7. CATEGORICAL LOGIC
Outline
James Fieser, UT Martin
updated 11/1/2023
The system of categorical logic was developed by Aristotle in his book Prior Analytics, and from around 1100 to the early 1900s was the dominant system of logic 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, and categorical syllogisms. 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
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 the “quantity” (universal/particular) 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. 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.
8. Boolian Notation (***on homework, but not on test***)
A: SP
= 0 (the class of S and non-P is empty; there are no members in the class of S
and non-P)
E: SP = 0 (the class of S and P is empty; there are no members in the class of S and P)
I: SP ≠ 0 (the class of S and P is not empty; there is at least one member in the class of S and P)
O: SP
≠ 0 (the class of S and non-P is not empty; there is at least one member in
the class of S and non-P)
B. IMMEDIATE INFERENCES AND LOGICAL EQUIVALENCES (***not on homework or on test***)
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. Contrapositive:
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
4. Traditional Square of Opposition
5. 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
6. 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
7. 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
8. 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
9. Boolean Square of Opposition: includes only the contradictories; all others are undetermined
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:
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: first conclusion enthymeme stated. However, this revision does not follow standard syllogistic form. The reason is that the subjects indicated in each conclusion are in the major premises, rather than the minor premises where they belong. The more accurate revision, then, is revision 2.
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: 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)
