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逻辑与哲学

人工智能

一阶逻辑

命题逻辑

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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):

傻

/∴

2. Modus Tollens (MT):

傻

/∴ ~

3. Disjunctive Syllogism (DS):

∨

/∴

∨

/∴

4. Simplification (Simp):

⋅

/∴

⋅

/∴

5. Conjunction (Conj):

/∴

⋅

6. Hypothetical Syllogism (HS):

傻

/∴

傻

7. Addition (Add):

/∴

∨

8. Constructive Dilemma (CD):

∨

傻

/∴

∨

Valid Equivalence

Forms (Rule of

Replacement)

Conditional and

Indirect Proof

9. Double Negation (DN):

:: ~~

10. DeMorgan’s Theorem (DeM):

~ (

⋅

) :: (~

∨ ~

)

~ (

∨

) :: (~

⋅ ~

)

11. Commutation (Comm):

(

∨

) :: (

∨

)

(

⋅

) :: (

⋅

)

12. Assocation (Assoc):

[

∨ (

∨

)] :: [(

∨

) ∨

]

[

⋅ (

⋅

)] :: [(

⋅

) ⋅

]

13. Distribution (Dist):

[

⋅ (

∨

)] :: [(

⋅

) ∨ (

⋅

)]

[

∨ (

⋅

)] :: [(

∨

) ⋅ (

∨

)]

14. Contraposition (Contra):

(

傻

) :: (~

傻 ~

)

15. Implication (Impl):

(

傻

) :: (~

∨

)

16. Exportation (Exp):

[(

⋅

) 傻

] :: [

傻 (

傻

)]

17. Tautology (Taut):

:: (

⋅

)

:: (

∨

)

18. Equivalence (Equiv):

(

⬅

) :: [(

傻

) ⋅ (

傻

)]

(

⬅

) :: [(

⋅

) ∨ (~

⋅ ~

)]

Conditional Proof

→

AP /∴

傻

Indirect Proof

→ ~

AP /∴

⋅ ~

Rule UI: (

)(. . .

. . .) /∴ (. . .

. . .)

Rule EI: (∃

)(. . .

. . .) /∴ (. . .

. . .)

Rule UG: (. . .

. . .) /∴ (

)(. . .

. . .)

Rule EG: (. . .

. . .) /∴ (∃

)(. . .

. . .)

Rule QN: (

)(. . .

. . .) :: ~ (∃

) ~ (. . .

. . .)

(∃

)(. . .

. . .) :: ~ (

) ~ (. . .

. . .)

(

) ~ (. . .

. . .) :: ~ (∃

)(. . .

. . .)

(∃

) ~ (. . .

. . .) :: ~ (

)(. . .

. . .)

Rule ID: (. . .

. . .) (. . .

. . .)

/∴ (. . .

. . .)

/∴ (. . .

. . .)

Rule IR: /∴ (

)(

)

Provided:

1. (. . .

...) results from replacing each occur-

rence of

free in (. . .

...) with a

that is

either a constant or a variable free in (. . .

...)

(making no other changes).

Provided:

is not a constant.

does not occur free previously in the proof.

3. (. . .

...) results from replacing each occur-

rence of

free in (. . .

...) with a

that is free

in (. . .

...) (making no other changes).

Provided:

is not a constant.

does not occur free previously in a line

obtained by EI.

does not occur free previously in an assumed

premise that has not yet been discharged.

4. (. . .

...) results from replacing each occur-

rence of

free in (. . .

...) with a

that is free

in (. . .

...) (making no other changes) and

there are no additional free occurrences of

already contained in (. . .

...).

Provided:

1. (. . .

...) results from replacing at least one

occurrence of

where u is a constant or a vari-

able free in (. . .

...) with a

that is free in

(. . .

...) (making no other changes) and there

are no additional free occurrences of

already

contained in (. . .

...).

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

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

Cengage Learning products are represented in Canada by

Nelson Education, Ltd.

For your course and learning solutions, visit

www.cengage.com

Purchase any of our products at your local college store or at our

preferred online store www.ichapters.com

For product information and technology assistance, contact us at

Cengage Learning Customer & Sales Support, 1-800-354-9706

For permission to use material from this text or product,

submit all requests online at www.cengage.com/permissions.

Further permissions questions can be e-mailed to

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Printed in Canada

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