9780872208131

Formal Logic : Its Scope and Limits

by ;
  • ISBN13:

    9780872208131

  • ISBN10:

    0872208133

  • Edition: 4th
  • Format: Hardcover
  • Copyright: 3/30/2006
  • Publisher: Hackett Pub Co Inc

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Summary

The first beginning logic text to employ the tree method -- a complete formal system of first-order logic that is remarkably easy to understand and use -- this text allows students to take control of the nuts and bolts of formal logic quickly, and to move on to more complex and abstract problems. The tree method is elaborated in manageable steps over five chapters, in each of which its adequacy is reviewed; soundness and completeness proofs are extended at each step, and the decidability proof is extended at the step from truth functions to the logic of non-overlapping quantifiers with a single variable, after which undecidability is demonstrated by example. The first three chapters are bilingual, with arguments presented twice, in logical notation and in English. The last three chapters consider the discoveries defining the scope and limits of formal methods that marked logic's coming of age in the 20th century: Godel's completeness and incompleteness theorems for first and second-order logic, and the Church-Turing theorem on the undecidability of first order logic. This new edition provides additional problems, solutions to selected problems, and two new Supplements: 'Truth-Functional Equivalence' reinstates material on that topic from the second edition that was omitted in the third, and 'Variant Methods', in which John Burgess provides a proof regarding the possibility of modifying the tree method so that it will always find a finite model when there is one, and another, which shows that a different modification -- once contemplated by Jeffrey -- can result in a dramatic speed-up of certain proofs.

Author Biography

Richard Jeffrey (1926-2002) was Professor of Philosophy, Princeton University.John P. Burgess is Professor of Philosophy, Princeton University.

Table of Contents

Preface to the Fourth Edition xi
Truth-Functional Logic
1(20)
``Not,'' ``And''
2(1)
``Or''
3(1)
Is This Argument Valid?
4(1)
A Bad Argument
4(1)
Soundness
5(1)
``If''
6(1)
Denial, Conjunction, Disjunction
6(2)
Conditionals
8(1)
Counterfactual Conditionals
9(1)
Biconditionals and Logical Equivalence
10(1)
Rules of Valuation
10(1)
Oddities of ``If''
11(2)
Rules of Formation
13(1)
Consistency and the Science of Refutation
14(1)
Tautologies
15(1)
Context Dependency
15(2)
Formal Validity
17(1)
Problems
18(3)
Truth Trees
21(14)
A Closed Tree
22(2)
An Open Finished Tree
24(1)
Double Denial
24(1)
Flowchart for ``--'' and ``→'' with Examples
25(2)
Rules of Inference, with Flowchart
27(2)
Problems
29(2)
Adequacy of the Tree Test
31(1)
Decidability
32(1)
Soundness
33(1)
Completeness
34(1)
Generality
35(24)
Universal Instantiation (``UI'')
37(1)
Existential Instantiation (``El'')
38(1)
UI Again---Closure
39(1)
Examples
40(1)
Rules of Formation
41(3)
The Complete Method, with Flowchart
44(2)
Logical Structure
46(2)
Problems
48(1)
Interpretations
49(2)
Rules of Interpretation
51(1)
Counterexamples
52(2)
``Some S's Are P''
54(1)
Decidability
55(1)
Completeness
55(1)
Soundness
56(1)
``Some S's Are P'': Solution
57(2)
Multiple Generality
59(16)
Example
59(1)
Example
60(1)
Logic into English
61(1)
Linkage
62(1)
Rules of Formation
63(1)
English into Logical Notation
64(1)
Example: Alma's Narcissism Inflames the Baron
65(1)
Example: Alma Inflamed by Her Own Narcissism
66(1)
Amor Vincit Omnia
66(1)
Problems
67(1)
Infinite Counterexamples
68(2)
More Problems
70(1)
Undecidability
70(1)
Soundness
70(1)
Completeness
71(1)
Translation Drill
72(2)
Exercises
74(1)
Identity
75(10)
Rules of Inference for Identity
76(2)
Saying of What Is Not That It Is Not
78(1)
Definite Descriptions
79(1)
Number
80(1)
Problems
81(1)
Rule of Interpretation for Identity
81(1)
Soundness
81(1)
Completeness
82(1)
Examples of Modified Interpretations
83(1)
Problems
84(1)
Functions
85(14)
Rule of Formation
86(1)
Rule of Interpretation
87(1)
El and UI Revised
88(1)
Regulating UI, with Flowchart
89(1)
Adequacy
90(3)
Problems
93(1)
Mathematical Reasoning and Groups
93(2)
Problems
95(1)
Robinson Arithmetic
96(1)
Problems
97(2)
Uncomputability
99(14)
How to Program a Register Machine
100(2)
Problems
102(1)
Register Machine Tree Tests
102(3)
The Church-Turing Thesis
105(1)
Unsolvability of the Halting Problem
106(3)
Problems
109(1)
Programs in Logical Notation
110(2)
Problems
112(1)
Undecidability
113(12)
The Decision Problem
114(1)
A Routine Test for Halting
115(1)
The Argument Is Valid iff the Program Halts
116(3)
Focusing the Undecidability Result
119(2)
A Solvable Case of the Decision Problem
121(1)
Undecidability without Function Symbols
121(1)
Undecidability of Two-Place Predicate Logic
122(1)
Problems
123(2)
Incompleteness
125(16)
Second-Order Logic
125(3)
Problems
128(1)
Logical Types
128(2)
Russell's Paradox
130(1)
Second-Order Formation and Valuation Rules
131(1)
Mathematical Induction
132(2)
Isomorphism, Categoricity, Completeness
134(1)
Incompleteness of Validity Tests for Second-Order Logic
135(3)
Problems
138(1)
Some History
139(2)
Supplement A Truth-Functional Equivalence
141(10)
Venn Diagrams
141(2)
Laws of Equivalence
143(1)
Normal Form
144(3)
Expressive Completeness
147(1)
Simplification
147(1)
Logic Circuits
148(2)
Problems
150(1)
Supplement B Variant Methods
151(10)
Looking for Finite Models
152(1)
Finding Finite Models
153(1)
Adequacy of the Method
154(1)
The Cut Rule
155(1)
Speedy Closure with Cut
156(1)
Slow Closure without Cut
157(1)
Problems
158(3)
Solutions 161(10)
Index 171

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