Contrary

A

E

T

T

Co

nt

ra

ra

nt

Co

Subalternation

ry

to

dic

dic

to

r

Subalternation

y

F

F

I

O

Subcontrary

Logically Equivalent Statement Forms

Conversion

Given statement

E: No S are P.

I: Some S are P.

Converse

No P are S.

Some P are S.

Obversion

Given statement

A: All S are P.

E: No S are P.

I: Some S are P.

O: Some S are not P.

Obverse

No S are non-P.

All S are non-P.

Some S are not non-P.

Some S are non-P.

Contraposition

Given statement

A: All S are P.

O: Some S are not P.

Contrapositive

All non-P are non-S.

Some non-P are not non-S.

Valid Syllogistic Forms

Unconditionally Valid Forms

Figure 1

AAA

EAE

AII

EIO

Figure 2

EAE

AEE

EIO

AOO

Figure 3

IAI

AII

OAO

EIO

Figure 4

AEE

IAI

EIO

Conditionally Valid Forms

Figure 1

AAI

EAO

Figure 2

AEO

EAO

Figure 3

Figure 4

AEO

Required

condition

S exist

AAI

EAO

EAO

M exist

AAI

P exist

Rules for Categorical Syllogisms

Rule 1:

Fallacy:

The middle term must be distributed at least once.

Undistributed middle

Rule 2:

Fallacy:

If a term is distributed in the conclusion, then it must be distributed in the premise.

Illicit major; illicit minor

Rule 3:

Fallacy:

Two negative premises are not allowed.

Exclusive premises

Rule 4:

A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise.

Drawing an afﬁrmative conclusion from a negative premise; drawing a negative conclusion from afﬁrmative premises

Fallacy:

Rule 5:

Fallacy:

If both premises are universal, the conclusion cannot be particular. Existential fallacy

NOTE: If only Rule 5 is broken, the syllogism is valid from the Aristotelian standpoint if the critical term denotes actually existing things.

Truth Tables for the Propositional Operators

p

q

~p

p•q

pvq

p⊃q

p≡q

T

T

F

F

T

F

T

F

F

F

T

T

T

F

F

F

T

T

T

F

T

F

T

T

T

F

F

T

Rules for the Probability Calculus

1.

2.

3.

4.

5.

6.

7.

P (A or not A) = 1

P (A and not A) = 0

P (A and B) = P (A) P (B)

(when A and B are independent)

P (A and B) = P (A) P (B given A)

P (A or B) = P (A) + P(B)

(when A and B are mutually exclusive)

P (A or B) = P (A) + P (B) – P (A and B)

P (A) = 1 – P (not A)

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A Concise Introduction to Logic

NINTH EDITION

Patrick J. Hurley

University of San Diego

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