sbccPhilosophy-111 Critical Thinking And Writing:

Categorical Syllogisms

 

Video Intro

 

The Formal Structure of Syllogistic Reasoning:

In its most general meaning a 'syllogism' is a formal deductive argument consisting of just two premises, one called the 'major' premise, and the other called the 'minor' premise, claiming that a 'conclusion' follows if the premises are true. Since the 'form' of the four types of categorical propositions, A E I & O, express either category inclusion or exclusion of members in one class with members of another class, it is possible to harness this power of 'distribution' to formulate a conclusion from only two premises with absolutely certain validity.

Thus, 'validity' is understood as a purely 'formal' deduction of a new truth claim from the distribution of terms contained in two other truth claims. Validity of deductive categorical arguments does not rely on the factual truth of the premises from which the conclusion is drawn. Validity of deductive categorical arguments relies solely on the 'form' of the argument and not it's content. It is absolutely essential to understand this in order to proceed in mastering the power of syllogistic reasoning.


1. Precise Definition of a Standard Form Categorical Syllogism:

A Standard Form Categorical Syllogism is any formal argument consisting of exactly three categorical propositions (two premises and one conclusion) containing exactly three categorical terms, each of which is used exactly twice in exactly two categorical propositions.


2. Major, Minor and Middle Categorical Terms in a Categorical Syllogism:

To identify the Major Term, look at the conclusion and find the predicate term. To find the Minor Term, look at the conclusion and find the subject term. The remaining term of the three categorical terms is the Middle Term. (NOTE: The Middle term never appears in the conclusion)

Example:
All light bulbs are human.
All Bostonians are light bulbs.
Therefore, All Bostonians are human.
(Major term = 'human', Minor term = 'Bostonians', Middle term = 'light bulbs')


3. Major Premise & Minor Premise:

In any Standard Form Categorical Syllogism, the premise proposition that expresses the 'distribution' of terms' between the Major Term and the Middle Term is called the Major Premise and it is always stated as the first premise.

In any Standard Form Categorical Syllogism, the premise proposition that expresses the 'distribution of terms' between the Minor Term and the Middle Term is called the Minor Premise, and it is always stated as the second premise.

Using the example here are the major and minor premises with the conclusion all Type A standard form categorical propositions from the Aristotelian Square of Opposition.

Example:
All light bulbs are human. (MAJOR PREMISE = Type A)
All Bostonians are light bulbs. (MINOR PREMISE = Type A)
Therefore, All Bostonians are human. (CONCLUSION = Type A)
(NOTE THE POSITION OF THE MIDDLE TERM, SUBJECT OF MAJOR PREMISE AND PREDICATE OF MINOR PREMISE )


4. Mood & Figure of Standard Form Categorical Propositions:

Obviously, using just the three categorical terms in the example ( 'humans', 'Bostonians', & 'light bulbs') it is possible to formulate literally hundreds of Standard Form Categorical Syllogisms using all four types of standard form categorical propositions AND MOVING THE MIDDLE TERM in four different permutations in the Major and Minor premises.



Consider this permutation of the three categorical terms:
No humans are light bulbs. (MAJOR PREMISE = TYPE E)
Some Bostonians are light bulbs. (MINOR PREMISE = TYPE I )
Therefore, some Bostonians are not human. (CONCLUSION = TYPE O)
(NOTE: THE MIDDLE TERM 'LIGHT BULBS' OCCUPIES A DIFFERENT POSITION IN THIS SYLLOGISM, PREDICATE OF BOTH MAJOR AND MINOR PREMISE)

Clever Medieval monks formulated a simple way to label the many permutations and combinations of a syllogisms possible configuration by propositional TYPE and POSITION OF THE MIDDLE TERM as subject or predicate in each of the premise propositions. Thus, of the hundreds of possible configurations by type and middle term position each can by labeled by its MOOD ( the propositional TYPES of the premises and conclusion propositions) and its FIGURE ( the FOUR possible positions the middle term might occupy in the premises) Consider that any MIDDLE TERM may be in one of four positions in the premises of any Standard Form Categorical Syllogism as the following demonstrates:



Figure 1: Middle term is SUBJECT of the MAJOR PREMISE and PREDICATE of the MINOR PREMISE.

Figure 2: Middle term is PREDICATE of the MAJOR PREMISE and PREDICATE of the MINOR PREMISE.

Figure 3: Middle term is SUBJECT of the MAJOR PREMISE and SUBJECT of the MINOR PREMISE.

Figure 4: Middle term is PREDICATE of the MAJOR PREMISE and SUBJECT of the MINOR PREMISE.

With 64 different permutations of syllogistic MOODS and 4 different permutations of FIGURES for each MOOD, doing the math results in exactly 256 combinations of MOOD AND FIGURE that embrace and exhaust all possible forms of Standard Form Categorical Syllogisms. As will be demonstrated, ALL BUT 15 are INVALID.


5. The Relation of Logical Form and Validity:

Since we know that an argument is VALID, if and only if, its conclusion follows from its premises by logical NECESSITY, then if any of the 256 permutations of Standard Form Categorical Syllogisms are demonstrably and verifiably VALID then ANY ARGUMENT OF THAT PARTICULAR FORM will also be valid, regardless of the truth or falsity of the premises. When a syllogism is VALID it is not logically possible for its conclusion to be factually false while its premises are factually true. In any Standard Form Categorical Syllogism, the factual truth of the premises entails the factual truth of the conclusion by logical necessity.

By use of Venn Diagrams below, it will be proven that any syllogism of the Mood & Figure AAA-1 is valid regardless of term content, while all syllogisms of the Mood & Figure AAA-3 are invalid regardless of term content.

Clever students of syllogistic reasoning quickly realize how to use this to refute arguments in debate. They can demonstrate an opponent's argument to be invalid simply by formulating another argument with precisely the same MOOD and FIGURE of their opponent's argument yet contains an obviously FALSE conclusion. This method of 'refutation by logical analogy' is a powerful tool in debate with stunning results when used with great skill and art.

Here is a classic example of a debatable argument that might seem valid, but it commits a FORMAL FALLACY:

All professors of Philosophy are educated people. (Type A)
All professors of Philosophy are respected members of society. (Type A)
Therefore, all respected members of society are educated people. (Type A)
Figure = 3

Note that this particular form AAA-3 has the MIDDLE TERM as SUBJECT in both premises, Major and Minor. Note also that the MINOR TERM is NOT DISTRIBUTED in EITHER major or minor premise. Yet the conclusion asserts that the MINOR TERM is distributed. Since the conclusion claims more distribution than the premises assert, then the argument is invalid. And if this argument is invalid, then ANY ARGUMENT OF THIS FORM, AAA-3 will also be INVALID. Here's one that refutes the original argument by logical analogy:

All drag queens are impersonators of the opposite sex. (Type A)
All drag queens are mammals. (Type A)
Therefore, all mammals are impersonators of the opposite sex. (Type A)
Figure = 3

This obviously absurd counter example to the original AAA-3 syllogism about professors of Philosophy sufficiently proves that any AAA-3 is invalid on the grounds that: the MINOR TERM is NOT DISTRIBUTED in EITHER major or minor premise. Yet the conclusion asserts that the MINOR TERM is distributed. Since the conclusion claims more distribution than the premises logically entail, then this argument is invalid. And since the FORM of this argument is invalid, then any argument with the same FORM, including the one about professors of Philosophy above, will also be invalid. It should also be noted that since no rational zoologist would advance the second absurd argument, it is a neat 'reductio ad absurdum' refutation of the first argument.


6. Testing, Proving and Verifying Validity with Venn Diagrams:

There is a purely graphical way to test the validity of any Standard Form Categorical Syllogism that was not available to the clever monks of the Middle ages. In the mid-nineteenth century, John Venn (1834-1923), a mathematician at Cambridge University, invented a graphical scheme for visualizing logical 'distribution' of categorical terms in categorical propositions. While Venn made no significant contribution to mathematics, his single invention of the Venn Diagram has made it much easier for logic and math students to understand term 'distribution'.

CAUTION: There are several other versions of Venn Diagrams, especially in mathematics classes, that look quite different from the original. In philosophy courses of Logic and Critical Thinking, the original is the standard to be followed as it will be here.

A Venn Diagram is simply a field within which overlapping circular areas represent categorical terms that either share members between categories or exclude members between categories. Hence, the diagram graphically illustrates DISTRIBUTION among the categories members. Areas of the overlapping circles where no members exist are SHADED. Areas where there is at least one member have a single 'X" in that area.

Here are examples of single categorical propositions with two overlapping circles graphing the distribution (or lack of distribution) between the categorical terms expressed in the propositions.

This is a Venn Diagram of any Type A proposition: All S is P. The shaded blue area graphically indicates that; If something is a member of the category 'S' (whatever 'S' might be) then that something must also be a member of the category 'P' (whatever 'P' might be). Another way to say the logically equivalent thing (by via negative) is; "It is not the case that something is a member of the category 'S' without at the same time being a member of the category 'P'." Or, "There are no members of the category 'S' outside the category 'P'." Thus, S is distributed and P is not.

This is a Venn Diagram of any Type I proposition: Some S is P. The 'X' in the overlapping area graphs the lack of distribution for both S and P. The logical import of this graph is; There exists at least one member of the category 'S' that is also a member of the category 'P'. Or, (again using via negativa); It is not the case that there exists at least one member of the category 'S' that is outside the category 'P'. It also states the converse of the original Some S is P. There exists at least one member of the category 'P' that is a member of the category 'S'.

Using the same Venn technique, here is an example of any Type E categorical proposition No S is P






Lastly, using the same Venn technique, here is an example of any Type O categorical proposition Some S is not P.





Now that the Venn technique has been mastered to graph single categorical proposition, it is simple to apply this technique to the three categorical propositions found in any Standard Form Categorical Syllogism in four easy steps to test the validity of those syllogisms.

For example, how it could be applied, step by step, to an evaluation of a syllogism of the EIO-3 mood and figure:
No M are P.
Some M are S.
Therefore, Some S are not P.

Step 1. First draw three overlapping circles and label them to represent the major, minor, and middle terms of the syllogism.

(Note: Right mouse-click on this image and download it for use in your term paper for this course)





Step 2. Using the Venn technique above, graph each premise proposition, starting with the UNIVERSAL premise.

(Note: In any syllogism that has a universal and a particular in the premises, always graph the UNIVERSAL FIRST. In syllogisms with TWO UNIVERSAL PREMISES, it does not matter which one you graph first.)


Since the major premise is a universal proposition, we may begin with it. The diagram for "No M are P" must shade in the entire area in which the M and P circles overlap.

(Note: Ignore the S circle by shading on both sides of it.)

Step 3: Using the Venn technique above, graph the minor premise. The diagram for "Some M are S" puts an inside the area where the M and S circles overlap.

(Note that this area (the portion also inside the P circle) has already been shaded, so the must be placed in the only remaining portion.)



Step 4: Stop drawing and merely look at our result. Ignoring the M circle entirely, we need only ask whether the drawing of the conclusion "Some S are not P" has already been drawn.

(Note that the graph will be like the one at left, in which there is an in the area inside the S circle but outside the P circle. Does that already appear in the previous graph above in Step 3? Yes, if the premises have been drawn correctly, then the conclusion is already drawn and validity has been proven for this syllogism and for any standard form categorical syllogism of the mood and figure EIO-3.)


7. Venn Diagram Examples

Here are the diagrams for several other syllogistic forms. Some are VALID forms, some are INVALID. In each case, both of the premises have already been drawn in the appropriate way, so if the drawing of the conclusion is already drawn, the syllogism must be valid, and if it is not, the syllogism must be invalid.

AAA-1 (valid)

All M are P.
All S are M.
Therefore, All S are P.






AAA-3 (invalid)

All M are P.
All M are S.
Therefore, All S are P.






OAO-3 (valid)

Some M are not P.
All M are S.
Therefore, Some S are not P.






EOO-2 (invalid)

No P are M.
Some S are not M.
Therefore, Some S are not P.






IOO-1 (invalid)

Some M are P.
Some S are not M.
Therefore, Some S are not P.






Lists of all 15 Valid Syllogistic Forms with accompanying Venn diagrams:

Pages 85 through 88 of my textbook, Reason, Argue, Refute give all 15 Venn Diagrams for any VALID Standard Form Categorical Syllogism. Each is arranged by Figure and by anagrammed Latin name. Learn all 15 of these valid syllogistic forms. If you do, then you will ALWAYS be able to FORMULATE a VALID argument on any debatable topic.

Here's a useful link to practice your skill in evaluating various formulations of Categorical Syllogisms. It's on poweroflogic.com site and it will help you test and sharpen your categorical syllogistic skills for our exams. Syllogism Evaluator.