An Introduction to Formal Logic

Contents
Preface
1. What is deductive logic?
	1.1 What is an argument?
	1.2 Kinds of evaluation
	1.3 Deduction vs. induction
	1.4 Just a few more examples
	1.5 Generalizing
	1.6 Summary
	Exercises 1
2. Validity and soundness
	2.1 Validity defined more carefully
	2.2 Consistency and equivalence
	2.3 Validity, truth, and the invalidity principle
	2.4 Inferences and arguments
	2.5 'Valid' vs 'true'
	2.6 What's the use of deduction?
	2.7 An illuminating circle?
	2.8 Summary
	Exercises 2
3. Forms of inference
	3.1 More forms of inference
	3.2 Four basic points about the use of schemas
	3.3 Arguments can instantiate many patterns
	3.4 Summary
	Exercises 3
4. Proofs
	4.1 Proofs: First examples
	4.2 Fully annotated proofs
	4.3 Glimpsing an ideal
	4.4 Deductively cogent multi-step arguments
	4.5 Indirect arguments
	4.6 Summary
	Exercises 4
5. The counterexample method
	5.1 'But you might as well argue . . . '
	5.2 The counterexample method, more carefully
	5.3 A 'quantifier shift' fallacy
	5.4 Summary
	Exercises 5
6. Logical validity
	6.1 Topic-neutrality
	6.2 Logical validity, at last
	6.3 Logical necessity
	6.4 The boundaries of logical validity?
	6.5 Definitions of validity as rational reconstructions
	6.6 Summary
	Exercises 6
7. Propositions and forms
	7.1 Types vs tokens
	7.2 Sense vs tone
	7.3 Are propositions sentences?
	7.4 Are propositions truth-relevant contents?
	7.5 Why we can be indecisive
	7.6 Forms of inference again
	7.7 Summary
	Exercises 7
Interlude: From informal to formal logic
8. Three connectives
	8.1 Two simple arguments
	8.2 'And'
	8.3 'Or'
	8.4 'Not'
	8.5 Scope
	8.6 Formalization
	8.7 The design brief for PL languages
	8.8 One PL language
	8.9 Summary
	Exercises 8
9. PL syntax
	9.1 Syntactic rules for PL languages
	9.2 Construction histories, parse trees
	9.3 Wffs have unique parse trees!
	9.4 Main connectives, subformulas, scope
	9.5 Bracketing styles
	9.6 Summary
	Exercises 9
10. PL semantics
	10.1 Interpreting wffs
	10.2 Languages and translation
	10.3 Atomic wffs are true or false
	10.4 Truth values
	10.5 Truth tables for the connectives
	10.6 Evaluating molecular wffs: two examples
	10.7 Uniqueness and bivalence
	10.8 Short working
	10.9 Summary
	Exercises 10
11. 'P's, 'Q's, 'α's, 'β's – and form again
	11.1 Styles of variable: object languages and metalanguages
	11.2 Quotation marks, use and mention
	11.3 To Quine-quote or not to Quine-quote?
	11.4 How strict about quotation do we want to be?
	11.5 Why Greek-letter variables?
	11.6 The idea of form, again
	11.7 Summary
	Exercises 11
12. Truth functions
	12.1 Truth-functional vs other connectives
	12.2 Functions and truth functions
	12.3 Truth tables for wffs
	12.4 'Possible valuations'
	12.5 Summary
	Exercises 12
13. Expressive adequacy
	13.1 Conjunction and disjunction interrelated
	13.2 Exclusive disjunction
	13.3 Another example: expressing the dollar truth function
	13.4 Expressive adequacy defined
	13.5 Some more adequacy results
	13.5 Some more adequacy results
	13.6 Summary
	Exercises 13
14. Tautologies
	14.1 Tautologies and contradictions
	14.2 Generalizing examples of tautologies
	14.3 Tautologies, necessity, and form
	14.4 Tautologies as analytically true
	14.5 Summary
	Exercises 14
15. Tautological entailment
	15.1 Three introductory examples
	15.2 Tautological entailment defined
	15.3 Brute-force truth-table testing
	15.4 More examples
	15.5 Ordinary-language arguments again
	15.6 Tautological consistency and tautological validity
	15.7 Summary
	Exercises 15
16. More about tautological entailment
	16.1 Extending the notion of tautological entailment
	16.2 Can there be a more efficient test?
	16.3 Truth-table testing and the counterexample method
	16.4 '⊨' and '∴'
	16.5 Generalizing examples of tautological entailment
	16.6 Tautological entailment and form
	16.7 Tautological equivalence as two-way entailment
	16.8 Summary
	Exercises 16
17. Explosion and absurdity
	17.1 Explosion!
	17.2 The falsum as an absurdity sign
	17.3 Adding the falsum to PL languages
	17.4 Summary
	Exercises 17
18. The truth-functional conditional
	18.1 Some arguments involving conditionals
	18.2 Four basic principles
	18.3 Introducing the truth-functional conditional
	18.4 Ways in which '→' is conditional-like
	18.5 'Only if'
	18.6 The biconditional
	18.7 Extended PL syntax and semantics, officially
	18.8 Contrasting '∴' and '⊨' and '→'
	18.9 Summary
	Exercises 18
19. 'If's and '→'s
	19.1 Types of conditional
	19.2 Simple conditionals as truth-functional: for
	19.3 Another kind of case where `if' is truth-functional
	19.4 Simple conditionals as truth-functional: against
	19.5 Three responses
	19.6 Adopting the material conditional
	19.7 Summary
	Exercises 19
Interlude: Why natural deduction?
20. PL proofs: conjunction and negation
	20.1 Rules for conjunction
	20.2 Rules for negation
	20.3 A double negation rule
	20.4 A more complex proof: thinking strategically
	20.5 Understanding proofs, discovering proofs
	20.6 'Given'
	20.7 'We can derive'
	20.8 Putting things together
	20.9 Explosion and absurdity again
	20.10 Summary
	Exercises 20
21. PL proofs: disjunction
	21.1 The iteration rule
	21.2 Introducing and eliminating disjunctions
	21.3 The disjunction rules, a diagrammatic summary
	21.4 Two more proofs
	21.5 Disjunctive syllogisms
	21.6 Summary
	Exercises 21
22. PL proofs: conditionals
	22.1 Rules for the conditional
	22.2 More proofs with conditionals
	22.3 The material conditional again
	22.4 Summary
	Exercises 22
23. PL proofs: theorems
	23.1 Theorems
	23.2 Derived rules
	23.3 Excluded middle and double negation
	23.4 Summary
	Exercises 23
24. PL proofs: metatheory
	24.1 Metatheory
	24.2 Putting everything together
	24.3 Vacuous discharge
	24.4 Generalizing PL proofs
	24.5 '⊨' and '⊢'
	24.6 Soundness
	24.7 Completeness
	24.8 Double negation and excluded middle again
	24.9 Summary
Interlude: Formalizing general propositions
25. Names and predicates
	25.1 Names and other 'terms'
	25.2 Predicates and their 'arity'
	25.3 Predicates, properties and relations
	25.4 Predicates: sense vs extension
	25.5 Sets
	25.6 Names: sense vs reference
	25.7 Reference, extension, and truth
	25.8 Summary
26. Quantifiers in ordinary language
	26.1 Which quantifiers?
	26.2 Every/any/all/each
	26.3 Quantifiers and scope
	26.4 Fixing domains
	26.5 Summary
	Exercises 26
27. Quantifier-variable notation
	27.1 Quantifier prefixes and 'variables' as pronouns
	27.2 Unary vs binary quantifiers
	27.3 Domains
	27.4 Quantifier symbols
	27.5 Unnamed objects
	27.6 A variant notation
	27.7 Summary
28. QL languages
	28.1 QL languages – a glimpse ahead
	28.2 Names, predicates and atomic wffs in QL: syntax
	28.3 Names, predicates, and atomic wffs in QL: interpretation
	28.4 One example: introducing QL₁
	28.5 Adding the connectives
	28.6 Syntax for the quantifiers
	28.7 An aside on scope again
	28.8 Interpreting the quantifiers
	28.9 Quantifier equivalences
	28.10 Summary
	Exercises 28
29. Simple translations
	29.1 Restricted quantifiers revisited
	29.2 Existential import
	29.3 'No'
	29.4 Translating via Loglish
	29.5 Translations into QL₂
	29.6 Moving quantifiers
	29.7 Summary
	Exercises 29
30. More on translations
	30.1 More translations into QL₂
	30.2 Translations from QL₂
	30.3 Choosing a domain
	30.4 'Translation' and 'logical form'
	30.5 Summary
	Exercises 30
Interlude: Arguing in QL
31. Informal quantifier rules
	31.1 Arguing with universal quantifiers
	31.2 Arguing with existential quantifiers
	31.3 Summary
	Exercises 31
32. QL proofs
	32.1 Dummy names in QL languages
	32.2 Schematic notation, and instances of quantified wffs
	32.3 Inference rules for '∀'
	32.4 Inference rules for '∃'
	32.5 Quantifier equivalences
	32.6 QL theorems
	32.7 Summary
	Exercises 32
33. More QL proofs
	33.1 The QL rules again
	33.2 How to misuse the QL rules
	33.3 Old and new logic: three proofs
	33.4 Five more QL proofs
	33.5 Summary
	Exercises 33
34. Empty domains?
	34.1 Dummy names and empty domains
	34.2 Preserving standard logic
	34.3 Summary
35. Q-valuations
	35.1 Q-valuations defined
	35.2 QL syntax and q-parse trees
	35.3 Evaluating quantified wffs: the headlines
	35.4 The official valuational semantics
	35.5 Toy examples
	35.6 Uniqueness of values
	35.7 The structure of valuations
	35.8 Summary
	Exercises 35
36. Q-validity
	36.1 Q-validity defined
	36.2 'All q-valuations'
	36.3 Establishing q-validity/q-invalidity: the headlines
	36.4 We can mechanically test for q-validity in simple cases
	36.5 'Working backwards'
	36.6 The Entscheidungsproblem
	36.7 Generalizing again
	36.8 Summary
	Exercises 36
37. QL proofs: metatheory
	37.1 The QL proof system reviewed
	37.2 Generalizing QL proofs
	37.3 Two turnstiles again
	37.4 Soundness
	37.5 Completeness
	37.6 Summary
Interlude: Extending QL
38. Identity
	38.1 Numerical vs qualitative identity
	38.2 Equivalence relations
	38.3 Identity as the smallest equivalence relation
	38.4 Leibniz's Law
	38.5 Summary
	Exercises 38
39. QL⁼ languages
	39.1 '=' as the identity predicate
	39.2 Translating into QL⁼
	39.3 Numerical quantifiers
	39.4 Existence claims
	39.5 Summary
	Exercises 39
40. Definite descriptions
	40.1 The project
	40.2 Russell's Theory of Descriptions
	40.3 Descriptions and existence
	40.4 Descriptions and scope
	40.5 More translations
	40.6 Summary
	Exercises 40
41. QL⁼ proofs
	41.1 Two derivation rules for identity
	41.2 More examples
	41.3 One and one makes two
	41.4 Metatheoretical headlines
	41.5 Summary
	Exercises 41
42. Functions
	42.1 Functions, informally again
	42.2 Function symbols, syntax
	42.3 Function symbols, semantics
	42.4 Functions, functional relations, and definite descriptions again
	42.5 Proofs involving functions
	42.6 ω-incompleteness!
	42.7 And where now?
	Exercises 42
Appendix: Soundness and completeness
	A1. Soundness for PL
	A2. Soundness for QL
	A3. Completeness: what we want to prove
	A4. PL completeness proved
	A5. QL completeness proved
	A6. A squeezing argument
The Greek alphabet
Further reading
	Parallel reading
	Philosophical matters arising
	Going further in formal logic
Index