7. CATEGORICAL LOGIC

Outline

James Fieser, UT Martin

updated 11/04/2025

The system of categorical logic was developed by Aristotle in his Prior Analytics (4th century BCE) and, from about the twelfth-century until the late nineteenth-century, remained the dominant framework for logical theory and teaching in Western philosophy. It is called “categorical” since it is based on how categories of things relate to each other. The system has two components: categorical propositions (an analysis of statements), and categorical syllogisms (an analysis of arguments). An example of a categorical proposition is the following:

All dogs are mammals

This involves an overlapping relation between the category “dogs” and the category “mammals”. A categorical syllogism is an argument built from three categorical propositions, where “syllogism” just means “argument”. An example is the following:

All dogs are mammals

All Dalmatians are dogs

Therefore, all Dalmatians are mammals

We begin by examining the structure of categorical propositions and will consider categorical arguments after that.

A. CATEGORICAL PROPOSITIONS

1. Four standard forms of categorical propositions (the four vowel letters are taken from the Latin words “affirm” and “nego”)

A:        All S is P   (All students are people)

E:        No S is P    (No students are pelicans)

I:          Some S is P    (Some students are pilots)

O:        Some S is not P   (Some students are not partiers)

2. Four requirements: each of the four forms must contain the following:

1. A quantifier (all, no, some)

2. A subject term (S)

3. A copula (is/are, is not)

4. A predicate term (P)

3. Venn Diagrams were introduced by John Venn in 1880 as a way to visualize the logical relations between terms in categorical propositions. There is one diagram for each of the four categorial forms:

How to construct the diagrams

• Shade areas where nothing is contained in the set.

• With “All S is P”, everything in the S circle is also in the P circle, so you shade the portion of S that is outside of P.

• With “No S is P”, nothing in S is also in P, so you shade the portion of S that overlaps with P.

• With I and E propositions, Place X within areas where something is contained in the set.

4. Quantity and quality of the four forms

Quality: a statement is either affirmative or negative

affirmative (A, I) negative (E, O)

Quantity: a statement is either universal or particular

universal (A, E) particular (I, O)

Each of the four forms has its own unique combination of quality and quantity, which exhausts all possibilities:

A:        All S is P (affirmative quality, universal quantity)

E:        No S is P (negative quality, universal quantity)

I:          Some S is P (affirmative quality, particular quantity)

O:        Some S is not P   (negative quality, particular quantity)

5. Translating from ordinary language

• Place asterisk around unit class (e.g., “All *Socrates* are men”)

• S and P terms must be nouns, so add "thing" to adjectives; for example, “some apples are red” translates into “Some apples are red things)

• Other noun possibilities are “times”, “places”, “cases”; for example, “Sometimes I am happy” translates into “some times are times when I am happy”

6. Distribution of the terms in the four forms

“Distribution” means when what's said about S or P applies to all S or P. It is like “quantity” (universal/particular) but applied to each S and P term, as opposed to the entire proposition. The distribution of the S and P terms in four forms make the most sense when comparing them to Venn diagrams of the four forms. Each of the four forms has its own unique combination of distribution and undistribution, which exhausts all possibilities:

                  d    u

A:        All S is P 

                  d    d

E:        No S is P

                     u    u

I:          Some S is P

          u       d

O:        Some S is not P

7. Existential import

“Existential import” means that a statement implies the existence of the thing it talks about. This pertains specifically to I and O statements which have the word “some”, whereby to assert “some S” presuppose that at least one S actually exists. This is relevant because universal statements (“All S are P,” “No S are P”) do not require that any S exist, for, they can be true even if there are no S at all. The notion of “existential import” with categorical statements is a recent invention. Aristotle himself did not explicitly distinguish between statements that imply existence and those that do not. Rather, in his system, even universal statements such as “All S are P” were naturally assumed to refer to existing things. However, in his book The Laws of Thought (1854), George Boole proposed that A and E statements do not imply that S exists, whereas while I and O statements do.

8. Definitions (summary so far)

• Form: A, E, I or O (form of a statement)

• Term: subject and predicate terms

• Quality: affirmative (A, I) negative (E, O)

• Quantity: universal (A, E) particular (I, O)

• Distribution: when what's said about S or P applies to all S or P

• Existential import: S term is committed to existence in I and O forms.

9. Boolian Notation (***on homework, but not on test***)

In his book Laws of thought (1854), Boole introduced and algebraic treatment of categorical propositions, which includes a special notation of the four categorical forms. However, his notation was quickly eclipsed by the rise of predicate logic in the late nineteenth centuries, and is today largely of is largely of historical interest only. His notation is as follows. Please note that P (strikethrough) means non-P.

A: SP = 0 (the intersection of S and non-P is empty; there are no members in that class)

E: SP = 0 (the intersection of S and P is empty; there are no members in that class)

I: SP ≠ 0 (the intersection of S and P is not empty; there is at least one member in that class)

O: SP ≠ 0 (the intersection of S and non-P is not empty; there is at least one member in that class)

B. IMMEDIATE INFERENCES AND LOGICAL EQUIVALENCES (***not on homework or on test***)

Presentations of categorical logic commonly describe how the four A, E, I and O forms can be legitimately transformed into each other in three possible ways: conversion, obversion and contraposition. Aristotle himself discussions conversion, whereas obversion and contraposition introduced by medieval logicians who followed Aristotle.

1. Conversion:

Switch the subject and predicate term

E and I: conversions are equivalent (have same Venn diagrams)

A and O: conversions are not equivalent (have different Venn diagrams)

A: All S are P ≠ All P are S

All Students are People ≠ All People are Students

E: No S are P = No P are S

No Students are Pelicans = No Pelicans are Students

I: Some S are P = Some P are S

Some Students are Pilots = Some Pilots are Students

O:  Some S are not P ≠ Some P are not S

Some Students are not Partiers ≠ Some Partiers are not Students

2. Obversion:

First, change quality; second, replace predicate with complement (i.e., negation in the form non-P)

A, E, I, and O: obversions are equivalent (have same Venn diagrams)

A: All S are P = No S are non-P

All Students are People = No Students are non-People

E: No S are P = All S are non-P

No Students are Pelicans = All Students are non-Pelicans

I: Some S are P = Some S are not non-P

Some Students are Pilots = some Students are not non-Pilots

O:  Some S are not P = Some S are non-P

Some Students are not Partiers = Some Students are non-Partiers

3. Contraposition:

First, switch subject and predicate; second, replace both terms with complement (similar to transposition)

A and O: contrapositives are equivalent;

E and I: contrapositives are not equivalent

A: All S are P = All non-P are non-S

All Students are People = All non-People are non-Students

E: No S are P ≠ No non-P are non-S

No Students are Pelicans ≠ No non-Pelicans are non-Students

I: Some S are P ≠ Some non-P are non-S

Some Students are Pilots ≠ Some non-Pilots are non-Students

O: Some S are not P = Some non-P are not non-S

Some Students are not Partiers = Some non-Partiers are not non-Students

C. TRADITIONAL SQUARE OF OPPOSITION

Aristotle discussed how the four A, E, I and O forms potentially opposed each other, and it was medieval philosopher Boethius who describes their relation in a Square of Opposition.

1. Contradictory: always have opposite truth values

A-O: All S are P // Some S are not P

E-I: No S are P // Some S are P

Validly infer opposition when either is true

Validly infer opposition when either is false

2. Contrary: at least one is false (both are not true)

A-E: All S are P // No S are P

Validly infer opposition when either true

No valid inference of opposition when either false

3. Subcontrary: at least one is true (both are not false)

I-O: Some S are P // Some S are not P

No valid inference of opposition when either is true

Validly infer opposition when either is false

4. Subalternations: truth flows down, falsehood flows up

A-I: All S are P // Some S are P

E-O: No S are P // Some S are not P

If universal is true, then particular is true, if particular is false, then universal is false

When A (or E) is true, I (or O) is true

When A (or E) is false, I is undetermined

When I (or O) is true, A (or E) is undetermined

When I (or O) is false, A (or E) is false

Boole’s reformulation of logic preserved only the contradictory relations (A–O and E–I) of the traditional square of opposition; because he denied existential import for universal statements, the relations of contrariety, subcontrariety, and subalternation became undetermined. Later logicians depicted this as a reduced or “Boolean” square containing only the contradictories.

C. CATEGORICAL SYLLOGISMS

1. Example of syllogism:

Major premise: All men are mortal (All men are mortal things)

Minor premise: Socrates is a man (All *Socrates* are men)

Conclusion: Socrates is mortal (All *Socrates* are mortal things)

2. Mood of Syllogisms

The three forms of the three propositions in a syllogism (e.g. AAA, EIO)

Example

A: All men are mortal things

A: All *Socrates* are men

A: All *Socrates* are mortal things

3. Figures of Syllogisms:

The order of the subject, predicate and middle terms in the premises

Chart:

1st Fig.    2nd Fig.    3rd Fig.    4th Fig.

M - P       P - M M - P       P - M

S - M       S - M M - S       M - S

S - P        S - P         S - P        S – P

Rules:

S and P are always in the conclusion in the same order

P and M are always in premise 1

S and M are always in premise 2

Tip: memorize the pattern for middle term: \ || /

4. Mood and Figure combined

Example: AAA-1 (i.e., Mood AAA, with Figure 1)

All M is P

All S is M

All S is P

Example: A00-2 (i.e., Mood A00, with Figure 2)

All P is M

Some S is M

Some S is P

D. VALIDITY OF SYLLOGISMS

There are 256 possible forms of syllogistic arguments, and there three different methods of indicating their validity or invalidity: (1) through intuitive lists, (2) through Venn diagrams, and (3) through five rules. Aristotle used the first, but later logicians devised the second and third.

1. Validity through intuitive lists (i.e., whether a syllogism intuitively seems valid)

Fifteen Unconditionally Valid (for both Aristotle and Boole)

Fig. 1: AAA-1, EAE-1, AII-1, EIO-1

Fig. 2: AEE-2, EAE-2, AOO-2, EIO-2

Fig. 3: AII-3, IAI-3, EIO-3, OAO-3

Fig. 4: AEE-4, IAI-4, EIO-4

Latin names of valid forms from a mnemonic poem by medieval logician William of Sherwood. The terms are only pseudowords, invented solely for this purpose, and not actual Latin words:

Fig. 1: barbara, celarent, darii, ferio

Fig. 2: camestres, cesare, baroco, festino

Fig. 3: datisi, disamis, ferison, bocardo

Fig. 4: camenes, dimaris, fresison

Nine Conditionally Valid (for only Aristotle, not Boole; these assumes that a term in the conclusion exists)

Fig. 1: AAI-1, EAO-1

Fig. 2: AEO-2, EAO-2

Fig. 3: AAI-3, EAO-3

Fig. 4: AEO-4, EAO-4, AAI-4

2. Validity with Venn Diagram:

Construct three overlapping circles for S P and M.

Diagram both premises (but not the conclusion), and see if diagram already contains the conclusion

If so then the argument is valid, if not then it is invalid

Diagram universal premise before particular

Examples:

AAA-1:

All M is P

All S is M

All S is P

A00-2

All P is M

Some S is not M

Some S is not P

When placement of X is ambiguous, put it on a line, which indicates that will be invalid since it could go in either direction (e.g., AOO-1, AOO-4, OAO-4)

3. Five Rules of Validity

The rules:

1. One distributed middle term: middle term must be distributed in at least one premise.

2. Distributed term-distributed term: term is distributed in conclusion iff it is distributed in premise.

3. One affirmative premise: must have at least one affirmative premise.

4. Negative-negative: negative conclusion iff negative premise.

5. Particular-particular: cannot conclude a particular from two universals

Examples:

AOO-3:

All M is P

Some M is not S

Some S is not P

Rule 1 OK: middle term distributed in premise 1

Rule 2 failed: P term distributed in conclusion but not in premise 1

Rule 3 OK: premise 1 is affirmative

Rule 4 OK: premise 2 negative, conclusion negative

Rule 5 OK: premise 2 particular, conclusion particular

EAE-3

No M is P

All M is S

No S is P

Rule 1 OK: middle term distributed in premise 1 and 2

Rule 2 failed: S term distributed in conclusion but not in premise 1

Rule 3 OK: premise 1 is affirmative

Rule 4 OK: premise 2 negative, conclusion negative

Rule 5 OK: no particulars

E. NON-STANDARD SYLLOGISMS (***not included in homework or on test***)

1. Enthymemes: syllogisms with an unstated premise

Example: Socrates is mortal because he his human

All men are mortal things (unstated)

All *Socrates* are men (stated)

All *Socrates* are mortal things (stated)

Example: Candide is a typical French novel, therefore it is vulgar

All French novels are vulgar things (unstated)

All *Candide* are French novels (stated)

All *Candide* are vulgar things (stated)

2. Sortes (polysyllogism): a chain of two or more syllogisms where the conclusion of first is a premise of the second

Example:

All lions are big cats.

All big cats are predators.

All predators are carnivores.

Therefore, all lions are carnivores.

Revision 1: In this version, the implied intermediate conclusion (‘All lions are predators’) is stated explicitly, which turns the argument into a chain of two linked syllogisms. However, it still does not follow the standard syllogistic form because, in each step, the subject term of the conclusion appears in the major premise rather than in the minor premise, where it should normally be. Thus, the structure violates the usual arrangement of terms in a categorical syllogism. The next version (Revision 2) corrects this by restating the two syllogisms separately in proper standard form.

All lions are big cats.

All big cats are predators.

[Therefore, all lions are predators.] (enthymeme)

All predators are carnivores.

Therefore, all lions are carnivores

Revision 2: in this version, the original argument becomes two separate standard form syllogisms. This revision has two independent arguments that are in standard form.

All big cats are predators.

All lions are big cats.

Therefore, all lions are predators

All predators are carnivores.

All lions are predators

Therefore, all lions are carnivores

Testing for validity

Method 1: test as two separate syllogisms (as per revision 2) with either Venn diagram or five rules

Method 2: test original version with four-term Venn diagram

All lions are big cats (remove all portions of lions outside big cats)

All big cats are predators (remove all portions of big cats outside predators)

All predators are carnivores (remove all predators outside carnivores)

Therefore, all lions are carnivores (all that remains of lions is within carnivores)