Intro to Logic

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  • Topic: Logic, Association of Public and Land-Grant Universities, American Association of State Colleges and Universities
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Traditional Square of Opposition
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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