Language, Proof, and Logic - Barwise_ Jon.pdf - Language, Proof, and Logic (1516) - superktos86

LANGUAGE,

PROOF AND

LOGIC

JON BARWISE & JOHN ETCHEMENDY

In collaboration with

Gerard Allwein

Dave Barker-Plummer

Albert Liu

SEVEN BRIDGES PRESS

NEW YORK • LONDON

CSLI Publications

Center for the Study of Language and Information

Leland Stanford Junior University

03 02 01 00 99 5 4 3 2 1

Library of Congress Cataloging-in-Publication Data

Barwise, Jon.

Language, proof and logic / Jon Barwise and John Etchemendy ;

in collaboration with Gerard Allwein, Dave Barker-Plummer, and

Albert Liu.

p. cm.

ISBN 1-889119-08-3 (pbk. : alk. paper)

I. Etchemendy, John, 1952- II. Allwein, Gerard, 1956-

III. Barker-Plummer, Dave. IV. Liu, Albert, 1966- V. Title.

IN PROCESS

99-41113

CIP

Acknowledgements

Our primary debt of gratitude goes to our three main collaborators on this

project: Gerry Allwein, Dave Barker-Plummer, and Albert Liu. They have

worked with us in designing the entire package, developing and implementing

the software, and teaching from and re¯ning the text. Without their intelli-

gence, dedication, and hard work, LPL would neither exist nor have most of

its other good properties.

In addition to the ¯ve of us, many people have contributed directly and in-

directly to the creation of the package. First, over two dozen programmers have

worked on predecessors of the software included with the package, both earlier

versions of Tarski's World and the program Hyperproof, some of whose code

has been incorporated into Fitch. We want especially to mention Christopher

Fuselier, Mark Greaves, Mike Lenz, Eric Ly, and Rick Wong, whose outstand-

ing contributions to the earlier programs provided the foundation of the new

software. Second, we thank several people who have helped with the develop-

ment of the new software in essential ways: Rick Sanders, Rachel Farber, Jon

Russell Barwise, Alex Lau, Brad Dolin, Thomas Robertson, Larry Lemmon,

and Daniel Chai. Their contributions have improved the package in a host of

ways.

Prerelease versions of LPL have been tested at several colleges and uni-

versities. In addition, other colleagues have provided excellent advice that we

have tried to incorporate into the ¯nal package. We thank Selmer Bringsjord,

Renssalaer Polytechnic Institute; Tom Burke, University of South Carolina;

Robin Cooper, Gothenburg University; James Derden, Humboldt State Uni-

versity; Josh Dever, SUNY Albany; Avrom Faderman, University of Rochester;

James Garson, University of Houston; Ted Hodgson, Montana State Univer-

sity; John Justice, Randolph-Macon Women's College; Ralph Kennedy, Wake

Forest University; Michael O'Rourke, University of Idaho; Greg Ray, Univer-

sity of Florida; Cindy Stern, California State University, Northridge; Richard

Tieszen, San Jose State University; Saul Traiger, Occidental College; and Lyle

Zynda, Indiana University at South Bend. We are particularly grateful to John

Justice, Ralph Kennedy, and their students (as well as the students at Stan-

ford and Indiana University), for their patience with early versions of the

software and for their extensive comments and suggestions.

We would also like to thank Stanford's Center for the Study of Language

and Information and Indiana University's College of Arts and Sciences for

iii

iv / Acknowledgements

their ¯nancial support of the project. Finally, we are grateful to our two

publishers, Dikran Karagueuzian of CSLI Publications and Clay Glad of Seven

Bridges Press, for their skill and enthusiasm about LPL.

Acknowledgements

Contents

Acknowledgements

iii

Introduction

The special role of logic in rational inquiry

. . . . . . . . . . . . . .

Why learn an arti¯cial language? . . . . . . . . . . . . . . . . . . . .

Consequence and proof . . . . . . . . . . . . . . . . . . . . . . . . . .

Instructions about homework exercises (essential! ) . . . . . . . . . .

To the instructor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Web address

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Propositional Logic

Atomic Sentences

1.1

Individual constants

. . . . . . . . . . . . . . . . . . . . . . . .

1.2

Predicate symbols

. . . . . . . . . . . . . . . . . . . . . . . . .

1.3

Atomic sentences . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4

General ¯rst-order languages

. . . . . . . . . . . . . . . . . . .

1.5

Function symbols (optional ) . . . . . . . . . . . . . . . . . . . .

1.6

The ¯rst-order language of set theory (optional )

. . . . . . . .

1.7

The ¯rst-order language of arithmetic (optional)

. . . . . . . .

1.8

Alternative notation (optional ) . . . . . . . . . . . . . . . . . .

The Logic of Atomic Sentences

2.1

Valid and sound arguments

. . . . . . . . . . . . . . . . . . . .

2.2

Methods of proof . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Formal proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

Constructing proofs in Fitch . . . . . . . . . . . . . . . . . . . .

2.5

Demonstrating nonconsequence . . . . . . . . . . . . . . . . . .

2.6

Alternative notation (optional ) . . . . . . . . . . . . . . . . . .

The Boolean Connectives

3.1

Negation symbol: : . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Conjunction symbol: ^ . . . . . . . . . . . . . . . . . . . . . . .

3.3

Disjunction symbol: _

. . . . . . . . . . . . . . . . . . . . . . .

3.4

Remarks about the game

. . . . . . . . . . . . . . . . . . . . .

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