CATEGORICAL LOGIC

Categorical logic was developed by Aristotle in
his book *Prior Analytics*, and was the dominant system of logic in
Western philosophy until the early 1900s. It is called “categorical” since it
is based on how categories of things relate to each other. It has two
components: categorical propositions, and categorical syllogisms. An example of
a categorical proposition is “all dogs are mammals” which involves an
overlapping relation between the category “dogs” and the category “mammals”. A
categorical syllogism as 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

CATEGORICAL PROPOSITIONS

Four standard forms of categorical propositions (vowels from “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)

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)

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.

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, universal quantity)

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 the 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

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 “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”

Boolian Notation

A: S~~P~~ = 0 (no
members in the class of S and non-P)

E: SP = 0 (no members in the class of S and P)

I: SP ≠ 0 (at least one member in the class of S and P)

O: S~~P~~
≠ 0 (at least one member in the class of S and non-P)

Definitions

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)

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

IMMEDIATE INFERENCES AND LOGICAL EQUIVALENCES

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 is 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

Obversion:

First, change quality; second, replace predicate with complement

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 is are P = Some S are not non-P

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

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

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

Contrapositive:

First, switch subject and predicate; second, replace both terms with complement

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

Traditional Square of Opposition

Contradictory: 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

Contrary: at least one false (both 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

Subcontrary: at least one true (both 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

Subalternations: truth flows up, falsehood down

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

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

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

Boolean Square of Opposition: includes only the contradictories; all others are undetermined

CATEGORICAL SYLLOGISMS

Example of syllogism:

1. All men are mortal (All men are mortal things)

2. Socrates is a man (All *Socrates* are men)

3. Socrates is mortal (All *Socrates* are mortal things)

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

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: \ || /

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

VALIDITY OF SYLLOGISMS

There are 256 possible forms of syllogistic arguments, and there are differing methods of indicating validity: (1) through intuitive lists, (2) through Venn diagrams, and (3) through five rules. Aristotle used the first, but today logicians use the second and third.

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; 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

Validity with Venn Diagram:

Construct three overlapping circles for S P and M.

Diagram both premises, see if diagram indicates conclusion

Diagram universal premise before particular

Examples:

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

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

All M is P

No S is M

No S is 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: no particulars