CHAPTER 5: THE LOGIC OF CATEGORICAL STATEMENTS

5.1 Categorical Statements

Four standard forms of categorical statements (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

1. A quantifier (all, no)

2. A subject term (S)

3. A copula (is, are)

4. A predicate term (P)

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

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

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.

Translating from ordinary language

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

Add "thing" to adjectives (e.g., “some apples are red things”)

Times, places, cases (e.g., “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)

5.2 Venn Diagrams

Diagraming categorical propositions

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 statements, Place X within areas where something is contained in the set.

5.3 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: only the contradictories; all others are undetermined

5.4 Constructing 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 statements 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

5.5 Validity of Syllogisms

Valid Syllogistic Forms (256 possible forms):

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