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This text is designed for those instructors who desire a comprehensive introduction to formal logic that is both rigorous and accessible to students encountering the subject for the first time. Numerous carefully crafted exercise sets accompanied by clear, crisp exposition give students a firm grasp of basic concepts and take the student from sentential logic through first-orde...
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Valid Forms for Sentential Logic
Valid Argument
Forms of Inference
1. Modus Ponens (MP):
p
傻
q
p
/∴
q
2. Modus Tollens (MT):
p
傻
q
~
q
/∴ ~
p
3. Disjunctive Syllogism (DS):
p
∨
q
~
p
/∴
q
p
∨
q
~
q
/∴
p
4. Simplification (Simp):
p
⋅
q
/∴
p
p
⋅
q
/∴
q
5. Conjunction (Conj):
p
q
/∴
p
⋅
q
6. Hypothetical Syllogism (HS):
p
傻
q
q
傻
r
/∴
p
傻
r
7. Addition (Add):
p
/∴
p
∨
q
8. Constructive Dilemma (CD):
p
∨
q
p
傻
r
q
傻
s
/∴
r
∨
s
Valid Equivalence
Forms (Rule of
Replacement)
Conditional and
Indirect Proof
9. Double Negation (DN):
p
:: ~~
p
10. DeMorgan’s Theorem (DeM):
~ (
p
⋅
q
) :: (~
p
∨ ~
q
)
~ (
p
∨
q
) :: (~
p
⋅ ~
q
)
11. Commutation (Comm):
(
p
∨
q
) :: (
q
∨
p
)
(
p
⋅
q
) :: (
q
⋅
p
)
12. Assocation (Assoc):
[
p
∨ (
q
∨
r
)] :: [(
p
∨
q
) ∨
r
]
[
p
⋅ (
q
⋅
r
)] :: [(
p
⋅
q
) ⋅
r
]
13. Distribution (Dist):
[
p
⋅ (
q
∨
r
)] :: [(
p
⋅
q
) ∨ (
p
⋅
r
)]
[
p
∨ (
q
⋅
r
)] :: [(
p
∨
q
) ⋅ (
p
∨
r
)]
14. Contraposition (Contra):
(
p
傻
q
) :: (~
q
傻 ~
p
)
15. Implication (Impl):
(
p
傻
q
) :: (~
p
∨
q
)
16. Exportation (Exp):
[(
p
⋅
q
) 傻
r
] :: [
p
傻 (
q
傻
r
)]
17. Tautology (Taut):
p
:: (
p
⋅
p
)
p
:: (
p
∨
p
)
18. Equivalence (Equiv):
(
p
⬅
q
) :: [(
p
傻
q
) ⋅ (
q
傻
p
)]
(
p
⬅
q
) :: [(
p
⋅
q
) ∨ (~
p
⋅ ~
q
)]
Conditional Proof
→
p
AP /∴
q
.
.
.
q
p
傻
q
CP
Indirect Proof
→ ~
p
AP /∴
p
.
.
.
q
⋅ ~
q
p
IP
Rule UI: (
u
)(. . .
u
. . .) /∴ (. . .
w
. . .)
Rule EI: (∃
u
)(. . .
u
. . .) /∴ (. . .
w
. . .)
Rule UG: (. . .
u
. . .) /∴ (
w
)(. . .
w
. . .)
Rule EG: (. . .
u
. . .) /∴ (∃
w
)(. . .
w
. . .)
Rule QN: (
u
)(. . .
u
. . .) :: ~ (∃
u
) ~ (. . .
u
. . .)
(∃
u
)(. . .
u
. . .) :: ~ (
u
) ~ (. . .
u
. . .)
(
u
) ~ (. . .
u
. . .) :: ~ (∃
u
)(. . .
u
. . .)
(∃
u
) ~ (. . .
u
. . .) :: ~ (
u
)(. . .
u
. . .)
Rule ID: (. . .
u
. . .) (. . .
u
. . .)
u
=
w
/∴ (. . .
w
. . .)
w
=
u
/∴ (. . .
w
. . .)
Rule IR: /∴ (
x
)(
x
=
x
)
Provided:
1. (. . .
w
...) results from replacing each occur-
rence of
u
free in (. . .
u
...) with a
w
that is
either a constant or a variable free in (. . .
w
...)
(making no other changes).
Provided:
1.
w
is not a constant.
2.
w
does not occur free previously in the proof.
3. (. . .
w
...) results from replacing each occur-
rence of
u
free in (. . .
u
...) with a
w
that is free
in (. . .
w
...) (making no other changes).
Provided:
1.
u
is not a constant.
2.
u
does not occur free previously in a line
obtained by EI.
3.
u
does not occur free previously in an assumed
premise that has not yet been discharged.
4. (. . .
w
...) results from replacing each occur-
rence of
u
free in (. . .
u
...) with a
w
that is free
in (. . .
w
...) (making no other changes) and
there are no additional free occurrences of
w
already contained in (. . .
w
...).
Provided:
1. (. . .
w
...) results from replacing at least one
occurrence of
u,
where u is a constant or a vari-
able free in (. . .
u
...) with a
w
that is free in
(. . .
w
...) (making no other changes) and there
are no additional free occurrences of
w
already
contained in (. . .
w
...).
Rules for Predicate Logic
Alan Hausman
Hunter College, City University of New York
Howard Kahane
Late of University of Maryland
Paul Tidman
Mount Union College
Logic and Philosophy
A Modern Introduction
Eleventh Edition
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
Logic and Philosophy:
A Modern Introduction,
Eleventh Edition
Alan Hausman, Howard
Kahane, Paul Tidman
Publisher: Clark Baxter
Sr. Sponsoring Editor:
Joann Kozyrev
Assistant Editor:
Nathan Gamache
Editorial Assistant:
Michaela Henry
Media Editor:
Diane Akerman
Marketing Manager:
Mark Haynes
Marketing Coordinator:
Josh Hendrick
Content Project Manager:
Alison Eigel Zade
Art Director: Faith Brosnan
Manufacturing Manager:
Marcia Locke
Senior Rights Account
Manager—Text: Bob Kauser
Production Service:
Scratchgravel Publishing
Services
Cover Designer: RHDG/
Christopher Harris
Cover image: David Muir/
Getty Images
Compositor: Macmillan
Publishing Solutions
Cover Designer:
RHDG/Christopher Harris
© 2010, 2007 Wadsworth, Cengage Learning
ALL RIGHTS RESERVED. No part of this work covered by the
copyright herein may be reproduced, transmitted, stored,
or used in any form or by any means graphic, electronic, or
mechanical, including but not limited to photocopying,
recording, scanning, digitizing, taping, Web distribution,
information networks, or information storage and retrieval
systems, except as permitted under Section 107 or 108 of the
1976 United States Copyright Act, without the prior written
permission of the publisher.
Library of Congress Control Number: 2008940689
Student Edition:
ISBN-13: 978-0-495-60158-6
ISBN-10: 0-495-60158-6
Wadsworth
20 Channel Center
Boston, MA 02210
USA
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