Download Language, proof, and logic

Contents
Acknowledgements
v
Introduction
The special role of logic in rational inquiry
Why learn an artificiallanguage? . . . . . .
Consequence and proof. . . . . . . . . . . .
Instructions about homework exercises (essential!)
To the instructor
Web address
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Propositional Logic
Atomic Sentences
1.1 Individ ual constants
1.2 Predicate symbols .
1.3 Atomic sentences . .
1.4 General first-order languages
1.5 Function symbols (optional)
1.6 The first-order language of set theory (optional)
1. 7 The first-order language of arithmetic (optional)
1.8 Alternative notation (optional) . . . . . . . . . .
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The
2.1
2.2
2.3
2.4
2.5
2.6
Logic of Atomic Sentences
Valid and sound argument s
Methods of proof . . . . . . .
Formal proofs . . . . . . . . .
Constructing proofs in Fitch
Demonstrating nonconsequence .
Alternative notation (optional)
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The
3.1
3.2
3.3
3.4
Boolean Connectives
Negation symbol: --; ..
Conjunction symbol: II
Disjunction symbol: V .
Remarks about the game
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Ambiguity and parentheses . . .
Equivalent ways of saying things
Translation...........
Alternative notation (optional) .
The
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Logic of Boolean Connectives
Tautologies and logical truth . . .
Logical and tautological equivalence
Logical and tautological consequence .
Tautological consequence in Fitch ..
Pushing negation around (optional) .
Conjunctive and disjunctive normal forms (optional)
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Methods of Proof for Boolean Logic
5.1 Valid inference steps . . . . . . . . .
5.2 Proof by cases . . . . . . . . . . . .
5.3 Indirect proof: proof by contradiction
5.4 Arguments with inconsistent premises (optional)
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Formal Proofs and Boolean Logic
6.1 Conjunction rule s
6.2 Disjunction rule s . . . . . . .
6.3 Negation rules . . . . . . . .
6.4 The proper use of subproofs .
6.5 Strategy and tactics . . . . .
6.6 Proofs without premises (optional)
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Conditionals
7.1 Material conditional symbol: ----t
7.2 Biconditional symbol: B
7.3 Conversational implicature . . .
7.4 Truth-functional completeness (optional)
7.5 Alternative notation (optional) . . . . . .
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The
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8.3
8.4
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CONTENTS
3.5
3.6
3.7
3.8
Logic of Conditionals
Informal methods of proof .
Formal rules of proof for ----t and B . . .
Soundness and completeness (optional)
Valid arguments: some review exercises
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II
Quantifiers
9 Introduction to Quantification
9.1 Variables and atomic wffs . .
9.2 The quantifier symbols: V,::3 .
9.3 Wffs and sentences . . . . . .
9.4 Semantics for the quantifiers
9.5 The four Aristotelian forms .
9.6 Translating complex noun phrases
9.7 Quantifiers and function symbols (optional)
9.8 Alternative notation (optional) . . . . . . . .
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10 The
10.1
10.2
10.3
10.4
10.5
10.6
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Logic of Quantifiers
Tautologies and quantification
First-order validity and consequence . . . . .
First-order equivalence and DeMorgan's laws
Other quantifier equivalences (optional)
The axiomatic met hod (optional) .
Lemmas . . . . . .
11 Multiple Quantifiers
11.1 Multiple uses of a single quantifier
11.2 Mixed quantifiers . . . . . . . . . .
11.3 The step-by-step met hod of translation
11.4 Paraphrasing Englishparaphrasing English
11.5 Ambiguity and context sensitivity . . . . .
11.6 Translations using function symbols (optional)
11. 7 Prenex form (optional) . . . . .
11.8 Some extra translation problems
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12 Methods of Proof for Quantifiers
12.1 Valid quantifier steps . . . . . . . . . . .
12.2 The method of existential instantiation .
12.3 The method of general conditional proof .
12.4 Proofs involving mixed quantifiers
12.5 Axiomatizing shape (optional)
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13 Formal Proofs and Quantifiers
13.1 Universal quantifier rules .
13.2 Existential quantifier rules
13.3 Strategy and tactics . . . .
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13.4 Soundness and completeness (optional)
13.5 Some review exercises (optional) . .
14 More about Quantification (optional)
14.1 Numerical quantification
14.2 Proving numerical claims . . . .
14.3 The, both, and neither . . . . . .
14.4 Adding other determiners to FOL
14.5 The logic of generalized quantification
14.6 Other expressive limitations of first-order logic
III
CONTENTS
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Applications and Metatheory
15 First-order Set Theory
15.1 Naive set theory ..
15.2 The empty set, singletons and pairs
15.3 Subsets . . . . . . . . .
15.4 Intersection and union . . . . . .
15.5 Ordered Pairs . . . . . . . . . .
15.6 Modeling relations in set theory
15.7 Functions . . . . . . . . . . . . .
15.8 The powerset of a set (optional)
15.9 Russell's Paradox (optional)
15.10 Zermelo Frankel set theory (ZFC) (optional)
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16 Mathematical Induction
16.1 Inductive definitions and inductive proofs
16.2 Inductive definitions in set theory . . . .
16.3 Induction on the natural numbers . . . .
16.4 Axiomatizing the natural numbers (optional)
16.5 Induction in Fitch . . . . . . . . . . . . .
16.6 Ordering the Natural Numbers (optional)
16.7 Strong Induction (optional) . . . . . . .
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17 Advanced Topics in Propositional Logic
17.1 Truth assignments and truth tables
17.2 Completeness for propositional logic
17.3 Hom sentences (optional)
17.4 Resolution (optional) . . . . . . . .
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18 Advanced Topics in FOL
18.1 First-order structures .
18.2 Truth and satisfaction, revisited
18.3 Soundness for FOL . . . . . . . .
18.4 The completeness of the shape axioms (optional)
18.5 Skolemization (optional) . . . .
18.6 Unification of terms (optional)
18.7 Resolution, revisited (optional)
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19 Completeness and Incompleteness
19.1 The Completeness Theorem for FOL
19.2 Adding witnessing constants
19.3 The Henkin theory . . . . .
19.4 The Elimination Theorem ..
19.5 The Henkin Construction ..
19.6 The Lowenheim-Skolem Theorem .
19.7 The Compactness Theorem . . . .
19.8 The Godel lncompleteness Theorem
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Summary of Formal Proof Rules
Propositional rules
First-order rules . . . . . . . . . .
lnduction rule s . . . . . . . . . . .
lnference Procedures (Con Rules)
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Glossary
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File Index
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Exercise Index
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General Index
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