Categorical Syllogisms - BLS LLB LOGIC -2

Categorical Syllogisms
We have learned a great deal about categorical propositions. In this lesson, we will learn how to construct valid arguments out of categorical propositions.
1. Standard Argument Form: First, recall that an argument is a set of premises which support some conclusion. In this section, we will be specifically concerned with the kind of argument called a “syllogism.”
Syllogism: An argument consisting of three statements: TWO premises and ONE conclusion.
Furthermore, we’ll specifically be concerned with what is known as a “categorical syllogism.”
Categorical syllogism: A syllogism consisting of three categorical propositions, and containing THREE DISTINCT TERMS, each of which appears in exactly two of the three propositions.
So, what are these 3 terms mentioned? Consider the following syllogism:
1. All mammals are creatures that have hair.
2. All dogs are mammals.
3. Therefore, all dogs are creatures that have hair.
See the three colors (red, blue, and green)? There are THREE different terms in this argument (besides the quantifiers and the copulas). The three different terms are called the “major term”, the “minor term”, and the “middle term.” Notice that the conclusion only contains TWO of the three terms (red and blue), but one of the terms (green) is found only in the premises. Here are some definitions:
Major Term: The predicate term of the conclusion
(above, the blue term, “creatures that have hair”)
Minor Term: The subject term of the conclusion
(above, the red term, “dogs”)
Middle Term: The term that does NOT appear in the conclusion
(above, the green term, “mammals”)
Standard premise order: Finally, note that premise 1 contains the major term, while premise 2 contains the minor term. Premise 1 is therefore called the major premise, while premise 2 is called the minor premise. The standard form demands that the major premise (i.e., the one containing the major term) ALWAYS be listed first.
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2. Mood and Figure: Now that we know the correct FORM of categorical syllogisms, we can learn some tools that will help us to determine when such syllogisms are valid or invalid. All categorical syllogisms have what is called a “mood” and a “figure.”
Mood: The mood of a categorical syllogism is a series of three letters corresponding to the type of proposition the major premise, the minor premise, and the conclusion are (A, E, I, or O).
When determining the mood of a categorical syllogism, you need to figure out which of the four forms of categorical proposition each line of the argument is (A, E, I, or O). For instance, in the argument above about dogs, ALL THREE statements are “A” propositions (of the form “All S are P”), so the mood of that argument would be “AAA”. Here is a syllogism that has a little more diversity:
1. No states with coastlines are states that are landlocked.
2. Some U.S. states are states that are landlocked.
3. Therefore, some U.S. states are not states with coastlines.
Let’s figure out the FORM of the premises and the conclusion:
1. No S are P (E)
2. Some S are P (I)
3. Some S are not P (O)
So, the mood of this proposition is “EIO”. Just to review, note also that the major term is states with coastlines, the minor term is U.S. states, and the middle term is states that are landlocked. Now, let’s learn about “figure.”
Figure: The figure of a categorical syllogism is a number which corresponds to the placement of the two middle terms.
For instance, consider the argument from earlier:
1. All mammals are creatures that have hair.
2. All dogs are mammals.
3. Therefore, all dogs are creatures that have hair.
Notice that the middle term in the major premise is on the LEFT, while the middle term in the minor premise is on the RIGHT. Whenever this happens, we say that the argument has “figure 1.” Altogether, there are four possible figures:
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Figure 1: The middle term is on the left in P1, and on the right in P2.
Figure 2: The middle term is on the right in both premises.
Figure 3: The middle term is on the left in both premises.
Figure 4: The middle term is on the right in P1, and on the left in P2.
Let’s look at the other argument from earlier:
1. No states with coastlines are states that are landlocked.
2. Some U.S. states are states that are landlocked.
3. Therefore, some U.S. states are not states with coastlines.
Note that the middle term appears on the right in BOTH premises. So, this argument has “figure 2.” We can remember the four figures more easily with the following diagrams. If we call the subject of the conclusion “S” (the minor term), and the predicate of the conclusion “P” (the major term), and the middle term “M”, then the four figures look like this:
We can draw lines through the middle terms in each of these four diagrams to create a collar-like shape, like this:
In order to memorize the four kinds of figures, picture this “collar flap” image. From left to right, we see the layout of figures 1, 2, 3, and 4.
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3. Valid Argument Forms: Now that we know about the proper FORMS of categorical syllogisms, and also how to assess what MOOD and FIGURE each argument has, we can use some charts to assess when an argument is valid or invalid.
Unconditionally Valid Forms: There are 15 combinations of mood and figure that are valid from the Boolean standpoint (we call these “unconditionally valid” argument forms). This chart depicts ALL of 15 the unconditionally valid argument forms
Recall this argument from earlier:
1. All mammals are creatures that have hair.
2. All dogs are mammals.
3. Therefore, all dogs are creatures that have hair.
Its mood was “AAA” since all three propositions are “A” propositions (i.e., they are all of the form “All S are P”). Its figure was “figure 1” since the middle term appears on the left and then on the right (picture the leftmost diagonal line of the “collar flap” diagram). Now we can look up “figure 1 – AAA” in the chart above. If it DOES APPEAR on the chart, then the argument is valid from the Boolean standpoint. If it DOES NOT APPEAR on the chart, then it is invalid from the Boolean standpoint. Since “figure 1 – AAA” DOES appear on the chart, the argument is valid!! Let’s try the other one:
1. No states with coastlines are states that are landlocked.
2. Some U.S. states are states that are landlocked.
3. Therefore, some U.S. states are not states with coastlines.
This argument’s mood is “EIO”. Its figure is “figure 2”. Let’s look that up on the chart. Sure enough, under “figure 2 - EIO” appears on the list!! This argument is valid.
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Conditionally Valid Forms: Now, recall that there were some inferences that were NOT valid from the Boolean standpoint which WERE valid from the Aristotelian standpoint—but only IF they were about existing things. The chart above listed argument forms that were valid from BOTH the Boolean AND the Aristotelian standpoint. The chart below lists argument forms that are ONLY valid: (1) from the Aristotelian standpoint, where universal propositions have existential import, and only IF (2) the term listed in the right-hand column actually exists. We call these “conditionally valid” argument forms. There are 9 conditionally valid argument forms for categorical syllogisms in addition to the 15 unconditionally valid argument forms:
Recall that the existential fallacy occurred when going from a universal premise to a particular conclusion. Similarly, all of the above “conditionally valid” argument forms have universal premises (“A” or “E”) and a particular conclusion (“I” or “O”). Consider the following argument:
1. All mammals are creatures that have hair.
2. All dogs are mammals.
3. Therefore, some dogs are creatures that have hair.
This argument is an “AAI” argument with “figure 1”. This argument does NOT appear on the “unconditionally valid” (Boolean) chart, because it goes from universal premises (which do NOT have existential import) to a particular conclusion (which DOES have existential import), and this sort of inference commits the existential fallacy according to Boole.
But, notice that this form (“figure 1 – AAI”) DOES appear on the “conditionally valid” (Aristotelian) chart. So, it IS conditionally valid on the Aristotelian interpretation.
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“Conditionally Valid” arguments are ones that are valid according to the Aristotelian interpretation, but this happens ONLY WHEN the relevant term EXISTS. So, in order to determine whether or not a conditionally valid argument is actually invalid on the Aristotelian interpretation, we need to determine whether the thing listed in the right-hand column of the chart above exists (this is listed on the chart under “Required Condition”).
Let’s return to our AAI-1 argument about dogs that we just looked at. The right-hand column for AAI-1 propositions says “S exists”—so, this means that the subject of the conclusion has to exist in order for the argument to be valid. The subject is dogs, and dogs DO exist—so the argument IS conditionally valid (according to Aristotle).
Note: Remember that “S” refers to the minor term (subject of the conclusion), “P” refers to the major term (predicate of the conclusion), and “M” refers to the middle term (the term that does not appear in the conclusion).
However, IF the subject does NOT exist, this argument would be INVALID (according to Aristotle). For instance, imagine that we replace dogs with unicorns, as follows:
1. All mammals are creatures that have hair.
2. All unicorns are mammals.
3. Therefore, some unicorns are creatures that have hair.
This is still a “figure 1 – AAI” categorical syllogism, but the subject of the conclusion (unicorns) does NOT exist. Therefore, the argument is actually INVALID (because, EVEN THOUGH it appears on the list of conditionally valid argument forms, it does NOT meet the required condition in the right-most column of that chart).
Note: Do homework for section 5.1 at this time.

LOGIC 2 - BLS LLB SEM IV, SYLLOGISM

Syllogistic Terminology, Part I

Abstract:  Today's class introduces the logical terms used to describe  two-premiss arguments composed of categorical statements.  As a stalking horse, we analyze two arguments and test them for validity by means of Venn Diagrams.

The Notes to Syllogistic Terminology are in two parts:

Part I:  Introduction of the Terms of the Syllogism
Part II: Review and Practice with Syllogistic Terminology

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Part I:  Introduction of the Terms of the Syllogism

Let us evaluate the following argument offered by Councilman William Meyers: "University students should have the right to graduate, if they dress decently. When you accept the responsibility of graduation into our society, you should look like a citizen." 

As we analyze Councilman Meyers' argument, we will introduce the nomenclature of syllogistic arguments.

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In order to evaluate this casual argument with charity, we need to be able to translate  the argument reliably into standard-form categorical propositions.

1. The first step is to find the conclusion: "University students should have the right to graduate, if they dress decently."
1. The conclusion has the form: p if q, where p and q stand for different statements. Such conditional statements are often best handled with symbolic logic, but here we want to analyze the argument in terms of categorical form.
2. Let's step back and see what "p if q" means. Suppose p stands for "the bird is black" and q stands for "the bird is a raven." Our statement form, "p if q," in this case, would be, "The bird is black, if it is a raven." In standard categorical form, we could translate, "All ravens are black."
3. Using this model, we can translate Meyers' conclusion as "All decently dressed persons are persons with the right to graduate."
1. The subject of the conclusion is called the minor term of the syllogism: "decently dressed students."
2. The predicate of the conclusion is called the major term of the syllogism: "persons with the right to graduate."
2. The second step is to find the premisses and put the syllogism into standard order and form.
1. Standard form indicates that all the statements are standard-form categorical propositions (A, E, I, or O).
2. Standard order indicates that the statements are put in the sequence of the major premiss first, the minor premiss second, and the conclusion third. Thus, to find the standard order of a syllogism, we need to first find out what the major and minor premiss are.
1. Not surprisingly, the major premiss is the premiss containing the major term. The major premiss is conventionally labeled with the letter 'P."
2. Likewise, the minor premiss is the premiss containing the minor term. The minor term is labeled by convention with the letter "S."
3. Mr. Meyers gives only one premiss: "When you accept the responsibility of graduation, you should look like a citizen."
1. The first part of the premiss, "When you accept the responsibility of graduation..." is meant to represent  the same class as "persons with the right to graduate."
2. Since "persons who have the right to graduate" is the predicate term of Mr. Meyers' conclusion, this premiss is our major premiss.
3. We can now translate the premiss to read in categorical form: "All persons with the right to graduate are persons who look like citizens."
4. Mr. Meyers' argument can now be put as follows:
   
   All persons with the right to graduate are persons who look like citizens.
   
   ....{no minor premiss present yet}....
   ___________________________________
   All decently dressed people are persons with the right to graduate.
1. Since, his premiss contains the major term, it is called the major premiss and is put first in the argument.
2. Mr. Meyers is assuming his listeners will supply the missing premiss. When an argument is elliptical in this manner (i.e., with a missing statement), the argument is called an enthymematic argument.
3. The missing premiss is the minor premiss and thus contains the minor term, "decently dressed students."
4. So the only statement which makes sense as the minor premiss is the statement, "all decently dressed students are persons who look like citizens." 
   
   (By  the principle of charity, we suppose Mr. Meyers is assuming the most reasonable premiss.)
5. The term occurring in both premisses, but not in the conclusion is called the middle term and is symbolized by the letter "M."
6. The categorical syllogism can now be put as follows.

 

	P--MAJOR TERM		M--MIDDLE TERM
All	[persons with the right to graduate]	are	[persons who look like citizens].
	S--MINOR TERM		M--MIDDLE TERM
All	[decently dressed students]	are	[persons who look like citizens].
	S--MINOR TERM		P--MAJOR TERM
All	[decently dressed students]	are	[persons with the right to graduate].

7. The form of the syllogism can be conveniently put as
   
   All P is M.
   All S is M.
   All S is P.
8. Two more terms are worth noting in our analysis of this syllogism. Logicians call the order of the names of the statements the mood of the syllogism. Mr. Meyers' syllogism is an AAA syllogism. Note that the mood does not uniquely describe the form of the syllogism, even though, by convention, the conclusion has the S and P term and the premisses contain the M term.
1.  In other words, if all we knew about Mr. Meyers' syllogism was that it was an AAA syllogism, we can conceive of the following possibilities for the position of the middle term.

Figure 1		Figure 2		Figure 3		Figure 4
**M -- P		**P -- M		**M-- P		**P -- M
**S -- M		**S -- M		**M-- S		**M -- S
**S -- P		**S -- P		**S -- P		**S -- P

2. Figure: The position of the middle term is described by the figure of the syllogism. The figures are named "1," "2," "3," and "4." They are easily remembered because they form the shape of a flying brick.

 

Think of the M's being solid in the center with no other terms between them. Mr. Meyer's syllogism is an AAA-2 syllogism since the M term is in the predicate of both premisses.

10. The mood and figure uniquely describe the form of the syllogism. Any syllogism of this form will have the same degree of validity or invalidity. I.e., if Mr. Meyer's AAA-2 syllogism is invalid, then any other syllogism of the same form is invalid.

3. The third step is to test the syllogism by means of Venn Diagrams or the rules for validity. This might be a good time to review the symbols used to diagram the standard-form propositions.
1. The idea is to look at the logical geography of the premisses. If the argument is valid, the premisses should mark out the conclusion beyond doubt, without further markings.
2. . The major premiss, "All P is M" would be diagrammed as the picture below. The diagram has been slanted so that it can be superimposed on the diagram for all three classes later.

3.  
4. The minor premiss, "All S is M" would be diagrammed as the picture below. It also has been slanted so it can be superimposed in the diagram above.

5.  
6.  Putting both diagrams together on the representation of the S, P, and M classes would give a picture like the one below. Can we "read off" the conclusion without further markings? Is there any possibility of an S not being a P? Do diagramming the premisses without additional marking produce a diagram of the conclusion?
7. Since there is the possibility of an "S" being in the area marked, and it is outside of the P-area, the syllogism is invalid. Mr. William Meyer's syllogism is invalid.  He might have a false premiss as well.

Syllogistic Terminology, Part II Continued from Syllogistic Terminology, Part I
Abstract: Today's class introduces the logical terms used to describe  two premiss arguments composed of categorical statements.  As a stalking horse, we the second argument and test it for validity by means of Venn Diagrams.

Part II: Review and Practice with Syllogistic Terminology

1. Let us review the foregoing terms and the procedures by evaluating another argument in summary fashion.
1. While Mary Chaney was changing a flat tire, the car rolled forward off the jack bending the axle. The estimate to fix the car was $1,580. Mary's car was insured, so she filed a claim. The insurance adjuster said the claim could not be paid because the vehicle only had three wheels at the time of the accident and so was not an "auto." An "auto" is defined in the insurance policy as "a land motor vehicle with at least four wheels designed for use on public roads." Is the claim adjuster's argument valid?
2. We will follows the rules of thumb described above to analyze the argument.
1. First, find the conclusion. The adjuster concludes, "Miss Chaney's vehicle is not an insured auto." This is a singular statement and is, in effect, an E statement because it is universal negative with the subject and predicate undistributed. Usually singular statements are left as such rather than awkwardly translating into something like the following:

"No things which are Miss Chaney's vehicle are insured autos." We will follow the former practice here.

2. Second, put the syllogism into standard order and form.
1. The reasons given for the conclusion are the statements taken from the insurance adjuster's claims that an automobile  must have at least four wheels and Miss Chaney's didn't.
2. The first premiss, the major premiss, has to have the predicate term of the conclusion. It would be "All insured autos are land vehicles with at least four wheels.
3. The second premiss, the minor premiss, has the subject term of the conclusion. It would be "Miss Chaney's vehicle is not a land vehicle with at least four wheels.  In sum, we have the following syllogism:

	P--MAJOR TERM		M--MIDDLE TERM
All	[insured autos]	are	[land vehicles with at least four wheels.]
	S--MINOR TERM		M--MIDDLE TERM
	[Miss Chaney's vehicle]	is not	[a land vehicle with at least four wheels.]
	S--MINOR TERM		P--MAJOR TERM
	[Miss Chaney's vehicle]	is not	[an insured auto.]

4. A moment's reflection gives us the following summary of the major parts of the argument and the common terms used to describe our two-premiss argument. (When analyzing syllogisms, one usually identifies the terms in the order sequenced here.)
   
   Categorical syllogism: The argument contains two premisses and a conclusion, and the argument contains three terms, each of which is used twice in the argument.
   
   Conclusion: "Miss Chaney's vehicle is not an insured auto.
   
   Major term: "insured autos.
   
   Minor term: "Miss Chaney's vehicle.
   
   Middle term: "land vehicles with at least four wheels.
   
   Major premiss: All insured autos are land vehicles with at least four wheels.
   
   Minor premiss: Miss Chaney's vehicle is not a land vehicle with at least four wheels.
   
   Mood: AEE 
   Figure: 2 
   Form: AEE-2

3. Test the syllogism for validity. The Venn Diagram representation of the insurance adjuster's argument could be presented in the following manner. The form of the syllogism is

All P is M.
No S is M.
No S is P.

The major premiss, "All P is M," by itself can be diagrammed, as before separately.

The minor premiss, "No S is M," by itself can be diagrammed, separate from the whole, as well.

Putting both diagrams together, if the syllogism is valid, we ought to be able to read off the conclusion, No S is P." Especially note that we do not diagram the conclusion.

4. Since the lens area in common between the S and P classes is completely shaded, we can read off the conclusion from the completed diagram. The insurance adjuster gave a valid argument. It is now up to Miss Chaney to question its soundness if she wishes to pursue her claim. Is there a false premiss in the argument? If so, even though the argument is valid, the argument does not prove the conclusion true.

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Syllogistic Fallacies

I. Venn diagrams and logical analogies are two of the three most common methods to test syllogisms. A third method is based on derived rules of validity.
	A. We will look at some arguments that might initially seem to be valid, but are not so.
	B. In each case, an informal explanation of its invalidity is described.
	C. These six reasons are, in effect, rules of the syllogism. Here, again, we follow Copi's analysis. There are other sets of rules that equally apply to the analysis of the syllogism, and you might want to inquire into some of these other methods.
	D. Corresponding to each rule of the syllogism is a fallacy (or fallacies) which is applied to all arguments that do not follow that rule.
	E. Do not memorize the rules (or the rule numbers), but do learn the names of the fallacies. The names of the fallacies describe what it is that is mistaken about the argument.
	F.  One way to think about the way a syllogism works is to conceptualize the general idea that two things related to the same thing might be related to each other.  The following mnemonic model might be helpful. Since S is related to M, and P is related to M, then S ought to be related to P.

Syllogistic Fallacies: Four Term Fallacy

Abstract:  The Four Term Fallacy or Fallacy of Equivocation is explained.  Strictly speaking, an argument which commits this fallacy cannot be a syllogism by definition because the argument contains more than three terms.

I.. Consider the following argument:
"A poor lesson is better than a good lesson because a poor lesson is better than nothing, and nothing is better than a good lesson."
		A. Note how in the following argument we have an uncomfortable feeling that the argument seems good with true premisses, but the conclusion is obviously false. Often, we smile at arguments like these because we know something is drastically wrong, but it is not initially intuitively obvious what it is. Knowing that a valid argument cannot have true premisses and a false conclusion, and yet the argument appears to be perfectly valid, is a tip-off for the presence of the fallacy of equivocation.
		Nothing is better than a good lesson. A poor lesson is better than nothing. A poor lesson is better than a good lesson.
		B. Obviously, there is something wrong with this syllogism; this is evident from its humorous appearance. When we sketch a diagram, without attending to the meaning of the classes, it is clear that the diagram would appear valid. How is this possible?
		C. Although the argument does not translate very well into standard form categorical propositions, if we attempt to do so, we can see that the classes do not match. The word "nothing" is being used in two different senses. One attempt at translation yields:
		No [lessons] are [things better than good lessons.] All [poor lessons] are [things better than no lessons at all.] All [poor lessons] are [things better than good lessons.]
		Notice that we have more than three terms--our middle term does not match. Hence, we cannot get a valid diagram:
		D. Fallacy of Four Terms occurs when a categorical syllogism contains more than three terms. More commonly, the fallacy of four terms is called from the point of view of informal logic, the fallacy of equivocation.
		1. Rule: A valid standard from categorical syllogism must contain exactly three terms, each of which is used in the same sense throughout the argument.
		2. With more than three terms, no connection can be established from which a conclusion can be drawn. Informally, the idea of the syllogism is that two things related to the same thing ought to be related to each other.
		3. If, for example, the M term is being used in two different senses, then the M term denotes two different classes and so cannot link together the S and P terms.  Note:  Not just the middle term is subject to equivocation, as in this example; any of the terms in a syllogism might have be used in two different senses.
		4.  Before testing any syllogism, be sure to read and understand what is being adduced; otherwise, the four term fallacy could possibly be overlooked.

Fallacy of the Undistributed Middle Term

Abstract:  The Fallacy of the Undistributed Middle Term is discussed and illustrated.

I. We continue our study of the syllogistic fallacies with a second common fallacy.
	A. Note, how in the following argument, about the only persons likely to be sympathetic are those who dislike Senator Jones.  (Notice that singular statements are treated as universal affirmative propositions.)
		All [Communists] are [believers in heavy taxes]. [Senator Jones] is a [believer in heavy taxes]. [Senator Jones] is a [Communist].
		The Venn Diagram would be sketched like this:
	B. It is fairly evident that for the conclusion to follow logically, one would have to presuppose instead that "All believers in heavy taxes are Communists," not "All Communists are believers in heavy taxes."  Notice that the former statement  would distribute the term "believers in heavy taxes." But this distribution is not what is asserted in the original argument.   In the original argument, the middle term is undistributed in both premisses.
	C. Fallacy of the Undistributed Middle Term occurs when the middle term is undistributed in both premisses.
		1. Rule: In a valid standard form categorical syllogism, the middle term must be distributed in at least one premiss.
		2. Reason: for the two terms of the conclusion to be connected through the third, as in the mechanism sketched below, at least one of them must be related to the whole of the class designated by the middle term. Otherwise, the connection might be with different parts of the middle term, as illustrated below, and no connection can be made.
		2.      Note:  Remember for the Fallacy of the Undistributed Middle Term to occur, the middle term must be undistributed in both premisses, not just one premises

Syllogistic Fallacies: Fallacy of the Illicit Minor Term and Fallacy of the Illicit Major Term

I.  We continue our discussion of the syllogistic fallacies with the third and fourth fallacies on our list.  Consider the following argument.
		All [subversives]D are [radicals]U. No [Republicans]D are [subversives]D. No [Republicans]D are [radicals]D.
	A. We can see from the Venn Diagrams corresponding to this argument that this argument is fallacious.
	B. When we plug in the distribution statuses for the classes in each argument from the chart learned when we studied categorical propositions, we notice something interesting.
	C. Notice how in the argument, the major term "P-radicals" is undistributed in the major premiss, but is distributed in the conclusion.
		1. Since a term is said to be "undistributed" when not every member of the class is being referred to, and a term is said to be "distributed" when each and every member of the class is being referred to, we are reasoning from information about part of a class to information about the whole of the class.
		2. When reasoning from a few instances to a conclusion involving all instances, we are, metaphorically speaking, committing the fallacy of converse accident.
		That is, in the premiss, we are referring to "some radicals" and then reasoning to "all radicals" in the conclusion.
		Another way of looking at this fallacy is to compare the process with subalternation on the Square of Opposition.
		We are moving from a subaltern being true (some radicals) to a superaltern being undetermined (all radicals) in truth value .
		3. Since this fallacious reasoning involves the major term in the syllogism, the fallacy committed there is termed the Illicit Process of the Major Term or Illicit Major, for short.
	D. The Fallacy of the Illicit Major occurs when the major term is undistributed in the premiss but is distributed in the conclusion (but not vice versa!).
	E. The second argument is as follows.
		All [good citizens]D are [nationalists]U All [good citizens]D are [progressives]U All [progressives]D are [nationalists]U
		1. We can see from the Venn Diagram for this argument that it is fallacious.
		2. When we plug in the distribution statuses for the classes in each argument from the chart learned when we studied categorical propositions, we notice something interesting.
	F. Notice how in the argument, the minor term "S-progressives" is undistributed in the minor premiss, but is distributed in the conclusion.
		1. As in the first argument above, we are moving from referring to some of the progressives in the premiss to referring to all of the progressives in the conclusion
		3. Since this fallacious reasoning involves the minor term in the syllogism, the fallacy committed there is termed the Illicit Process of the Minor Term or Illicit Minor, for short.
	G. The Fallacy of the Illicit Minor occurs when the minor term is undistributed in the premiss but is distributed in the conclusion (but not vice versa).
		Rule: In a valid standard form categorical syllogism no term can be distributed in the conclusion unless it is also distributed in the premisses...
		Reason: ...otherwise the conclusion would assert more than what is contained in the premises.

Syllogistic Fallacies: Exclusive Premisses

Abstract:  The Fallacy of Two Negative Premisses or Exclusive Premisses is illustrated and explained.

I. We continue our study of fallacies with a fifth fallacy. Consider the following argument.

"No internal combustion engines are nonpolluting power plants, and no nonpolluting power plants are safe devices. Therefore, no internal combustion engines are safe devices."

A. First, let's put the argument in standard form:

M

P

No [nonpolluting power plants] are [safe devices].

S

M

No [internal combustion engines] are [nonpolluting power plants].

S

P

No [internal combustion engines] are [safe devices].

1. The Venn diagram shows this argument to be invalid.

2. Note that both premisses are negative. As most people are intuitively aware, about what a thing is not, do not carry much information about what that thing is. If I say I am thinking of something that is not a tree, you would not know very much about what I am thinking.

3. By referring to the mnemonic of the mechanism of the syllogism sketched here, we can surmise that the basis of the syllogism is captured by noting that two things related to the same thing should be somehow related to each other, if at least one of them is totally related.

4. However, when both premisses are negative, our mnemonic shows the classes are not related in some way to each other, and this information is of no use to see how the terms in the conclusion are related. This state of affairs can be illustrated as follows.

B. This Rule of Quality states that no standard form syllogism with two negative premisses is valid.

1. The fallacy is called either the Fallacy of Exclusive Premisses or the Fallacy of Two Negative Premisses.

2. Reason: When a syllogism has exclusive premisses, all that is being asserted is that S is wholly or partially excluded from part or all of the M class, likewise for the P class; but since this statement is true for every possible syllogism, the premisses entail no information.

3. Note that you can detect the fallacy of Exclusive Premisses merely by inspecting the mood of the syllogism. Test yourself on the following examples.

EOI-2

AAA-2

		OOI-3

OEO-1

		IEO2

EEE-2

Syllogistic Fallacies: Affirmative Conclusions from a Negative Premiss Abstract:  The Fallacy of the Affirmative Conclusion From a Negative Premiss is explained and illustrated.   I. The following argument illustrates another one of our syllogistic fallacies. "Some laborites are democrats, because All blue collar workers are laborites, and some blue collar workers are not democrats. A. When we set up this argument in a standard form and order syllogism, we obtain ... Some [blue-collar workers] are not [democrats]. All [blue-collar workers] are [laborites]. Some [laborites] are [democrats]. 1. The Venn diagram shows this argument to be invalid. 2. The mnemonic mechanism of the syllogism suggests why this argument is invalid. We can't make the affirmative link between S and P. B. Intuitively, most of us think that if a conclusion is negative, then one premiss must be negative as well, and if the conclusion is affirmative, neither premiss could be negative in a valid argument. This intuition is correct. II. The second quality rule is if either of the premisses of a valid standard form syllogism is negative, then the conclusion must also be negative. A. Reason: If an affirmative conclusion is entailed, then both premisses must be statements of class inclusion. Since class inclusion is only obtained by affirmative statements, if the conclusion has one class is partly or wholly contained in the other, then the premisses must assert that the middle class is contained by the minor class and contained in the major class.                              C.            The Fallacy of Drawing an Affirmative Conclusion from a Negative Premiss is the resultant fallacy, if the rule does not hold. C. Note: the syllogism does not have to be in standard form for use to be able to spot this fallacy. All that we need to see is the mood. Test your understanding by trying the following problems. AEA-2 EEA-4 IOI-3 EAE-1 IAO-1 OIO-3

Syllogistic Fallacies: Existential Fallacy

Abstract:  The Existential Fallacy is illustrated and explained.

I. The final fallacy of the syllogistic fallacies is illustrated in the following argument:
	"Since no rigid levers are flexible things, Some rigid levers are not elastic bars because all elastic bars are flexible things."
	A. When set up in standard form and order the syllogism looks like this:
		All [elastic bars] are [flexible things]. No [rigid levers] are [flexible things]. Some [rigid levers] are not [elastic bars].
		1. The Venn Diagram for this argument raises some interesting issues. How would you evaluate the following argument? Is it valid?	Top of Form Bottom of Form

		2. According to our interpretation of the symbols used in Venn diagrams, we would have to have an "X" in the SMP area, but there is no "X" there. The blank space indicates no information is known about that area.
		3. If we had independent information concerning the existence of rigid levers, we would know that at least one rigid lever existed, and this one would have to be in the SMP area of the diagram.
		4. However, if we are evaluating the argument as given and we do not assume anything else, we cannot validly get to the conclusion from these premisses.
	B. On the Boolean interpretation of categorical syllogisms, we cannot assume the existence of individuals mentioned in universal statements. If our language, if we want to assert that individuals exist, we must say so by adding a particular statement.
		1. On this convention, the word "some" when used in a particular statement is taken to imply at least one of the individuals exists.
		2. In sum, then, universal statements do not imply that the classes exist, whereas particular statements do imply that the classes exist.
		3. We take this interpretation in our logic here so that arguments can be presented concerning subjects about ideal or nonexistent objects such as frictionless planes, ideal gasses, and black bodies.
II. Rule (Boolean Interpretation): No valid standard form categorical syllogism with a particular premiss can have both premisses universal.
	A. Reason: If the rule were not followed, then we would go from premisses which have no existential import to a conclusion that does have existential import. The problem of existential import can be illustrated by Venn Diagrams.
	B. The Existential Fallacy occurs whenever a standard form syllogism has two universal premisses and a particular conclusion.
	C. See if you can determine merely by inspection if the following syllogisms are valid or invalid.
		AAI-3	Top of Form Bottom of Form

EEO-4

Top of Form Bottom of Form

		EAO-1	Top of Form Bottom of Form

EAI-3

Top of Form Bottom of Form

		AE0-1	Top of Form Bottom of Form

AEA-2

Top of Form Bottom of Form

		E00-2	Top of Form Bottom of Form

AOI-3

Top of Form Bottom of Form

		OEO-4	Top of Form Bottom of Form

OOO-1

Top of Form Bottom of Form

	D. Note: If your logic presupposes existence, you cannot simply discard this rule, since the remaining rules would not be complete.

Quantity, Quality, and Distribution of Standard Form Categorical Propositions

Abstract:  The most important properties of standard form categorical propositions are explained and illustrated. I. Categorical propositions and classes.
	A. The long range goal is to give a theory of deduction, i.e., to explain the relationship between the premisses and conclusion of a valid argument and provide techniques for the appraisal of deductive arguments. Hence, we will be distinguishing between valid and invalid arguments.
		1. A deductive argument is defined as one whose premisses are claimed to provide conclusive evidence for the truth of its conclusion.
		2. A valid deductive argument is one in which it is impossible for the premisses to be true without the conclusion being true also.
	B. Our study of deduction, for the present, will be about arguments stated in categorical propositions, e.g.,
		No honest people are persons who embroider the truth. Some politicians are persons who embroider the truth. Some politicians are not honest people.
		1. A categorical proposition is defined as any proposition that can be interpreted as asserting a relation of inclusion or exclusion, complete or partial, between two classes.
		2. A class is defined as a collection of all objects which have some specified characteristic in common. This is no more complicated than observing that the class of "lightbulbs" all have the common characteristic of "being a lightbulb."
			Thus, we can have four class relations in the various kinds of categorical propositions:
			Utilizing the classes, "people" and "good beings":
			a. complete inclusion>>>"All people are good beings."
			b. complete exclusion>>>"No people are good beings."
			c. partial inclusion>>>"Some people are good beings."
			d. partial exclusion>>>"Some people are not good beings."
			We can also describe these four kind of statements respectively as
			a. universal affirmative
			b. universal negative
			c. particular affirmative
			d. particular negative
		3. Often, it is convenient to look at the general form of the statements given above. To these forms, special names are given: A, E, I, and O.
			A: All S is P.
			E: No S is P.
			I: Some S is P.
			O: Some S is not P.
			...where S and P stand for the logical subject and the logical predicate of the statement respectively.
		4. A mnemonic device for the four kinds of statements is to remember Affirmo and Nego.
		5. Note, here, the logical subject differs from the grammatical subject of a statement.
			For example, in the statement, "All (unfledged floithoisters) are (things apt to become unflaggled)," the logical subject is everything between the "all" and the "are," and the logical predicate is everything after the "are."
		6. Also note that the word "some" is taken to mean "at least one." This meaning differs somewhat from ordinary language.
		7. A model statement, then, can be represented as
			Quantifier [subject term] copula [predicate term].
II. Analysis of the Categorical Proposition: Quality, Quantity, and Distribution
	A. The quantity of a categorical proposition is determined by whether or not it refers to all members of its subject class (i.e., universal or particular). The question "How many?" is asking for quantity.
	B. The quality of a categorical proposition is determined by whether the asserted class relation is one of exclusion or inclusion (i.e., affirmative or negative).
	C. Indicators of "how much" are called quantity indicators (quantifiers) and specifically are "all," "no," and "some."
	D. Indicators of affirmative and negative are quality indicators (qualifiers) and specifically are "are," "are not," "is," "is not," and "no,"
		Note that "no" is both a quantifier and a qualifier.
	E. Memorize the following table:
Name	Form				Quantity	Quality	Distribution
							Subject	Predicate
A	All S is P				universal	affirmative	distributed	Undistributed
E	No S is P				universal	negative	distributed	Distributed
I	Some S is P				particular	affirmative	undistributed	Undistributed
O	Some S is not P				particular	negative	undistributed	Distributed
	F. Distribution of a term.
		1. A distributed term is a term of a categorical proposition that is used with reference to every member of a class. If the term is not being used to refer to each and every member of the class, it is said to be undistributed.
		2. Consider the following propositions:
			A: All birds are winged creatures.
			E: No birds are wingless creatures.
			I: Some birds are black things.
			O: Some birds are not black things.
Read the above statements and see how the following chart represents distribution.
	Subject					Predicate
	A: refers to all birds					does not refer to every member, e.g., bats, flying fish.
	E: refers to all birds by indicating that they are not part of the predicate class					refers to all wingless creatures by indicating that they are not part of the subject class
	I: refers only to some birds					refers only to some black things, viz., those which are birds
	O: refers only to some birds, not all of them					refers to all members of the class! Viz., not one of them is in the class referred to by "some birds"
		3. For the predicate of the O proposition, consider the following analogy. If we know that there is a book not in a bookcase, then we know something about each and every shelf in that bookcase-- the book is not on that shelf.
		4. There are three ways to remember the distribution status of subject and predicate for standard form categorical propositions:
			a. Memorize it.
			b. Figure it out from an example (as was done above).
			c. Remember the following rule:
				The quantity of a standard form categorical proposition determines the distribution of the subject (such that if the quantity is universal, the subject is distributed and if the quantity is particular, the subject is undistributed), and ...
				the quality of a standard form categorical proposition determines the distribution status of the predicate (such that if the quality is affirmative, the predicate is undistributed, and if the quality is negative, the predicate is distributed).