated as an I-proposition.
For example, "Businessmen are not seldom honest." is to be reduced to "Some businessmen are honest", which is an I - proposition. This is so because it involves a double negation which is equivalent to affirmation.

CHAPTER 1: NATURE OF LOGIC

CHAPTER 1. NATURE OF LOGIC

A) Traditional and Modern definitions of Logic
B) Basic features of Inductive and Deductive reasoning. Their uses in Law Courts
C) Some basic logical concepts –Form, Content, Truth , Validity, Inference, Implication.

Logic is a science of valid reasoning.

Logic is a word that comes from the Greek word λογική pronounced as logikē that means, the study of reasoning.

All the places where reasoning is needed, logic is needed. The more accurately we use our reasoning, the more effective is our work in that area. So all those who think, use logic knowingly or unknowingly.

Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic falls in the area of epistemology, which asks: "How do we know what we know?" In mathematics, it is the study of valid inferences within some formal language.

Logic has origins in several ancient civilizations, including ancient India, China and Greece. In west, Logic was established as a discipline by Aristotle, who established its fundamental place in philosophy.

The study of logic was part of the classical trivium. Averroes defined logic as "the tool for distinguishing between the true and the false." Richard Whately, defined logic as "the Science, as well as the Art, of reasoning." Frege, defined logic as "the science of the most general laws of truth."

The concept of logical form is central to logic, it being held that the validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic.

Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato are good examples of informal logic.

Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language.

Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is often divided into two branches: propositional logic and predicate logic.

Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.

A) Traditional and Modern definitions of Logic

Traditional Logic is the type of logic propagated by Aristotle. This is popularly known as traditional formal logic. This is because here, the form of statements used in arguments is given total importance. The traditional formal logic is general designation for the systems of deductive logic that do not involve the use of formal languages, or the apparatus of mathematical logic. The basis of traditional logic is syllogistic reasoning.

Traditional logic is defined as “a system of formal logic mainly concerned with the syllogistic forms of deduction that is based on Aristotle and includes some of the changes by contemporary logicians.”

Modern Logic on the other hand contains more form based relationships in the logical thinking. So, the modern logic is not limited to syllogism based arguments, but it goes much beyond them. It becomes mathematical and symbolic. This is the reason why after the modern logic was developed, it started being used practically in every science where thinking in the right way is needed.

Modern logic is defined as “logic where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematics.” The development of the modern "symbolic" or "mathematical" logic is the most significant in the history of logic, and in human intellectual history.

B) Basic features of Deductive and Inductive reasoning.

Logic is divided into two types, these are inductive and deductive reasoning.

The Deductive Reasoning is a reasoning where the conclusion stays within the scope of its supporting statements. It is said that deductive reasoning involves in drawing logical conclusions from definitions and axioms. We can also say that deductive Reasoning involves in deriving known conclusions from known facts. As a result, the conclusions of deductive reasoning are certain.

The Inductive reasoning is a reasoning where conclusion goes beyond the scope of its supporting statements. Many say that inductive reasoning involves in drawing general conclusions from specific examples. We also say that Inductive Reasoning involves in deriving unknown conclusion from known facts. This is the reason why conclusions of Inductive reasoning are probable and not certain.

Though induction and deduction are the two types of logical reasoning, they are not watertight compartments. The general statements that we use as supporting statements in deductive arguments, actually are result of inductive inferences.

So, we find that Induction is of two types, perfect and imperfect.

In perfect induction, we get conclusion from supporting statements, i.e. premises, this conclusion goes beyond the scope of premises, and then we verify and test this conclusion by using some methods that are similar to deduction.

In imperfect induction, we get conclusion from the supporting statements, i.e. premises, this conclusion says something beyond the scope of premises, but then we do not verify and test this conclusion. We just leave it as it is. So, the conclusion of imperfect induction is always probable.

Deduction versus Induction

Aristotle used to classify the type of arguments using syllogism. But all arguments cannot be fitted in the type of syllogism. We can use a simple test for inductive and deductive arguments. The premises in a deductive argument guarantee the truth of the conclusion, so, if the premises are true, the conclusion must be true. The premises in an inductive argument provide some degree of support for the conclusion, but it is possible to have true premises false conclusion.

Aristotle views inductive syllogism as scientific induction and therefore as a more rigorous form of inductive argument. The logical form of the inductive syllogism, after the convertibility maneuver, is the same as the deductive syllogism.

In this sense, induction and deduction possess the same (final) logical form. But, of course, in order to successfully perform an induction, one has to know that convertibility is possible, and this requires an act of intelligence which is able to discern the metaphysical realities between things out in the world. We discuss this issue under non-discursive reasoning below.

Uses of logical reasoning in Law Courts:

In the field of law, we need reasoning in order to present the matter of any litigant effectively, so that we can help him get justice in the existing frames of law. But even the opposite side litigant who has a contrary view also wants justice, so the aim of a good lawyer is always to disclose the truth and just situation. Logical reasoning is absolutely necessary for this as without logical reasoning we cannot find out the truth behind the stories told by the litigants.

C) Some basic logical concepts – Form, Content, Truth , Validity, Inference, Implication.

Logic as we saw, is a science of valid reasoning. In order to know what is valid reasoning, we must first have some concepts clarified. Also, Logic can be best best understood if we understand the basic concepts of logic. So, let us see the definitions of some basic concepts in logic:

Word, is a meaningful group of alphabets used in any language.

Syncatagormatic word, are the words that are used to enhance the meaning of words that can stand on their own. So, the Syncatagormatic words do not make any meaning on their own. They depend on other words for their meaningfulness.

Catagormatic word, is a word that has its own meaning, so it can stand on its own in the process of expressing a meaningful concept.

Sentence, is a meaningful group of words used to convey any meaning.
Statement, is a sentence that asserts some affirmative or negative fact.
Proposition, is a statement used in logical arguments. This means, in logic, a statement is called a proposition.
Form, stands for the relationship of various parts of a statement within itself and in a set of statements called argument.
Content, is the matter of facts mentioned in the statement or argument.
Truth, is the agreement of facts mentioned in an argument with reality.
Validity, is the appropriateness of relationship between various parts of argument.
Inference, is a set of propositions or statements where, on the basis of one or more statements one statement is obtained as a conclusion.

Implication, is a type of statement where the truth of one component is indicated or suggested by the truth of another. Here, the component on which the truth of another component depends is the first component called antecedent and the component that follows from the first component is the consequent.

CHAPTER 2 TERMS

TERMS

a) Meaning of Terms- Connotation and Denotation of terms- Positive and Negative terms, Contrary and Contradictory terms.
b) Distinctions between – Proposition and Sentence, Proposition and Judgment, Proposition and Fact, Constituent and Component.

c) Distribution of terms- for universal, particular, affirmative and negative terms.

The TERM is a word that is independently used in logical arguments and term is word that can stand on its own and express a meaning. Naturally, all words cannot become terms. Let us see the meaning and types and classification of terms in details:

a) Meaning of Terms- Connotation and Denotation of terms- Positive and Negative terms, Contrary and Contradictory terms.

Term is a “Word” that can stand on its own and so can become a subject or predicate of proposition in logic. We know that “Word” is a meaningful combination of alphabets.

Words are classified into two types on the basis of their function of expressing or enhancing the meaning. The types of words are, categormatic & syntacategormatic.

Let us see the definitions of the types of words:

Catergormatic are the words that can express some meaning or their own, so, they can be used as terms in any proposition.

Syncategormatic are the supporting words that are used to connect or enhance the catergormatic words. i.e. Terms.

The syncategormatic words cannot express any meaning on their own, so they do not become Terms. Only categormatic words have capacity to become terms.

Any word or term differs in its meaning as per its use. Some times a word has one dictionary meaning, but is used in a different sense. This time, if we do not understand the correct meaning, we may get confused. So, we must note that the Terms have Two senses, on the basis of the meaning indicated by them. These senses are called Connotation & Denotation.

Let us see these types in details:

Connotation indicates the meaning of a word accepted by Custom or Community. Connotation is a commonly understood association that a word or phrase carries, in addition to the explicit or literal meaning of that word or phrase that is called its Denotation. Connotation is either positive or negative.

Denotation indicates the Technical Meaning of a word accepted & listed in a Dictionary. Denotation is the transition of a sign to its meaning that dictionaries try to define.

Sometimes, denotation is contracted to connotation. e.g. “You are brilliant” in sarcastic way means, “You are an Idiot!” So its Connotation becomes opposite to its Denotation, i.e. actual meanings.

The terms are seen to indicate two things, quality and quantity. Quality indicates the presence or absence of things stated while quantity indicates number of group member that possess that quality.

The terms indicate either the presence or absence of something. According to their function, they are classified in to positive and negative.

Let us see how:

Positive Term is the term which affirms some thing or quality in Something.

Negative Term is the term which denies something or quality in something.

The positive and negativeness of a term is called its quality.

The terms indicate either one individual, or a small part of group indicated by the word or the whole group indicated by the word. According to the number of individuals indicated in the term, we have singular, particular and universal terms.

Let us see how:

A singular term is a term that speaks something about one single individual person, thing or entity. The fact stated here can be either positive or negative.

A particular term is a term that speaks about a small part of the group indicated by a term. The thing spoken can be either positive or negative.

A universal term is a term that speaks about the entire group indicated by it. This statement can be either positive or negative.

The singularity, particularness or universalness of a term is called its quantity.

Generally we find that in subject predicate class membership propositions that are used in inferences, the predicate term is always universal, and we check the quantity of the subject term to classify the proposition.

On the basis of the quality and quantity indicated in terms, the terms are classified into three more types. These classifications depend on the difference in quality, or quantity or both. When only the quality is different, the terms are called contrary, when only quantity is different, the terms are called sub-alternate and when both the quality and quantity is different, the terms are called contradictory.

Let us see this classification in details:

Contrary Terms are the Terms that have the same quantity. Generally, the contrary relationship indicated the relation between two Universal terms. If the terms that are same in quantity and differ in quality, but are Particular, the relationship is called sub–contrary. In short, contrary relation exhibits the pairs of terms that are same in quantity and different in quality.

Sub alternate or sub altern terms are the terms that have the same quality but different quantity. This means, when a pair of affirmative or negative terms has one universal and one particular term, the pair indicates a sub altern relationship. This means, the universal affirmative and particular affirmative terms indicate a sub altern relationship and so do the universal negative and particular negative terms.

Contradictory Terms are the terms that differ both in Quality & Quantity. Thisx means, in a pair of two contradictory terms, if one is universal affirmative, the other will be particular negative and if one is particular affirmative, the other will be universal negative.

Table to explain opposition of terms at a glance

Type of term	Quality	Quantity
Contradictory	X	X
Contrary	X	same
Sub Contrary	X	same
Subaltern	same	X

b) Distinctions between –

Proposition and Sentence,

Proposition and Judgment,

Proposition and Fact,

Constituent and Component.

To classify the propositions and compare them further with sentence, judgment, fact and so on, we must first note the basics of an expression.

Every time when we try to express some meaningful thing, we use a language. A language is made up of alphabets and connecting punctuation symbols. The first thing we get in any language is a basic meaningful combination of alphabets. A Word is a meaningful combination of Alphabets. Then we combine these meaningful combinations of alphabets to make more sense. This time we get a sentence. Sentence is a meaningful Combination of Words. In a sentence, as per the requirement of its meaning, we also use different punctuation marks. We have many different types of sentences, but all do not have the capacity to be used in logical arguments. Only the statements that state the presence of something or absence of something, that means, only the assertive sentences, are the ones that can be used in logical arguments. These are also called statements or propositions.

Statement or Proposition is any subject less or subject predicate, relational or class membership. universal, particular or singular, simple or compound, Affirmative or negative; assertive sentence.

i) Proposition and Sentence

As we have seen above, Sentence is a meaningful Combination of Words where as, Statement or Proposition is any subject less or subject predicate, relational or class membership. universal, particular or singular, simple or compound, Affirmative or negative; assertive sentence.

So, any sentence has a power to make a meaning, but it does not have a power to be a part of an argument. On the other hand, the sentences that can be used in an inference, are called statements or propositions.

Proposition is a sentence that asserts some thing in positive or negative manner. Propositions are of two types, simple and compound. Simple statements are either subject-less or with a subject and predicate. In either case, the verb seen in simple statements is only one. On the other hand, when two or more simple statements are combined together & form a statement we get compound statement.

ii) Proposition and Judgment

Proposition is a statement that states a matter of fact. It does not carry any opinion or view of the person making the statement. It just states what is. This means, a proposition or statement states only pure undiluted non-tampered facts without and smell of right and wrong, good or bad, proper or improper, desirable or undesirable, nice or not nice etc. etc.

Judgment is a statement Expressing the Opinion or View of Someone about some event or situation. This view may or may not indicate the fact or truth. Those who give a judgment, state if something is good or bad, right or wrong, and desirable or undesirable. This means, what they say, is not what is, but what they feel. In logic, we value what is, and not what we feel. So, proposition is something that is used in logic and not judgement.

Iii) Proposition and Fact.

Fact is the Actual Event or Things that can be objectively verified by any one. This means, fact is never true or false, right or wrong. Fact is just fact. Just as we do not need any one's opinion to check what is the time right now, watch in hand tells it to anyone, we need not check whether fact is true or not. It is always true.

Proposition is statement that states something as a matter of fact. This means, though the proposition is stating something, it may or may not be agreeing with actual fact. Sometimes, a statement says something as a matter of fact, but is actually not so, like, in a statement, “Chintu met a man who had a tail and two horns on his head.” here, the proposition is stating something as a fact, but actually, it is an imagination that is contrary to fact. The argument based on such statement may be logically correct, but still may not be stating facts.

iv) Constituent and Component

This is like any other thing or class, every proposition has two main parts.

One is the part without which a statement cannot stand. This is the constituent.

The other is a part that enhances the statement, but the statement can still stand without this part. Such part is called a component.

In short, constituent is like the vital parts of a person without which one cannot be alive, while, components are like the body parts, without which, one may be disabled, incomplete or handicapped, but still will be alive.

Exactly like that, a proposition can become a proposition just because a constituent, and that proposition gets enhanced in the presence of its components.

So, a Constituent is the integral part of any proposition and gives meaning to the proposition, without which the proposition cannot exist.

Component is the part of a proposition that enhances the proposition by adding itself to the constituent, but which can be detached from the proposition without extinguishing the existence of the proposition.

CHAPTER 3 PROPOSITIONS

PROPOSITIONS

a) Traditional classification of proposition into Categorical and Conditional

b) four- fold classification.
c) Reduction of sentences to their logical forms.
d) Distribution of terms in A, E, I, O propositions.

Propositions are sentences used in logic. These are of various types, as we can express a matter of fact in any way. To convey any meaning, we may just use a subject less statement like 'it rains.' or we may use just a subject predicate relational statement like 'India is larger than Japan in land area.' we may also use subject predicate class membership statement like 'some subjects are easy,' or 'no subjects are easy to study for exam.' These propositions are either simple, i.e. have only one subject and predicate, or compound, i.e. having at least two subjects and two predicates and connecting words that join the two or more simple propositions in a link to convey some meaningful relationship between them.

In traditional logic, given by Aristotle, only one type of propositions were treated useful in logical arguments. They were subject predicate class membership type of statements. Aristotle had classified these subject predicate class membership propositions that he used to call propositions; into two types on the basis of their attributes. Attributes are characteristics that are basis of any piece of expression.

These attributes were, quality and quantity.

Quality states assertion or denial of information indicated in piece of expression.

Quantity states number of individuals indicated by subject term in a proposition.

Each attribute has two sub attributes.

So, quality is of two types, affirmative and negative.

Also, quantity is of two types they are, singular and general.

General propositions are further classified in two types, universal and particular.

Let us see this classification of traditional propositions at a glance:

Each proposition has to have at least one quality & at least one quantity. So, we have six types of traditional propositions. These proposition types are as follows; singular affirmative, singular negative, universal affirmative, universal negative, particular affirmative, particular negative.

a) Traditional classification of proposition into Categorical and Conditional

in traditional logic given by Aristotle, we study Only the simple subject predicate propositions that indicate class membership. Here, we have two types of relationships in the subject and predicate. Depending on these relations, the propositions are classified into categorical and conditional.

If the affirmation or denial in a proposition depends on any condition, then the proposition is conditional. If it is not dependent on any condition, then it is categorical.

According to some other definition, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the subject term) are included in another (the predicate term). Categorical propositions can be simple or compound, but they are necessarily class membership type propositions. In simple categorical proposition, we have one subject and one predicate, in a compound, we have two subjects & two predicates.

A conditional proposition is a proposition that has only one affirmation in which one part of the proposition depends on the other part.

Some others say that Conditional propositions are the compound propositions that contain at least two subjects and two predicates, where we have two simple statements such that the truth of one depends on the truth of another.

In either case, we must admit that conditional proposition is the proposition that talks of some condition under which one part of the proposition is true.

So, when we go further to convert the traditional propositions into propositional form, we may be able to say that the universal propositions are conditional in nature and particular are categorical in nature.

b) Four- fold classification.

We have seen above that propositions have a quality and quantity. The quantity of a proposition depends on its subject. It is this quantity that decides whether the proposition is going to be just categorical or conditional.

When subject of proposition represents single individual, it is singular proposition. When the subject of a proposition represents a group of individuals, it is a general proposition. General propositions are further classified into Universal and Particular.

When the subject tells something about the whole group represented by it, the proposition is known to be universal. When the subject tells something about some members of the group represented by it, the proposition is known to be particular.

The propositions are also classified using another criteria of quality and this makes them affirmative or negative.

So, both singular and general propositions are either affirmative or negative.

Aristotle in his classification used to count singular proipositions in the category of universal propositions. Only difference they had with normal universal propositions was that these did not have their particular counterparts.

As a result, we have four types of general propositions, on the basis of both the quality as well as quantity.

This is how the four types of general propositions are; A, E, I, & O.

The quantity and quality of these are as follows:

Type	S	P
A	Universal	Affirmative
E	Universal	Negative
I	Particular	Affirmative
O	Particular	Negative

We can express any relationship of subject and predicate into these four ways.

The relationship between propositions having same subject and predicate but having different quality and / or quantity is called as the relation of opposition of propositions. Relation of opposition between these propositions is as follows:

When two universal propositions differ in quality, they are called CONTRARY. When two particular propositions differ in quality, they are SUB-CONTRARY. When two propositions of same quality, differ in quantity, they are SUB-ALTERN. When propositions differ both in quality and quantity they are CONTRADICTORY.

This relationship is also shown in a square of opposition.

The square of opposition of proposition shows two universals on the top and two particulars at the bottom. On the left side of the square, we have affirmative type of proposition and on the right side we have negative type of proposition. So, TOP is universal, BOTTOM is particular, LEFT is affirmative, RIGHT is negative. Since all 4 sides of square are connected, each side shows one quality & one quantity.

Each proposition is either true or false. So, if a given proposition is true, the proposition having same subject and predicate and that differs in quality or quantity or both may be true or false or uncertain. This relationship is called as the relation of truth value between various propositions having opposition relation.

The relation of truth values between these opposite propositions is as follows:

If a universal proposition is true, its contrary is false, its sub-altern is true and its contradictory is false.

If a universal proposition is false, its contrary is uncertain, its sub-altern is uncertain and its contradictory is true.

If a particular proposition is true, its sub-contrary is uncertain, its sub-altern is uncertain and its contradictory is false.

If a particular proposition is false, its sub-contrary is true, its sub-altern is false and its contradictory is true.

The opposition relation of proposition is the foundation of all traditional logic, syllogism, and various types of mediate and immediate arguments used in traditional logic. The immediate inferences having relationship based on opposition of proposition, is also called the relationship of EDUCTION where we can convert one type of proposition into another maintaining its meaning and validity.

But to do all this, it is necessary to understand the relationship of subject and predicate in a given proposition in a perfect way. This can be done easily by taking help of mathematical logic and set theory.

The expression of proposition in such a format by using set theory to indicate the relationship between subject and predicate is called venn diagram.

Venn Diagrams:

Venn diagram (also known as a set diagram or logic diagram) is a diagram that shows all possible logical relations between a finite collection of different sets. Venn diagrams were conceived around 1880 by John Venn.

The general propositions represent the relation of two groups indicated by the subject and predicate, and so, they can be represented symbolically using the venn diagram method used in mathematics.

To do this, we use two intersecting circles. The circle on the left represents the subject, and the one on the right, represents the predicate.

To represent "A" proposition, we shade the part of the circle of subject, that is outside that of predicate. This shows that set of subject outside predicate is empty.

ATo represent "E" proposition, we shade the part of the circle of subject, that is inside predicate. This shows that the set of subject inside the predicate is empty.

To represent "I" proposition, we put a cross in part of the circle of subject, that is inside that of predicate. This shows that set of subject inside predicate is not empty.

To represent "O" proposition, we put a cross in part of subject circle, outside that of predicate. This shows that the set of subject outside the predicate is not empty.

When these propositions are symbolized, we change them in a specific format so that we can show the class membership of the subject and the predicate terms.

c) Reduction of sentences to their logical forms.

We have four standard forms of categorical propositions A, E, I and O. They have following structure & form.

A = 'All S is P'

E = 'No S is P’

I = 'Some S is P'

O = 'Some S is not P'

The method of symbolizing propositions using their quality and quantity is known as the method of Quantification and the symbols used to indicate the quantity of the subject are known as quantifiers.

Let us see how this is done:

Singular propositions:

Affirmative:

“Ramu is a boy” is symbolized as Br

Negative:

“Sita is not a boy” is symbolized as ~Bs

General propositions:

These are of four kinds as we have seen earlier. They are symbolized as follows:

"A" proposition:

Subject-less:

“Everything perishes” will be written as “Given any x, x is Perishable.”

This is symbolized as follows:

Given any x = (x), x is perishable = (Px),

so whole proposition is, (x)(Px)

Subject-predicate:

“All crows are birds” will be written as

“Given any x, if x is a crow, then x is a bird”

This is symbolized as follows:

Given any x = (x), If x is a crow = Cx, then = , , x is a bird = Bx

So, the whole proposition is,

(x)(Cx Bx)

"E" proposition:

Subject-less:

“Nothing is Permanent” will be written as Given any x, x is not Permanent.

This is symbolized as follows

Given any x = (x), x is not permanent = (~Px),

so whole proposition is, (x)(~Px)

Subject-predicate:

“No crows are red” will be written as

“Given any x, if x is a crow, then x is not red”

This is symbolized as follows:

Given any x = (x), If x is a crow = Cx, then = , , x is not red = ~Rx

So, the whole proposition is,

(x)(Cx ~Rx)

"I" proposition:

Subject-less:

“Lions exist” will be written as There is an x, such that, x is a Lion.

This is symbolized as follows:

There is an x such that = (x), x is a lion = Lx

So, the whole proposition is, (x)(Lx)

Subject-predicate:

“Some roses are red” will be written as

“There is an x such that, x is a rose, and x is red”

Here, since both words begin with R, we take R for subject & D for predicate,

This is symbolized as follows:

There is an x such that, = (x), x is a rose = Rx, then = ., x is red = Dx

So, the whole proposition is,

(x)(Rx . Dx)

"O"proposition:

Subject-less:

Ghosts do not exist is be written as There is an x, such that, x is not a Ghost.

This is symbolized as follows:

There is an x such that = (x), x is not a ghost = ~Gx

So, the whole proposition is, (x)(~Gx)

Subject-predicate:

“Some buses are not red” will be written as

“There is an x such that, if x is a bus, then x is not red”

This is symbolized as follows:

There is an x such that = (x), x is a bus = Cx, and = . , x is not red = ~Rx

So, the whole proposition is,

(x)(Bx . ~Rx)

When we symbolize, the first letter of the subject term is taken as a capital letter and small x is written after it to indicate the singular variable that is quantified in the beginning. This is how we symbolize general propositions in traditional classification

Thus, we know that the logical structure of any categorical proposition exhibits the following four items Quantifier (Subject term) copula (predicate term) in order.

Here the first item is the 'quantifier' or more precisely the words expressing the quantity of the proposition. This is attached to the subject term.

The second item in any regular logical proposition is the subject term.

The third item is copula, placed in between the subject and predicate term. The quality of the proposition is expressed in and through the copula. .

The fourth item is predicate term, that expresses something about the subject. This comes after the copula in a proposition that has a regular order.Thus, a categorical proposition which is in standard form must exhibit explicitly the subject, the predicate, the copula, its quality and quantity. Let us call a categorical proposition regular if it is in its standard form, otherwise it is called irregular.

In our ordinary language most categorical propositions are irregular in nature. Even irregular categorical propositions can be put in their regular form. It should be noted that while reducing an irregular categorical proposition into its standard form, we should pay enough attention to the meaning of the proposition so that the reduced proposition is equivalent in meaning to its irregular counter-part.

Before describing the method of reduction of irregular propositions into their regular forms, it is good to understand the reasons for irregularity. The irregularity of any categorical proposition may be due to one or more of these following factors.

Copula is not clear or it is mixed with verb which forms part of predicate
Logical ingredients are not arranged in their proper logical order.
Quantity is not expressed by a proper word like 'all', 'no' (or none), 'some' etc.
All exclusive, exceptive and interrogative propositions are clearly irregular.
Quality is not specified by attaching the sign of negation to the copula.

In light of this, let us describe systematically the method of reduction of an irregular categorical proposition into its standard form (or into a regular proposition).

Let us see with examples the method of reduction.

I. Reduction of categorical propositions whose copula is not stated explicitly.

Let us consider an example of irregular proposition, where copula is not explicit.

"All sincere students deserve success".

This is an irregular proposition. Here, the copula is mixed with main verb.

The method of reducing such irregular sentences into regular ones is as follows:

The subject and the quantifier of the irregular proposition should remain as they are, while the rest of the proposition may be converted to a class forming property (i.e. term) which would be our logical predicate.

In our above example 'All' is the quantifier attached to the subject 'sincere students'. We should not touch the quantifier nor the subject term of the proposition, they should remain where they are.

On the other hand, the rest of the proposition 'deserve success' should be converted into a class forming property 'success deserving'.

This should be our logical predicate. Then we link the subject term with the predicate term with a standard copula.

Thus, "All sincere students deserve success." Irregular proposition.

"All sincere students are success deserving." = A - Proposition.

"All people seek power." Irregular proposition.

"All people are power seekers." A – Proposition.

"Some people drink Coca Cola." Irregular proposition.

"Some people are Coca Cola drinkers." I – proposition

II. Irregular propositions where the usual logical ingredients are all present but are not arranged in their logical order.

Consider the following examples of irregular propositions.

"All is well that ends well" and "Ladies are all affectionate."

In these cases, first we have to locate the subject term and then rearrange the words occurring in the proposition to obtain the regular categorical proposition.

Such reductions are usually quite straight forward.

Thus we reduce the above two examples as given below.
"All is well that ends well." Irregular proposition
"All things that end well are things that are well." A - Proposition
"Ladies are all affectionate." Irregular proposition
"All ladies are affectionate." A – Proposition

III. Statements in which quantity is not expressed by proper quantity words.

Some propositions do not contain word like 'All', 'No', 'some' or contain no words to indicate the quantity. We reduce such a type of irregular proposition into its logical form as explained below.

Here we have to consider two sub-cases :

(i) where there is indication of quantity but no proper quantity words like 'All', 'No', on 'Some' are used

(ii) where the irregular proposition contains no word to indicate its quantity.

These errors are of the following types:

(a) Affirmative sentences that begin with words like 'every', 'any', 'each' are to be treated as A-propositions, where such words are to be replaced by the word "all" and rest of the proposition remains as it is or may be modified as necessary. The followings are some of the examples of this type.

"Every man is liable to commit mistakes." Irregular proposition.

"All men are persons who liable to commit mistakes." A – Proposition.

"Each student took part in the competition." Irregular proposition.

"All students are persons who took part in the competition." A – Proposition

"Any one of my students is laborious." Irregular proposition.

"All my students are laborious." A – Proposition.

A negative sentence that begins with a word like 'every', 'any', 'each', or 'all' is to be treated as an O-proposition. Any such proposition may be reduced to its logical form as shown below.

"Every man is not honest". Irregular proposition

"Some men are not honest." O – Proposition

"Any student cannot get first class." Irregular proposition.

"Some students are not persons who can get first class." O – Proposition.

"All is not gold that glitters." Irregular proposition.

"Some things that glitter are not gold." O - Proposition.

(b) Sentences with singular term or definite singular term without the sign of negation are to be treated as A-proposition.

For example, "Ram is mortal.",

"The oldest university of Orissa is in Bhubaneswar." are A-propositions.

Here the predicate is affirmed of the whole of the subject term. On the other hand, sentences with singular term or definite singular term with the sign of negation are to be treated as E-propositions.

For example, "Ram is not a student" and "The tallest student of the class is not a singer" are to be treated as E-propositions. These are cases where the predicate is denied of the whole of the subject term.

IV. Sentences beginning with the words like 'no', 'never', 'none' are to be treated as E-propositions.

The following sentence is an example of this type.

"Never men are perfect." Irregular proposition

"No man is perfect." E – Proposition

V. Affirmative sentences with words, like 'a few', 'certain', 'most', 'many' are to be treated as I-propositions, while negative sentences with these words are to be treated as O-propositions.

Since the word 'few' has a negative sense, an affirmative sentence beginning with the word 'few' is negative in quality. A negative sentence beginning with the word 'few' is affirmative in quality because it involves a double negation that amount to affirmation. The following are examples of above type.

"A few men are present." Irregular proposition.

"Some men are present." I – proposition.

"Certain books are good." Irregular proposition.

"Some books are good." I – proposition.

"Most of the students are laborious." Irregular proposition.

"Some students are laborious." I – proposition.

Here 'most' means less then 'all' and hence it is equivalent to 'some'.

"Many Indians are religious." Irregular proposition.

"Some Indians are religious." I – proposition.

"Certain books are not readable." Irregular proposition

"Some books are not readable." O – Proposition

"Most of the students are not rich." Irregular proposition.

"Some students are not rich." O – Proposition

"Few men are above temptation." Irregular proposition

"Some men are not above temptation." O – Proposition

"Few men are not selfish." Irregular proposition

"Some men are selfish.'

VI. Any statement whose subject is qualified with words like 'only', 'alone', 'none but', or 'no one else but' is called an exclusive proposition.

Here, the term qualified by any such word applies exclusively to the other term.

In such cases the quantity of the proposition is not explicitly stated.

This is the reason why such statements are tricky and they can mislead or indicate a contrary meaning if not reduced to correct form in the right way.

The propositions beginning with words like 'only', 'alone', 'none but' etc are to be reduced to their logical form by the following procedure.

While converting such statements, first interchange the subject and the predicate.

Then replace the words like 'only', 'alone' etc with 'all'.

Now it will become a regular proposition.

For example,

"Only Oriyas are students of this college." Irregular proposition.

"All students of this college are oriyas." A – Proposition.

"The honest alone wins the confidence of people." Irregular Proposition.

"All persons who win the confidence of people are honest." A-proposition.

VII. Propositions in which the predicate is affirmed or denied of the whole subject with some exception is called an exceptive proposition.

An exceptive proposition may be definite or indefinite. If the exception is definitely specified as in case of "All metals except mercury are solid" then the proposition is to be treated as universal and if the exception is indefinite, as in case of "All metals except one is solid", the proposition is to be treated as particular.

"All metals except mercury are solid." is a universal proposition.

It means, "All non-mercury metals are solid."

Now let us consider an example where the exception is indefinite.

For example, "All students of my class except a few are well prepared".

This is to be reduced to an I-proposition as given below.

"All students of my class except a few are well prepared" is Irregular proposition.

"Some students of my class are well prepared." is an I – proposition.

VIII. There are impersonal propositions where the quantity is not specified.

Consider for example, "It is cold", "It is ten O'clock".

In such cases propositions in question are to be reduced to A-proposition because the subject in each of these cases is a definite description.

"It is cold". Irregular proposition

"The whether is cold." A – Proposition.

"It is ten O'clock." Irregular proposition.

"The time is ten O'clock." A – Proposition.

There are some propositions where the quantity is not specified. In such cases we have to examine the context of its use to decide the quantity.

For example, consider following sentences

(1) "Dogs are carnivorous",

(2) "Men are mortal",

(3) "Students are present."

In first two examples, the quantity has to be universal but in the third case, it is particular. Thus, their reductions into logical form are as follows.

"Dogs are carnivorous." Irregular proposition.

"All dogs are carnivorous." A – Proposition.

This is so because we know that "being carnivorous' is true of all dogs.

"Men are mortal." Irregular proposition.

"All men are mortal." A – Proposition

Here 'being mortal' is generally true of men.

But in the proposition "Students are present",

we mean to assert that some students are present".

So the proposition "Men are mortal" is reduced to

"All men are mortal"

But in the example "Students are present",

'being present' is not generally true of all students.

So the proposition "Students are present" is reduced to

"Some dents are present" which is an I-proposition.

Thus the context of use of a proposition determines the nature of the proposition.

IX. Problematic propositions are particular in meaning.

For example "The poor may be happy" should be treated as a particular proposition, because what such a proposition asserts is that it is sometimes true and sometimes false.

Thus, "The poor may be happy" is reduced to "Some poor people are happy", which is an I-proposition.

X. Similarly, there are propositions where the quantity is not specified but their predicates are qualified by the words like 'hardly', 'scarcely', 'seldom'.

Such propositions should be treated as particular negative.

For example, "Businessmen are seldom honest", is an irregular proposition.

It is reduced to "Some businessmen are not honest".

If such a proposition contains the sign of negation that these proposition is to be treated as an I-proposition.

For example, "Businessmen are not seldom honest." is to be reduced to "Some businessmen are honest", which is an I - proposition.

This is so because it involves a double negation which is equivalent to affirmation.

d) Distribution of terms in A, E, I, O propositions.

Distribution of Terms : When we state something about the entire group indicated by the Terms, the Term is distributed. In a universal proposition Subject is Distributed and in a negative proposition Predicate is Distributed.

Quantity of Proposition : It is the quantity of the group of the subject of a proposition. This is of two types. Universal & Particular. The Universal quantity distributes the subject not the particular.

Quality of Proposition : - It is the quality of the Predicate of the proposition. This is affirmative or negative. Affirmative says that subject or its group belongs to the group of predicate. Here the predicate terms is not distributed. Negative quality says that the subject or its group does not belongs to the group of predicate. Here the predicate is distributed.

TABLE explaining the DISTRIBUTION of terms

Type	S	P
A	Universal	Affirmative X
E	Universal	Negative
I	Particular X	Affirmative X
O	Particular X	Negative

Distribution of terms - universal, particular, affirmative, negative terms.

Distribution of Terms : When we state something about the entire group indicated by the Terms, the Term is distributed. In a universal proposition Subject is Distributed and in a negative proposition Predicate is Distributed.

TABLE explaining the DISTRIBUTION of terms

Type	S	P
A	Universal	Affirmative
E	Universal	Negative
I	Particular	Affirmative
O	Particular	Negative

\ For the proposition types:

A S X

E S P

I X X

O X P

CHAPTER 4. MODERN CLASSIFICATION OF PROPOSITIONS

CHAPTER – 4
MODERN CLASSIFICATION OF PROPOSITIONS
a) Aim of Modern classification, kinds of Simple and Compound propositions

b) Basic Truth Tables for Compound propositions.

Traditional logic deals with limited types of propositions. So, it was difficult to deal with many types of propositions. This is the reason why Modern Logic or formal logic came into existence. It follows and expands on Logic given by Aristotle.

This logic simplifies the way in which we reason. It also makes difference between form and content of propositions and arguments. This logic has introduced mathematical formal methods in logic and with the help of these methods, we can test the valid relationships between terms and propositions in no time.

Let us see the modern classification and its details:

a) Aim of Modern classification,.

Modern logic aims at re-organizing the logical concepts and expanding the boundaries of logical thinking. While doing so, we look at the statements used in logic with a different perspective.

This is the reason why we classify them a bit differently here on the basis of terms, verbs and connectives used in them. This way to classify the propositions makes it easy to understand the relationship between parts of the propositions in an argument as here we make them have objective and mathematical appearance.

Modern classification tries to simplify our thinking and also organize it more effectively so that more types of reasoning can be included in the classification.

Kinds of Simple and Compound propositions & basic Truth Tables

In modern logic, simple proposition is defined as one with only one verb in it. Such a proposition has no connective in it. The simple proposition have no connective. They have only one verb and do not indicate any complicated meaning.

The Simple propositions are classified into two types,

a) subject-less propositions, b) subject-predicate propositions,

The subject-predicate propositions are further classified into

i) relational propositions and ii) class membership proposition.

Let us see the simple proposition types in details:

a) Subject-less propositions, are propositions that have only predicate and no subject. These are symbolized by using single alphabet that stands for predicate.

b) Subject-predicate propositions, are the propositions that have a subject, a predicate and a verb. The subject-predicate propositions are further classified into two types. Relational and class-membership. Let us see these types:

i) Relational propositions are the propositions that show some type of relationship between the term of subject and that of predicate. This means in this type, both the subject and predicate are singular terms.

ii) Class membership proposition shows that the subject term belon gs to the class indicated by predicate. So, here, predicate term is general.

Modern logic also defines a compound proposition that has one or more components connected using one or more connectives.

The compound propositions have at least one connective used in them. They have one or more component that connectives join meaningfully.

When we express these propositions in an objective way, we can explicitly state whether the given compound proposition is true or not on the basis of truth or falsity of the components it connects and the type of connective used.

In modern logic the connecting words, commonly called as connectives, are classified into two types, viz. Monadic and Diadic.

Monadic connective is a connective that works on only one proposition.

The class of monadic connectives has only one connective in it.

This is negation.

This means in modern logic, negative proposition is no more with different quality.

It is a compound proposition.

A negation is expressed by words like 'no, never, not' etc.

While symbolizing a negation, we use the symbol ' ~ ' that is called curl or tilde.

A negation is true when the component to which it is attached is false.

Diadic connectives are connectives that work on two propositions. We have four diadic connectives. They are; conjunction, disjunction, implication and equivalence.

Conjunction is expressed by words like 'and, but'.

While symbolizing this, we use the symbol ' . ' called a dot.

A proposition with conjunction is true only when both its components are true.

Dis-junction is expressed by words like 'either, or.'

While symbolizing this, we use the symbol ' v ' called a vedge.

A proposition with disjunction is false only when both its components are false.

Implication is expressed by words like 'If...then, unless...'

While symbolizing this, we use the symbol ' ' called a horse-shoe.

A proposition with implication is false only when its antecedent, i.e. the first component is true and the consequent, i.e. the second component is false.

Equivalence is expressed by words like 'if and only if... then.'

While symbolizing this, we use the symbol ' ' called a dot.

A proposition with conjunction is true only when both its components are true.

Let us see this classification at a glance:

Proposition

Sentence that asserts

| |

Simple Compound (with connective)

No connective one or more components

| |

| | | |

Subject-less Subject-predicate Monadic Diadic

No subject | one component two component

| | …........................|

| | Negation 1 = Conjunction = .

Relational Class-membership =No, Not 2 = Dis junction = V

= ~ 3 = Implication =

4 = Equivalence=

b) Basic Truth Tables for Compound propositions

We saw the connectives and their symbols. Now let us see how the propositions are symbolized in modern classification.

Compound propositions are symbolized in modern classification by taking a capital alphabet for the first letter of the predicate of first component simple statement, and a capital alphabet for the first letter of the predicate of the second component simple statement.

Between these two alphabets, we put the symbol for the connective that is connecting these two components.

This means, if we have a proposition,

'If Logic is easy, then many will learn it.'

we take 'E' for 'logic is easy' and ' L' for 'many will learn it'.

The connective here is implication. The symbol for it is, .

We write this in between E and L. This reads as 'E L'

This is how we can symbolize any given proposition in modern logic.

So, if we take standard alphabets P for first component and Q for second, we can express all compound proposition types as follows:

Negation: ~P

Conjunction: P . Q

Dis-junction: P v Q

Implication: P Q

Equivalence: P Q

The method we use to check the validity of their relations is called the method of constructing truth tables. While doing this, we check the possibilities of truth and falsity in both the components.

We arrange these possibilities here using the 2ⁿmethod of calculating the possibilities. Here 2 stands for the two truth value options, viz. True and false. The alphabet 'n' stands for number of variables present in the compound proposition.

If a proposition has only one variable, that means only one simple proposition, even if it is repeated, then we have 2¹= 2 possibilities of truth value combinations.

If a proposition has two different simple statements as components, then we have 2²= 4 possibilities of truth value combinations.

If a proposition has three different simple statements as components, then we have 2³= 8 possibilities of truth value combinations.

If a proposition has four different simple statements as components, then we have 2⁴= 16 possibilities of truth value combinations.

Of course, for learning the basic truth-functional tables, we need to see only the first two options, i.e. the statements with 2 and 4 combination options.

When we have a single component as in ~P, we write the truth table as:

P ~P

T F

F T

When we have two components as in P . Q, P v Q, P Q, P Q, we make the truth tables by using the terms of validity of each connective as follows:

Let us write possibilities for all proposition types together for easy understanding.

P Q P . Q P v Q P Q P Q

T T T T T T T T T T T T T T

T F T F F T T F T F F T F F

F T F F T F T T F T T F F T

F F F F F F F F F T F F T F

On the basis of the above table, we can pick up the table for any relavent proposition type to be symbolized and form a truth table for it.

While doing this, follow the following steps:

Write the first part of 'P Q' and the truth values under it

then write the proposition type as per the connective.

Like,

Negation: ~P

Conjunction: P . Q

Dis-junction: P v Q

Implication: P Q

Equivalence: P Q

Then form the relevant truth table for it.

Suppose we have a proposition like, 'Law is useful and Religion is peaceful”

We symbolize it as 'U . P' Then we form a truth table for it as:

U P U . P

T T T T T

T F T F F

F T F F T

F F F F F

Suppose we have a proposition like, 'Law is useful or Religion is peaceful”

We symbolize it as 'U v P' Then we form a truth table for it as:

U P U v P

T T T T T

T F T T F

F T F T T

F F F F F

Suppose we have a proposition like, 'If Law is useful then Religion is peaceful”

We symbolize it as 'U P' Then we form a truth table for it as:

U P U P

T T T T T

T F T F F

F T F T T

F F F T F

Suppose we have proposition, 'If & only If Law is useful then Religion is peaceful”

We symbolize it as 'U P' Then we form a truth table for it as:

U P U P

T T T T T

T F T F F

F T F F T

F F F T F

CHAPTER – 5

COMPARATIVE STUDY OF TRADITIONAL AND MODERN CLASSIFICATION OF PROPOSITIONS

a) Distinction between the Traditional and Modern General propositions.
b) Meaning of prediction with special reference to the Copula.
c) Failure of Traditional classification of propositions.

a) Distinction between the Traditional and Modern General propositions.

According to Traditional Logic general propositions are classified in four categories.

These are:

A = Universal affirmative

E = Universal negative

I = Particular affirmative

O = Particular negative

We have already studied them in details in earlier chapters.

General Propositions in modern Logic are similar to those in traditional logic.

‘All mobile phones are electronic gadgets’ is simple proposition. In such proposition we find the relation of different classes.

In the above proposition the subject term refers to a class of objects ‘mobile phones’ & the predicate term refers to another class of objects ‘electronic gadgets’.

So, a general proposition is a proposition which asserts that one class is wholly or partly included in or excluded from another class.

A general proposition, therefore, makes an assertion about all or about some of the members of a class.

The method of symbolizing with Quantifiers, seen in chapters above is actually the method used in Modern Logic, after the concept of symbolizing the propositions became popular.

b) Meaning of prediction with special reference to the Copula.

Traditional logicians have divided propositions into singular and general. Singular propositions have a single individual as a subject. This means, in a singular proposition, the subject is a singular individual thing and predicate is a class of individuals.

General propositions have a group of individuals as a subject. This means, in a General proposition, we have a group of individuals as a subject as well as a group of individuals as a predicate.

The general propositions are of two types, universal and general.

When the general proposition says something about the entire group indicated in the subject, it is known as a universal proposition.

When the general proposition says something about a part of the group indicated in the subject, it is known as a particular proposition.

According to this, the general propositions are classified into four categories.

These are:

A = Universal affirmative

E = Universal negative

I = Particular affirmative

O = Particular negative

c) Failure of Traditional classification of propositions.

The problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if:

“Some cat is feared by every mouse”

then it follows logically that:

All mice are afraid of at least one cat

The syntax of traditional logic (TL) permits exactly four sentence types:

"All As are Bs",

"No As are Bs",

"Some As are Bs" and

"Some As are not Bs".

Each type is a quantified sentence containing exactly one quantifier.

Since the sentences above each contain two quantifiers; 'some' and 'every' in the first sentence and 'all' and 'at least one' in the second sentence, they cannot be adequately represented in TL.

The best TL can do is to incorporate the second quantifier from each sentence into the second term, thus rendering the artificial-sounding terms 'feared-by-every-mouse' and 'afraid-of-at-least-one-cat'. This in effect "buries" these quantifiers, which are essential to the inference's validity, within the hyphenated terms.

Hence the sentence "Some cat is feared by every mouse" is allotted the same logical form as the sentence "Some cat is hungry". And so the logical form in TL is:

Some As are Bs

All Cs are Ds

which is clearly invalid.

The first logical calculus capable of dealing with such inferences was Gottlob Frege's Begriffsschrif, the ancestor of modern predicate logic, which dealt with quantifiers by means of variable bindings.

Modestly, Frege did not argue that his logic was more expressive than extant logical calculi, but commentators on Frege's logic regard this as one of his key achievements.

Using modern predicate calculus, we quickly discover that the statement is ambiguous.

Some cat is feared by every mouse

could mean

Some cat is feared by every mouse, i.e.

For every mouse m, there exists a cat c, such that c is feared by m,

$\forall m. \, (\, \text{Mouse}(m) \rightarrow \exists c. \, (\text{Cat}(c) \land \text{Fears}(m,c)) \, )$

in which case the conclusion is trivial.

But it could also mean Some cat is (feared by every mouse), i.e.

There exists one cat c, such that for every mouse m, c is feared by m.

$\exists c. \, ( \, \text{Cat}(c) \land \forall m. \, (\text{Mouse}(m) \rightarrow \text{Fears}(m,c)) \, )$

This example illustrates the importance of specifying the scope of quantifiers as for all and there exists.

CHAPTER 6. INFERENCE
INFERENCE

a) Kinds of inference- Immediate and Mediate.
b) Opposition of proposition- Types of opposition- inference by opposition of propositions- opposition of Singular propositions.
AN INFERENCE is a mental process by which we pass from one or more statements to another that is logically related to the former.

a) Kinds of inference –

Inferences are classified on the basis of their scope into Deductive and Inductive. Deductive Inference have a conclusion that stays within the scope of premises. Inductive Inferences are the ones that go beyond the scope of the premises.

The Deductive Inferences are of two types, Mediate and Immediate.

Inductive Inferences are of two types, perfect induction and imperfect induction.

Immediate & Mediate

We are studying the Immediate and mediate inferences here.

Based on the number of their premise, inferences are basically classified into two types, immediate and mediate:

Immediate Inference consists in passing directly from a single premise to a conclusion. It is reasoning, without the intermediary of a middle term or second proposition, from one proposition to another which necessarily follows from it.

Ex: No Dalmatians are cats. Therefore, no cats are Dalmatians.

All squares are polygons. Therefore, some polygons are squares.

Mediate Inference consists in deriving a conclusion from two or more logically interrelated premises. Involving an advance in knowledge, it is reasoning that involves the intermediary of a middle term or second proposition which warrants the drawing of a new truth.

Ex: All true Christians are theists.

Paul is a true Christian.

Therefore, Paul is a theist.

Let us see the various types of inferences and their sub classes:

The following outline serves as a guide in understanding the different types of inference according to various classifications.

I. Induction

A. Perfect Induction

B. Imperfect Induction

II. Deduction

A. Immediate Inference

1. Oppositional Inference

a. Contrary Opposition

b. Contradictory Opposition

c. Subaltern Opposition

d. Subcontrary Opposition

2. Eduction

a. Obversion

b. Conversion

c. Contraposition

d. Inversion

3. Possibility and Actuality

B. Mediate Inference

1. Categorical Syllogism

2. Hypothetical Syllogism

a. Conditional Syllogism

b. Disjunctive Syllogism

c. Conjunctive Syllogism

3. Special Types of Syllogism

a. Enthymeme

b. Epichireme

c. Polysyllogism

d. Sorites

e. Dilemma

b) Opposition of proposition –

Opposition of propositions is the traditional way to classify general propositions into four types on the basis of their quality and quantity. We have already discussed this in details in earlier chapters.

Types of opposition –

The opposition relation is of three types.

And we have the oppositions on the basis of

quality = Contrary [ A-E] & sub-contrary [I-O], or

quantity = sub-altern [A-I, E-O] or

both = contradictory [A X O, E X I]

Inference by opposition of proposition –

Opposite or Opposed Propositions Are propositions that cannot be simultaneously true or that cannot be simultaneously false, or that cannot be either simultaneously true or simultaneously false.

This impossibility of being simultaneously true, or false, or either true or false is the essential note of logical opposition.

Propositions are opposed if they have the same subject and predicate but differ from one another in quality or quantity, or both in quality and quantity.

When we draw the opposite of any type as a conclusion on the basis of a proposition that is known, we have an inference by opposition of proposition.

The truth functional relationship between oppositions can help us know how this relation can be effective.

Let us see the table of truth and falsity of opposition relations:

Original \|\| Result → V	A	E	I	O
A	T / F	*F / T*	*T / ?*	*F / T*
E	*F / ?*	T / F	*F / T*	*T / ?*
I	*? / F*	*? / T*	T / F	*? / T*
O	*F / T*	*? / F*	*? / T*	T / F

Using the above table, we can infer the valid conclusions for the inferences based on the opposition relations of propositions.

Opposition of singular propositions

Singular proposition is the proposition having a singular term as its subject. In the four fold classification, this is treated as a universal proposition.

But the only difference is that unlike the general propiositions, the singular propositions do not have subalterns and contradictories. They have only contraries.

So, when we have an opposition relation of an affirmative singular proposition, taken as A, we get an E proposition. But we do not have any other variations in it.

Similarly, when we have an opposition relation of a negative singular proposition, taken as E, we get an A proposition. But we do not have any other variations in it.

This is known as opposition of singular propositions.

CHAPTER 7. EDUCTIONS

EDUCTIONS
a) Conversion and Obversion and other Immediate inferences.
b) Laws of Thought as applied to propositions.

a) Conversion and Obversion and other Immediate inferences.

A proposition that falls in the category of traditional classification, i.e. that is either universal affirmative, or universal negative or particular affirmative or particular negative, has seven more types of relations or ways to express the same subject and predicate terms. These relationships are called Eduction relations.

The concept of Immediate inferences or Eduction relations like obversion conversion etc. is based exactly on this. Let us see how this works:

Here, we are writing the original term relation in a proposition as S==P. To show that we are using the term that is opposite to the original one we are drawing a line above the term. So, when the negative of subject term is used, we write S. Similarly, when we are using the term that is negative of predicate term we write P.

by using the mathematical combination rule, we can get total eight combinations where subject term and predicate term appears once in a statement and each one is either affirmative or negative. This means, if we have s-p as original, it is one of the eight combinations. Rest seven are its relations. This can be written as follows:

S – P – – – P – S

S – ~P – – – P – ~S

~S – P – – – ~P – S

~S – ~P – – – ~P – ~S

The table below can explain these relations & names of each relation at a glance.

S==P	ORIGIONAL	P==S	Converse
S== P	Obverse	P==S	Obverted Converse
S==P	Partial Inverse	P==S	Partial Contrapositive
S==P	Full Inverse	P==S	Full Contrapositive

To understand how this is done, we must see how to check validity of proposition used in any relation of above types by taking example of each type of proposition and converting it in all the above relationships.

The conversion method and understanding of the meaning of the converted statements itself can explain why in some cases no conversion is possible.

Remember, for accepting any type as an equivalent expression of any type of proposition, it must follow the basic Logic rules.

It must clear the distribution test
It must not distort the original meaning.

Let us take A proposition;

e.g. let us say “All study is a useful thing”

We write it as 'S a P'

Let us see Eduction relations of this.

Here, we need to check for all the four proposition type options for each relation.

Original : All study is a useful thing S a P

Obverse: = S e P

All study is a non-useful thing. A

No study is a non-useful thing. E

Some study is a non-useful thing. I

Some study is not a non-useful thing. O

Converse: P i S

All useful thing is a study. A

No useful thing is a study. E

Some useful thing is a study. I

Some useful thing is not a study. O

Obverted Converse: P o S

All useful thing is non-study. A

No useful thing is non-study. E

Some useful thing is non-study. I

Some useful thing is not non-study. O

Partial Inverse: S o P

All non-study is a useful thing. A

No non-study is a useful thing. E

Some non-study is a useful thing. I

Some non-study is not a useful thing. O

Full Inverse: S i P

All non-study is a non-useful thing. A

No non-study is a non-useful thing. E

Some non-study is non-useful thing. I

Some non-study is not non-useful thing. O

Contra-positive (partial): P e S

All non-useful thing is a study. A

No non-useful thing is a study. E

Some non-useful thing is a study. I

Some non-useful thing is not a study. O

Contra-positive (full): P a S

All non-useful thing is a non-study. A

No non-useful thing is a non-study. E

Some non-useful thing is non-study. I

Some non-useful thing is not non-study. O

Let us take E proposition;

e.g. let us say “No study is a useless thing”

We write it as 'S e P'

Let us see Eduction relations of this.

Here, we need to check for all the four proposition type options for each relation.

Original : No study is a useless thing S e P

Obverse: = S a P

All study is a non-useless thing. A

No study is a non-useless thing. E

Some study is a non-useless thing. I

Some study is not a non-useless thing. O

Converse: P e S

All useless thing is a study. A

No useless thing is a study. E

Some useless thing is a study. I

Some useless thing is not a study. O

Obverted Converse: P a S

All useless thing is non-study. A

No useless thing is non-study. E

Some useless thing is non-study. I

Some useless thing is not non-study. O

Partial Inverse: S i P

All non-study is a useless thing. A

No non-study is a useless thing. E

Some non-study is a useless thing. I

Some non-study is not a useless thing. O

Full Inverse: S o P

All non-study is a non- useless thing. A

No non-study is a non-useless thing. E

Some non-study is non- useless thing. I

Some non-study is not non-useless thing. O

Contra-positive (partial): P i S

All non- useless thing is a study. A

No non- useless thing is a study. E

Some non- useless thing is a study. I

Some non- useless thing is not a study. O

Contra-positive (full): P o S

All non-useless thing is a non-study. A

No non-useless thing is a non-study. E

Some non-useless thing is non-study. I

Some non-useless thing is not non-study. O

Let us take I proposition;

e.g. let us say “Some study is a useful thing”

We write it as 'S i P'

Let us see Eduction relations of this.

Here, we need to check for all the four proposition type options for each relation.

Original : Some study is a useful thing S i P

Obverse: = S o P

All study is a non-useful thing. A

No study is a non-useful thing. E

Some study is a non-useful thing. I

Some study is not a non-useful thing. O

Converse: P i S

All useful thing is a study. A

No useful thing is a study. E

Some useful thing is a study. I

Some useful thing is not a study. O

Obverted Converse: P o S

All useful thing is non-study. A

No useful thing is non-study. E

Some useful thing is non-study. I

Some useful thing is not non-study. O

Partial Inverse: S x P

All non-study is a useful thing. A

No non-study is a useful thing. E

Some non-study is a useful thing. I

Some non-study is not a useful thing. O

Full Inverse: S x P

All non-study is a non-useful thing. A

No non-study is a non-useful thing. E

Some non-study is non-useful thing. I

Some non-study is not non-useful thing. O

Contra-positive (partial): P x S

All non-useful thing is a study. A

No non-useful thing is a study. E

Some non-useful thing is a study. I

Some non-useful thing is not a study. O

Contra-positive (full): P x S

All non-useful thing is a non-study. A

No non-useful thing is a non-study. E

Some non-useful thing is non-study. I

Some non-useful thing is not non-study. O

Let us take O proposition;

e.g. let us say “Some study is a not useless thing”

We write it as 'S o P'

Let us see Eduction relations of this.

Here, we need to check for all the four proposition type options for each relation.

Original : Some study is not a useless thing S o P

Obverse: = S i P

All study is a non-useless thing. A

No study is a non-useless thing. E

Some study is a non-useless thing. I

Some study is not a non-useless thing. O

Converse: P x S

All useless thing is a study. A

No useless thing is a study. E

Some useless thing is a study. I

Some useless thing is not a study. O

Obverted Converse: P x S

All useless thing is non-study. A

No useless thing is non-study. E

Some useless thing is non-study. I

Some useless thing is not non-study. O

Partial Inverse: S x P

All non-study is a useless thing. A

No non-study is a useless thing. E

Some non-study is a useless thing. I

Some non-study is not a useless thing. O

Full Inverse: S x P

All non-study is a non- useless thing. A

No non-study is a non-useless thing. E

Some non-study is non- useless thing. I

Some non-study is not non-useless thing. O

Contra-positive (partial): P i S

All non- useless thing is a study. A

No non- useless thing is a study. E

Some non- useless thing is a study. I

Some non- useless thing is not a study. O

Contra-positive (full): P o S

All non-useless thing is a non-study. A

No non-useless thing is a non-study. E

Some non-useless thing is non-study. I

Some non-useless thing is not non-study. O

Let us see EDUCTION at a glance in brief:

Original	Obverse	Partial Inverse	Full Inverse	Converse	Obverted Converse	Partial Contra-positive	Full Contra-positive
S - P	S - P	S - P	S - P	P - S	P - S	P - S	P - S
S a P	S e P	S o P	S i P	P i S	P o S	P e S	P a S
S e P	S a P	S i P	S o P	P e S	P a S	P i S	P o S
S i P	S o P	S x P	S x P	P i S	P o S	P x S	P x S
S o P	S i P	S x P	S x P	P x S	P x S	P i S	P o S

In detail:

Relation	Changed	Type	Original
Original = S-P			All S is P	No S is P	Some S is P	Some S is not P
			A	E	I	O
Obverse	All S is non P	A		All S is non P
S-P	No S is non P	E	No S is non P
	Some S is non P	I				Some S is non P
	Some S is not non P	O			Some S is not non P

Converse	All P is S	A				X
P-S	No P is S	E		No P is S		X
	Some P is S	I	Some P is S		Some P is S	X
	Some P is not S	O				X

Obv Converse	All S is non P	A		All S is non P		X
P-S	No S is non P	E				X
	Some S is non P	I				X
	Some S is not non P	O	Some S is not non P		Some S is not non P	X

Part Inverse	All non S is P	A			X	X
S-P	No non S is P	E			X	X
	Some non S is P	I		Some non S is P	X	X
	Some non S is not P	O	Some non S is not P		X	X

Full Inverse	All non S is non P	A			X	X
S-P	No non S is non P	E			X	X
	Some non S is non P	I	Some non S is non P		X	X
	Some non S is not non P	O		Some non S is not non P	X	X

Part Contra +ve	All non-P is S	A			X
P-S	No non P is S	E	No non P is S		X
	Some non P is S	I		Some non P is S	X	Some non P is S
	Some non P is not S	O			X

Full Contra +ve	All non P is non S	A	All non P is non S		X
P-S	No non P is non S	E			X
	Some non P is non S	I			X
	Some non P is not non S	O		Some non P is not non S	X	Some non P is not non S

b) Laws of Thought as applied to propositions.

In 18th, 19th, & early 20th Century, scholars who followed the Aristotelian and Medieval tradition in logic, spoke of the “laws of thought” as the basis of all logic.

The usual list of logical laws includes three axioms:

The law of identity,

The law of non-contradiction, and

The law of excluded middle.

The thinking in logic must have a solid base and these three laws provide this base. They are the foundation of logical thinking.

The law of identity could be summarized as the patently unremarkable but seemingly inescapable notion that things must be, of course, identical with themselves. Expressed symbolically: “A is A,” where A is an individual, a species, or a genus. Although Aristotle never explicitly enunciates this law, he does observe, in the Metaphysics, that “the fact that a thing is itself is [the only] answer to all such questions as why the man is man, or the musician musical.”

This suggests that he does accept, unsurprisingly, the perfectly obvious idea that things are themselves. If, however, identical things must possess identical attributes, this opens the door to various logical maneuvers.

One can, for example, substitute equivalent terms for one another and, even more portentously, one can arrive at some conception of analogy and induction. Aristotle writes, “all water is said to be . . . the same as all water . . . because of a certain likeness.” If water is water, then by the law of identity, anything we discover to be water must possess the same water-properties.

Aristotle provides several formulations of the law of non-contradiction, the idea that logically correct propositions cannot affirm and deny the same thing:

“It is impossible for anyone to believe the same thing to be and not be.”

“The same attribute cannot at the same time belong and not belong to the same subject in the same respect.” “The most indisputable of all beliefs is that contradictory statements are not at the same time true.” Symbolically, the law of non-contradiction is sometimes represented as “not (A and not A).”

The law of excluded middle can be summarized as the idea that every proposition must be either true or false, not both and not neither. In Aristotle’s words, “It is necessary for the affirmation or the negation to be true or false.” Symbolically, we can represent the law of excluded middle as an exclusive disjunction: “A is true or A is false,” where only one alternative holds. Because every proposition must be true or false, it does not follow, of course, that we can know if a particular proposition is true or false.

Despite challenges to these so-called laws, Aristotelians inevitably claim that such counterarguments have unresolved ambiguity equivocation, on a conflation of what we know with what is actually the case, on a false or static account of identity, or on some other failure to fully grasp the implications of what one is saying.

In short, we can say that our thinking naturally follows some thumb rules that are listed as the three main laws. They are called as laws of thought. These are, law of identity, law of non-contradiction, and law of excluded middle.

Let us see these laws in a simple way:

LAW OF IDENTITY: This law says that something is what it is. In short, we can say, “A is A”. That means, to prove or state the existence of something that already is, we need not have any other proof. The presence of anything is self-proven. This is where we say, “If I am, then I am. Or, I am existing, therefore I am existing. Or, I am myself.” Another common way of expressing law of identity is, “Sun is Sun”, “Moon is Moon”, “Tree is Tree” and so on.
LAW OF NON-CONTRADICTION: This law is also written as and called as Law of Contradiction by some people. This states a simple thing, a thing cannot be true and false at the same time at the same place. If someone is saying so, he is telling a lie. If a thing is existing, then it cannot be absent from the same place at the same time when and where it is claimed to exist. This means, two contradictory statements cannot be true together. For example, if I say, “I have Logic book in my hand” I cannot say at the same time, in the same place, “I do not have Logic in my hand.”
LAW OF EXCLUDED MIDDLE: This law states that there is no third option between a statement and its contradiction. This means, when we give two contradictory options for anything, there is no third way possible. This law is useful especially when we have to categorically state some options about something. Use of this law removes all ambiguity & vagueness of expression. For example, when I say, “Either I believe in what you say or I do not.” there ios no third way. The person to whom I am talking cannot say that I believe in him and both believe at the same time, he cannot talk of any third possibility.

This is how we describe the laws of thought. We must remember that these are the foundation of logical thinking and all of us have been using them in our thinking much before we learned that they are the basis of thinking. They form the basic foundation of any logical activity. Experiments may show that even animals and insects use these laws in their thinking when they think and choose to do anything.

CHAPTER 8. DEFINITION

DEFINITION

a) Its purpose- rules and fallacies as per Traditional Definition
b) Modern Definitions-kinds.

A definition is a statement which explains what a thing is. It is a statement that answers the question “What is this thing?”

In giving the definition of the term, it is presupposed that the comprehension of the term is understood, because the definition is based on its comprehension.

Real definition is one which explains & reveals complete nature of thing or object.

However, this is quite impossible since, we do not usually have a full grasp of the nature of things.

It therefore explains the normal acceptance of a simple description as definition of an object.

“Definition is an explanation of a thing, word, phrase or symbol that is used in order to explain the defined thing clearly.”

By using a definition, we explain actual things as well as abstract concepts. We can see that there are two parts in any definition. The first part consists of thing that is defined and second consists of words used to explain this thing.

These two parts have specific names in a definition.

The part of definition that is explained by rest of words is called the definindum.

The part of the definition that explains the definindum is called the definiens.

So, “a definindum is a thing, word, phrase or symbol that is defined in a definition.” whereas, “the set of words that are used to explain something, or some word or phrase or symbol are called the definiens.”

The term “definition” came from the Latin word “Definire” means, “to lay down the markers or limits.”

Definition is a conceptual manifestation either of the meaning of the term or of the formal features of an object. “ definire” meaning “ to lay down”

Thus, etymologically, to define means: Real Definition. A real definition is one which explains and reveals the complete nature of a thing or object.

However, this is quite impossible since, we do not usually have a full grasp of the nature of things. It therefore explains the normal acceptance of a simple description as definition of an object.

Purposes of Definitions

We use the method of definition in order to know things better. Yet, whenever we define, we always define anything with a purpose.

In order to understand a definition, we must first know why we define.

Let us understand the purposes of a definition. We define anything in order to;

1. Increase Vocabulary.

2. Explain anything clearly.

3. Reduce Ambiguity of word.

4. Eliminate ambiguity of any word.

5. Explain a word theoretically.

6. To Influence attitudes.

Let us see these purposes in details:

1. Increase Vocabulary.

When we are learning any new language, we need to define new words in order to know more words in the language and increase our vocabulary.

2. Explain anything clearly.

When we use any language, some words are not clear enough. At times just listening a word is not enough to understand it. So we need to define them.

3. Reduce vagueness of word.

Some times the meaning of a word depends on the context and without clearity about context, the word appears vague. Definition is necessary at such times.

4. Eliminate ambiguity of any word.

Some words have many meanings and at times are used ambiguously and one does not understand which meaning to use. At such times, definition is of help.

5. Explain a word theoretically.

We have a number of technical terms and words that cannot be understood without definition. It is a correct and clear definition that can help us understand these words and symbols and phrases correctly.

6. To Influence attitudes.

Definition also plays a very important role in the society where people gain by influencing the attitudes of others. At times for social good or for personal good, people define some words or terms in order to influence attitudes.

Rules of Definition:

definition has the power to explain something effectively only and only when the definition is perfect and complete and faultless.

Such a perfect complete faultless definition is called a good definition.

Whenever we want to define anything, we always want to give such perfect definitions, but we seldom know the basic rules of a good definition.

A good definition must follow certain rules in order to be effective.

These rules state that, a definition must set out the essential attributes of the thing defined.

A Definitions should avoid circularity. This means, a definition must not repeat same things in different ways without any meaning where we find that we cannot define "antecedent" without using the "consequent", nor conversely.

The definition must not be too wide or too narrow.

It must be applicable to everything to which it applies.

It must not miss anything out. Also, it must not include any things to which the defined term would not truly apply. The definition must not be obscure.

Definition is used to remove obscurity, so using obscure words in definition is meaningless. A definition should not be negative where it can be positive.

These Rules of Definition can be listed as follows:

1. The definition must be clearer than the term that is being defined. The purpose of the definition is to explain and must, therefore be easy to understand. It must not contain terms which will only make it less intelligible.

2. The definition must not contain the term being defined. The definition must use other terms in defining. It is supposed to explain a particular term and is not supposed to use the same term in the explanation.

3. The definition must be convertible with the term being defined. The purpose of this rule is to make sure that the definition is equal in extension with the term being defined. The definition must not be too narrow nor too broad. If the term and the definition are equal in extension, then, they are convertible.

4. The definition must not be negative but positive whenever possible. The definition is supposed to explain what a term or object is, and not, what it is not. Only when a tern is negative should the definition be negative.

Types of Definitions

Definitions are classified into various types by various logicians. At times, some of these types differ from each other so much that they appear to be contradictory to each other. Let us see some of these types classified by these logicians.

One classification is:

Nominal Definition is definition which speaks about a term but not declaring anything about it. This is done by considering the origin of the term, by describing the term, by giving the synonym of the term or by citing an example that will represent the term

Classification of Nominal Definition:

a.Nominal Definition by Etymology

– attained by tracing the origin of the term.

Ex.: Fraternity came from “frater”, which means “brother”.

b. Nominal Definition by Description

– attained by describing the term.

Ex.: A rose is a flower.

c. Nominal Definition by Synonym

– it is done by giving a word equivalent to the term.

Ex.: Being kind is being benevolent.

d. Nominal Definition by Example

– it is done by citing anything that will represent the term.

Ex.: Our Chief Executive is Benigno Simeon Aquino III.

2. Real Definition declares something about the term. This kind of definition serves to explain about the nature and to distinguish it from other terms.

Classification of Real Definition

a. Real Definition by Genus and Specific Difference

- a definition that explains the essence of a term by considering the intelligible elements that make up the term.

Ex.: A triangle is a figure with three sides.

“figure” – genus, “three sides” – specific difference

b. Real Definition by Description

- It is done by stating the genus of the term but altering the specific difference by giving the logical property, which belongs to the term to be defined.

Ex.: A Police Officer is a man bestowed with authority to enforce a law.

“man” – genus, “bestowed with authority to enforce a law” – logical property

c. Real Definition by Cause

-It is attained by stating the genus of the term but altering the specific difference by tracing its cause. A cause could be its purpose, function, reason for existence, make-up or origin.

Ex.: A book is a written material made-up of several pages and is a source of information.

“written material”– genus, “source of information”– cause or reason for existence

Second classification of definitions is as follows:

DENOTATIVE DEFINITIONS try to explain the meaning of a word by mentioning at least several objects it denotes.

Although we might not view these strictly as definitions, they are, nevertheless, frequently called "denotative definitions."

Among connotative definitions, two different kinds are worth mentioning,

Ostensive definition,
Definition by partial ennumeration

Among denotative definitions, ostensive definitions stand out as especially common and useful.

1. Ostensive definitions are definitions by pointing.

When a young child wants to know the meaning of the word “dog" we are apt to point to a dog and call out the word "dog."

This is an example of an ostensive definition.

2. A second type of denotative definition worth mentioning is a definition by partial enumeration.

Definitions by partial enumeration are simply lists of objects, or types of objects, to which the word refers.

The list, "beagle," "cocker spaniel," "dachshund," "greyhound," "poodle," provides an example of a definition of dog is by partial enumeration.

While denotative definitions might not really seem much like definitions, they do ultimately attempt to convey the meaning of a word, at least indirectly.

For the hope is that by citing the objects the word refers to, the people we are talking with will come to see what that word means.

However, let's turn now to definitions in the more ordinary sense of the term.

CONNOTATIVE DEFINITIONS are usually formulated in the following three ways:

X is Y. Example: A bachelor is an unmarried man.
The word "X" means Y. Example: The word "Bachelor" means unmarried man.
X =DF. Y. As an example: Bachelor =DF. unmarried man.

In all these cases the term on the left "bachelor" in the above examples is the one being defined, and we call it the "definiendum."

While we refer to the terms used to define this word "unmarried man" in our example, collectively as the "definiens."

Among connotative definitions, perhaps five different kinds are worth mentioning,

(1) persuasive definitions,

(2) theoretical definitions,

(3) precising definitions,

(4) stipulative definitions, and

(5) lexical definitions.

Let us see these definition types in details:

Persuasive Definitions: The purpose of a persuasive definition is to convince us to believe that something is the case and to get us to act accordingly. Frequently definitions of words like "freedom," "democracy," and "communism," are of this type. (E.g., taxation is the means by which bureaucrats rip off the people who have elected them.) While these sorts of definitions might be emotionally useful, we should avoid them when we are attempting to be logical.
Theoretical Definitions: Theoretical definitions explain by a theory. Whether they are correct or not will depend, largely, on whether the theory they are an integral part of is correct. Newton's famous formula "F = ma" (i.e. Force = mass x acceleration), provides a good example of such a definition.
Precising Definitions: Precising definitions attempt to reduce the vagueness of a term by sharpening its boundaries. For example, we might decide to reduce the vagueness in the term "bachelor" by defining a bachelor as an unmarried man who is at least 21 years old. We often encounter précising definitions in the law and in the sciences. Such definitions do alter the meaning of the word they define to some extent. This is acceptable, however, if the revised meaning they provide is not radically different from the original. Sometimes by providing précising definitions we can reduce the potential for verbal disputes that are based on a term's vagueness. When A and B begin argue about whether a bicycle is a vehicle we try to get them to recognize that term "vehicle" contains vagueness. Once they have seen this, we can make them agree to reduce it by providing a précising definition.
Stipulative Definitions: Stipulative definitions are frequently provided when we need to refer to a complex idea, but there simply is no word for that idea. A word is selected and assigned a meaning without any pretense that this is what that word really means. While we cannot criticize stipulative definitions for being incorrect, and so, the objection, "But that isn't what the word means" is inappropriate); we can criticize them as unnecessary, or too vague to be useful.
Lexical Definitions: Unlike stipulative definitions, lexical definitions do attempt to capture the real meaning of a word and so can be either correct or incorrect. When we tell someone that "intractable" means not easily governed, or obstinate, this is the kind of definition we are providing. Roughly, lexical definitions are the kinds of definitions found in dictionaries. Frequently words that are first introduced in the language as stipulative definitions become, over time, lexical definitions. (Consider, for example, Winston Churchill's famous use of the expression "iron curtain.") Besides synonymous definitions, definitions by genus and difference are perhaps the most common type of lexical definition. The essential characteristic of these definitions is we are defining the definiendum by using two terms in the definiens. For example, in the definition, "a bachelor is an unmarried man," we are defining the word "bachelor" in terms of "unmarried" and "man." In this definition the term "unmarried" is the difference, while term "man" is the genus. (The difference, or difference term, qualifies, or says what kind of thing, the genus is.)

Third classification of definition is as follows:

This list has seven kinds of definitions.

1. Stipulative Definitions stipulate, or specify, how a term is to be used.

Sometimes stipulative definitions are used to introduce wholly new terms, othertimes to restrict (or narrow) a meaning in a particular context.

The former use may be seen in the immediately preceding example, where the new term "oxycodone" is being introduced as an abbreviation (mercifully) for the mouthful "dihydrohydroxycodeinone".

2. Lexical definitions, or dictionary definitions, are reports of common usage.

Such definitions are said to be reportive or reportative definitions.

They are true or false depending on whether they do or do not accurately report common usage.

In addition, if the dictionary is published by a prestigious firm and is compiled by competent and respected lexicographers, then the definitions are normative.

The definitions both report and regulate common usage. It thus becomes possible to say of a given person that s/he is misusing a particular term.

If a person's use of a term is at great variance with how that term is regularly used, and if that person does not stipulate that the term is being used in a specialized nonstandard way, then she is using that term incorrectly.

3. Precising definitions are used to refine the meaning of an established term whose meaning is vague in a context and which needs improving.

4. Theoretical definitions is unique to science and philosophy and do not occur in ordinary prose. This is an overly restrictive analysis; theories are not unique to science but characterize virtually all our thinking.

5. Operational definitions explain the way in which a scientific function works. This type of definitions have disappeared in physics; occasionally, however, one will still find instances of them in psychology.

6. The definiens in a recursive definition is typically in two parts: a so-called 'basis' clause in which the definiendum does not occur, and a so-called 'inductive step' in which the definiendum does occur. At first the definition may appear to be circular since the definiendum explicitly occurs in the definiens. But the circularity is only apparent, since the basis clause offers a non-circular entry to – not a circle – but a 'chain' of an indefinite number of 'links'.

7. Persuasive definitions are simply intended to influence attitudes and generally do violence to the lexical definitions. When people begin to cite definitions in a heated argument, it is a good bet that they are making them up.

Fourth and all exhaustive classification:

In short, we can classify the definitions in the following manner:

1. Real ==== a) Ostensive, b) Extensive

2. Nominal = a) Lexical, b) Bi-verbal, c) stipulative, d) per genus et differentium

These types can be seen in details as follows:

1. Real definition: A Real definition is the definition of something that exists. This means, we can use the real definition for explaining things that exist and that can be objectively studied.

We have two sub classes of this definition type.

These are, a) Ostensive and b) Extensive. Let us see them in details:

a) Ostensive Definition is the method of defining any thing by pointing it out. When we show some object in order to define it, we use the ostensive definition.

b) Extensive definition is the definition where we give examples in order to explain something. When we want to define anything, we list out some of the members or things or types that belong to the group indicated by that word.

2. Nominal definition: A nominal definition is a definition of a word, phrase or symbol. When we wish to define or explain any word, phrase or symbol, we use this type of definition. This means, we use nominal definition when we are defining any concept created by human beings in any language of humans.

The nominal definitions have four sub-classes. These subclasses are, a) Lexical, b) Bi-verbal, c) stipulative, d) per genus et differentium.

Let us see these sub-classes in details.

a) Lexical definition gives a dictionary meaning of a word, or defines a word as it is used by any community or group of people.

b) Bi-verbal definition defines a word by using another word or a phrase by using another phrase. But if while doing this, the definition is not making the actual meaning adequately clear, the definition commits fallacy of synonymous definition.

c) Stipulative definition is given when someone is assigning a meaning to a word in order to influence attitudes of twist the actual meaning of the word. This definition may or may not tell the real nature of the word defined.

d) Per genus et differentium is the type of definition where we define a word by stating the group to which it belongs, i.e. the genus; and the factor that still differentiates the given word from rest of the group, i.e. the differentia. We use this definition when we are classifying something that is being defined and also showing that though this thing belongs to that group, it is still different from rest of the group members because it possesses some quality that makes it stand out.

Fallacies of definition.

When a definition is not appropriate, it commits a fallacy. Fallacies of definition are the various ways in which definitions can fail to explain terms. The phrase is used to suggest an analogy with an informal fallacy. "Definitions that fail to have merit because they are overly broad, use obscure or ambiguous language, or contain circular reasoning are called fallacies of definition."

The major fallacies are; overly broad or Too Wide, overly narrow or Too Narrow, Mutually exclusive definitions, Synonymus definitions, Obscure definitions, Self-contradictory definitions & circular definitions.

Fallacies in definitions are listed as follows:

1. Too Wide definition is the definition that applies to things or members to which that word actually does not apply.

2. Too Narrow definition is the definition that excludes many things to which the word being defined actually applies.

3. Mutually exclusive definitions are the definitions where we find some qualities that do not belong to the word defined. The definiens of mutually exclusive definitions list characteristics which are the opposite of those found in the definiendum. e.g. a cow is defined as a flying animal with no legs.

4. Synonyms definitions are the definitions where one word is defined by another without explaining any of them clearly.

5. Obscure definitions are definitions using inappropriate language or the language that feels odd, but does not explain anything about the word in question..

6. Self-contradictory definition occurs when the definindum used two contradictory qualities together in explaining the definiens.

7. Ambiguous definition is the definition where a word has many meanings & we are using an inappropriate meaning while defining it in some situation.

8. Figurative definition is the way to define something using decorative language. Such a language may or may not explain the word appropriately.

9. Circular definitions If one concept is defined by another, and the other is defined by the first, this is known as a circular definition where neither defenins nor definindum offers enlightenment about what one wanted to know

Limitations of definition

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

A question naturally arises when we start defining things. This is, if every term of every definiens must be defined, by itself, where at last should we stop?

A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions, must resort to circularity

Many philosophers have chosen instead to leave some terms undefined. The scholastic philosophers claimed that the highest genera; the so-called ten generalissima cannot be defined, since a higher genus cannot be assigned under which they may fall.

Thus being, unity and similar concepts cannot be defined.

John Locke supposes in An Essay Concerning Human Understanding that the names of simple concepts do not admit of any definition. More recently Bertrand Russell sought to develop a formal language based on logical atoms.

Other philosophers, notably Wittgenstein, rejected the need for any undefined simples. Wittgenstein pointed out in his Philosophical Investigations that what counts as a "simple" in one circumstance might not do so in another.

He rejected the very idea that every explanation of the meaning of a term needed itself to be explained: "As though an explanation hung in the air unless supported by another one", claiming instead that explanation of a term is only needed to avoid misunderstanding.

Locke and Mill also argued that individuals cannot be defined.

Names are learned by connecting an idea with a sound, so that speaker and hearer have the same idea when the same word is used. This is not possible when no one else is acquainted with the particular thing that has "fallen under our notice".

Russell offered his theory of descriptions in part as a way of defining a proper name, the definition being given by a definite description that "picks out" exactly one individual. Saul Kripke pointed to difficulties with this approach, especially in relation to modality, in his book Naming and Necessity.

There is a presumption in the classic example of a definition that the definiens can be stated. Wittgenstein argued that for some terms this is not the case.

The examples he used include game, number and family. In such cases, he argued, there is no fixed boundary that can be used to provide a definition.

Rather, the items are grouped together because of a family resemblance.

For terms such as these it is not possible and indeed not necessary to state a definition; rather, one simply comes to understand the use of the term.

CHAPTER 9. DEFINITIONS IN LAW
Definition and law

Precise definition with special reference to any specific definition- disablement, industry (Labour Law), Private and Public Nuisance (Laws of Torts), consent (Law of Contract), Medical- intervention, physician, terminally- ill

Law has definitions of concepts codified in various legal acts. Knowing these helps us understand the basics of law. Let us see some of these definitions.

Disablement is something that reduces the earning capacity of a workman in any employment in which he was engaged at the time of the accident resulting in the disablement,where the disablement is of a permanent nature, such disablement as reduces his earning capacity in every employment which he was capable of undertaking at that time.

Partial disablement means, where the disablement is of a temporary nature, such disablement as reduces the earning capacity of a workman in any employment in which he was engaged at the time of the accident resulting in the disablement, and, where the disablement is of a permanent nature, such disablement as reduces his earning capacity in every employment which he was capable of undertaking at that time : Provided that every injury specified in Part II of Schedule I shall be deemed to result in permanent partial disablement;

Total disablement means such disablement, whether of a temporary or permanent nature, as incapacitates a workman for all work which he was capable of performing at the time of the accident resulting in such disablement Provided that permanent total disablement shall be deemed to result from every injury specified in Part I of Schedule I or from injuries specified in Part II.

Medical intervention: Intervention is act of intervening, interfering or interceding with the intent of modifying the outcome. In medicine, intervention is usually undertaken to help treat or cure a condition. Such intervention by a registered medical practitioner in health condition of patient is Medical Intervention.

Physician is a person who is legally qualified to practice medicine, especially one who specializes in diagnosis and medical treatment as distinct from surgery. A physician is a professional who practices medicine, which is concerned with promoting, maintaining or restoring human health through the study, diagnosis, and treatment of disease, injury, and other physical and mental impairments.

Terminally ill person is a person who is sick and is diagnosed with a disease that will take their life. Such a person is usually told by doctors that they only have several months or years to live. This term is more commonly used for progressive diseases such as cancer or advanced heart disease than for trauma.

Consent is permission for something to happen or agreement to do something. According to Indian Contract Act, Two or more persons are said to consent when they agree upon the same thing in the same sense. Consent mentioned in the defition is assumed to be a free consent.

Free consent: Consent is said to be free when it is not caused by coercion, undue influence, fraud, misrepresentation, mistake. Consent is said to be so caused when it would not have been given but for the existence of such coercion, undue influence, fraud, misrepresentation or mistake.

Coercion is the committing, or threatening to commit, any act forbidden by the Indian Penal Code, or the unlawful detaining, or threatening to detain, any property, to the prejudice of any person whatever, with the intention of causing any person to enter into an agreement.

Undue influence defined:

(1) A contract is said to be induced by "undue influence” where the relations subsisting between the parties are such that one of the parties is in a position to dominate the will of the other and uses that position to obtain an unfair advantage over the other.

(2) In particular and without prejudice to the generality of the foregoing principle, a person is deemed to be in a position to dominate the will of another (a) where he holds a real or apparent authority over the other or where he stands in a fiduciary relation to the other; or (b) where he makes a contract with a person whose mental capacity is temporarily or permanently affected by reason of age, illness, or mental or bodily distress.

(3) Where a person who is in a position to dominate the will of another, enters into a contract with him, and the transaction appears, on the face of it or on the evidence adduced, to be unconscionable, the burden of proving that such contract was not induced by undue influence shall lie upon the person in a position to dominate the will of the other.

Fraud means and includes any of the following acts committed by a party to a contract, or with his connivance, or by his agent, with intent to deceive another party thereto of his agent, or to induce him to enter into the contract:-
(1) the suggestion, as a fact, of that which is not true, by one who does not believe it to be true;
(2) the active concealment of a fact by one having knowledge or belief of the fact;
(3) a promise made without any intention of performing it
(4) any other act fitted to deceive;
(5) any such act or omission as the law specially declares to be fraudulent.

Misrepresentation means and includes –
(1) The positive assertion, in a manner not warranted by the information of the person making it, of that which is not true, though he believes it to be true
(2) any breach, of duty which, without an intent to deceive, gains an advantage to the person committing it, or any one claiming under him, by misleading another to his prejudice or to the prejudice of any one claiming under him;
(3) Causing, however innocently, a party to an agreement to make a mistake as to the substance of the thing which is the subject of the agreement.

NUISANCE: Substantial interference with the right to use and enjoy land, which may be intentional, negligent or ultrahazardous in origin, and must be a result of defendant's activity. This word means literally annoyance; in law, it signifies, according to Blackstone, " anything that worketh hurt, inconvenience or damage." Nuisances are either public or common, or private nuisances.

A public or common nuisance is such an inconvenience or troublesome offence, as annoys the whole community in general, and not merely some particular person. To constitute a Public nuisance, there must be such 'a number of persons annoyed, that the offense can no longer be considered a private nuisance. The concept of nuisance is relative. A thing may be a nuisance in one place, which is not so in another; therefore situation or locality of the nuisance must be considered.

A private nuisance is anything done to cause hurt or annoyance of the lands, tenements, or hereditaments of another.

Industry means any systematic activity carried on by co-operation between an employer and his workmen for the production ,supply or distribution of goods or services with a view to satisfy human wants or wishes (not being wants or wishes which are merely spiritual or religious in nature), whether or not,

(i) any capital has been invested for the purpose of carrying on such activity; or

(ii) such activity is carried on with a motive to make any gain or profit, and includes (a) any activity of the Dock Labour Board established under section 5-A of the Dock Workers ( Regulation of Employment)Act,1948( 9 of 1948); (b) any activity relating to promotion of sales or business or both carried on by an establishment,
but does not include any agricultural operation except where such agricultural operation is carried on in an integrated manner with any other activity (being any such activity as is referred to in the foregoing provisions of this clause) and such other activity is the predominant one.

CHAPTER 10. DIVISION

DIVISION

Logical division - rules and fallacies of division - division by dichotomy.

Logical division:

Logical division is a simple method of dividing a class into its sub-classes in order to explain the or describe any class. This type of division is useful in explaining many concepts and making the understanding clear.

Division is useful for;

a] determination of exact relationships among related things,

b] formulation of definitions

When we divide, we use two main criteria. These are, Physical division and metaphysical division.

Physical division divides a whole into its parts
• e.g., a complex machine into its simple mechanical parts

Metaphysical division divides an entity into its qualities,

• e.g.,a species into its genus & difference
– man into animality & rationality

• a substance into its attributes
– sugar into color, texture, solubility, taste, etc.

• a quality into its dimensions
– sound into pitch, timbre, volume

Understanding Division:

Division is another way to explain any class by talking about its sub-groups and dividing the class into its sub groups. Here are its basic qualities:

• Logical Division

– begins with a summum genus
– proceeds through intermediate genera
– ends at the infimae species
– NB: It does not continue to individuals

• The results of division should meet these criteria:

1. The subclasses of each class should be coextensive with original class.
2. The subclasses of each class should be mutually exclusive.
3. The subclasses of each class should be jointly exhaustive.
4. Each stage of a division should be based on a single principle.

Kinds of Classification

Classification is the technique of inquiry in which similar individuals and classes are grouped into larger classes.

e.g., how are steam, diesel, & gasoline engines related to one another?

Natural Classification:
• Natural classification is a scheme that provides theoretical understanding of its subject matter e.g. classification of living things into monerans, protistans, plants, fungi and animals
• The concept “monerans” is now obsolescent because it does not provide sufficient theoretical clarity.

Artificial Classification:
• Artificial classification is a scheme established merely to serve some particular human purpose e.g. classification of plants as crops, ornamental, and weed

Classification and Division Compared

• The result of a classification will look like the result of a division.
• Classification begins with a individuals or small classes and works
towards a summum genus. It works in the direction opposite to that of division
• Classification begins with a set of apparently related things found in
the world based on experience and builds from there. Hence, it is well-suited to natural objects. But it will work with any kind of object.

Two Overly Ambitious Ideals

the divisions by a few things can never encounter any fallacy.

In logic as well as in any reasoning, if we are using division to explain something, we all aim at making divisions that will have no fallacies. In order to have a perfect flawless division we must divide using one of the following methods.

• Pure division
– begins with the summum genus and
– divides on the basis of a priori considerations
• i.e., it is based on logical possibility, not experience

• Dichotomous division
– divides on the basis of the presence or absence of a particular feature
• Classification can also be dichotomous.
• Striving for these ideals
– works well with mathematical objects,
– does not work well with natural objects
– guarantees a division that meets criteria
– sometimes provides more insight than alternative divisions.
• But “ dichotomous division is often difficult and often impracticable”
• Sometimes, class Rules notification is more practical.

RULES OF DIVISION:

When we are using logical division, we need to follow certain rules. thesde are as follows:

One division must follow only one criteria. It must be either physical or metaphysical.
The division criteria must be mutually exclusive and collectively exhaustive.
All the parts of an entity being explained must be covered by the division.
No extra members must be suggested as parts of the entity explained during the process of division.

FALLACIES OF DIVISION:

When we fail to follow above rules, we end up in committing following fallacies:

Division by cross criteria: When we divide something by using two or more criteria at the same time, we commit this fallacy. e.g. when we divide Indians into "Hindus, Muslims, Christians, Sikh, Rich, poor, Tall, short, Fair, Dark, introverts and extroverts"; we are committing this fallacy as we are using many criteria, both of physical as well as metaphysical divisions at the same time. at the same time.
Too narrow division: when we exclude some of the members from the group or some qualities of the entity being explained, we commit this fallacy. e.g. Quadrilateral into, square and rectangle. Here we exclude many other types of quadrilaterals and so the division becomes too narrow as it leaves out many other members that actually belong to this group.
Too wide division: when we include some members that actually do not belong to the group as we are dividing, our division becomes too wide. e.g. birds into single coloured & multicolored. Here, many other single coloured and multicolored things and beings get indicated as part of the group of bird, so it is a too wide division.

CHAPTER 11. INDUCTION

INDUCTION

a) Simple Enumeration as a form of induction.
b) Analogy – characteristic of a good and bad analogy.

4. c) Use of simple enu,eration and analogy in law – circumstantial evidence.

5. Induction is a type of inference where we go from known to unknown or from less general to more general. Here, from the things that are known, we say something about things that are not known. This is the reason why in induction we always say something more than what we already know of.

6. So, Induction, a form of argument in which the premises give grounds for the conclusion but do not make it certain. Induction is contrasted with deduction, in which true premises imply a definite conclusion, the conclusion of Induction is always probable. The probability rate changes as per strength of evidence.

7. Unlike deductive arguments, inductive reasoning allows for the possibility that the conclusion is false, even if all of the premises are true.

8. Induction is of two types, perfect and imperfect. Perfect induction takes support of deduction in later stages to establish a certain conclusion, while imperfect induction does not do this.

9. The two types of imperfect induction are, Simple enumeration and Analogy.

10. a) Simple Enumeration as a form of induction.

11. Simple enumeration is a method of arriving at a generalization on the basis of uniform uncontradicted observation of something.

12. While using this method, we observe a number of instances that agree in some quality. During our observation, we do not find any contrary instance. So, we arrive at a conclusion that as far as that thing is concerned, there are no contrary instances. Then we get a general proposition as a conclusion.

13. We do not verify our conclusion further or try to analyze the events in order to find any logical relationship in these common similar events.

14. This is the reason why even when our observation is wide, it still stays imperfect. This is because our method is a method of SIMPLE enumeration and not COMPLETE enumeration. In complete enumeration, since we have observed all instances from a group about which we are talking, there is no chance of coming across a contrary instance. But this is not the condition of simple enumeration.

15. In simple enumeration, conclusion can be disproved by observing just one single contrary instance. So, wider the observation, greater is the probability of an inference by simple enumeration.

16. The conclusion by simple enumeration is highly probable when the number of observed instances is really high.

17. But if one is arriving at a conclusion on the basis of very limited observation, the conclusion is less probable and hence, it is termed as hasty generalization or illicit generalization.

18. Many times we find that people arrive at hasty generalizations in determining some vital things in their daily life.

19. b) Analogy –

20. Analogy is a type of imperfect induction where we are comparing two things, persons, groups or classes. while doing so, we observe some similarities and on the basis of these, we infer some further similarity, as we find an additional quality in one of the two compared things, persons, groups or classes.

21. Many times, we observe or compare two things, events, groups, individuals, things, etc. etc, observe some similarities, and then, infer some further similarity. We have no logical reason why we get such a conclusion, but we simply rely on our observation. This is how analogy works.

22. Characteristic of a good and bad analogy.

23. Here, if the observed similarities are relevant to the additional quality, then our conclusion is likely to be true and we may say that Analogy is good Analogy.

24. But if the observed qualities are not relevant to the additional quality, then our conclusion about predicting the additional similarity is not likely to be true, so, we say that such an analogy is Bad Analogy.

25. c) Use of Simple Enumeration and Analogy in law:

26. in circumstantial evidence & getting precedents.

27. In law, we need to use simple enumeration and Analogy to infer things from circumstantial evidence. Of them analogy is more useful in legal matters. Also, while using precedent law, we use analogy to indicate the support of past decided cases in our matter.

28. When we see a person following some pattern of behavior or thinking or actions, while talking of the Modus Operandi of that person, we use simple enumeration as we talk of the generalized pattern of behavior of that person.

29. This is the method followed by criminal investigators quite often.

30. They determine the Modus Operandi of a criminal to find out the criminal and / or to track the criminals. This is a very common practice used by the police in registering the crime record of certain criminals while maintaining their files.

31. While contesting any matter, the lawyers use analogy in arguing about similar matters, or actions done by an individual in similar situations, to infer about the truth of the statement given by any witness.

32. For example, if it is shown that the witness had reacted in a particular way in the past in similar situations, or has reacted in a particular way in similar situation created in court, then, one can infer that he must have reacted exactly in same way when the actual event had happened that the witness was witnessing.

33. This type of inference adds to the weight-age in argument in court.

34. Similarly, when we are arguing any matter, we may come across previously decided matters of same type in the same court, or higher court or another court. We use the citation of these matters as case law or precedent law to lead the judge to the conclusion we want, and the procedure of inductive argument that we use in this type of matter is of analogy. This is why is is said that Analogy is of great use in legal arguments.

BLS LLB

Sunday, 27 November 2016

FAMILY LAW - HINDU LAW (SUCCESSION)

FAMILY LAW - HINDU LAW (MINORITY AND GUARDIANSHIP)

FAMILY LAW - HINDU LAW (MARRIAGE)

FAMILY LAW - HINDU LAW (ADOPTATION AND MAINTENANCE)

Saturday, 19 November 2016

bls llb - logic (sem -1)

DEDUCTION VERSUS INDUCTION

LAWS OF THOUGHT

INDUCTION:

CHAPTER 1: NATURE OF LOGIC

Deduction versus Induction

CHAPTER 4. MODERN CLASSIFICATION OF PROPOSITIONS

CHAPTER 8. DEFINITION

DEFINITION

a) Its purpose- rules and fallacies as per Traditional Definition
b) Modern Definitions-kinds.

Classification of Nominal Definition:

Limitations of definition

About Me

Sunday, 27 November 2016

FAMILY LAW - HINDU LAW (SUCCESSION)

FAMILY LAW - HINDU LAW (MINORITY AND GUARDIANSHIP)

FAMILY LAW - HINDU LAW (MARRIAGE)

FAMILY LAW - HINDU LAW (ADOPTATION AND MAINTENANCE)

Saturday, 19 November 2016

bls llb - logic (sem -1)

DEDUCTION VERSUS INDUCTION

LAWS OF THOUGHT

INDUCTION:

CHAPTER 1: NATURE OF LOGIC

Deduction versus Induction

CHAPTER 4. MODERN CLASSIFICATION OF PROPOSITIONS

CHAPTER 8. DEFINITION

DEFINITION

a) Its purpose- rules and fallacies as per Traditional Definition b) Modern Definitions-kinds.

Classification of Nominal Definition:

Limitations of definition

a) Its purpose- rules and fallacies as per Traditional Definition
b) Modern Definitions-kinds.