An Introduction to Formal Logic

Cover

About the Book and the Author

An Introduction to Formal Logic

© Peter Smith 2003
     ISBN 978-0-521-80133-3 hardback
     ISBN 978-0-521-00804-4 paperback

Contents

Preface

1 What is logic?
     1.1 What is an argument?
     1.2 What sort of evaluation?
     1.3 Deduction vs. induction
     1.4 More examples
     1.5 The systematic evaluation of arguments
     1.6 Summary
     Exercises 1

2 Validity and soundness
     2.1 Validity and possibility
     2.2 What\'s the use of deduction?
     2.3 The invalidity principle
     2.4 Inferences and arguments
     2.5 What sort of thing are premisses and conclusions?
     2.6 Summary
     Exercises 2

3 Patterns of inference
     3.1 More patterns
     3.2 Three simple points about inference patterns
     3.3 Generality and topic neutrality
     3.4 Arguments instantiate many patterns
     3.5 \'Logical form\'
     3.6 \'Arguments are reliable in virtue of their form\'
     3.7 Summary
     Exercises 3

4 The counterexample technique
     4.1 The technique illustrated
     4.2 More illustrations
     4.3 The technique described
     4.4 More examples
     4.5 Countering the counterexample technique?
     4.6 Summary
     Exercises 4

5 Proofs
     5.1 Two sample proofs
     5.2 Fully annotated proofs
     5.3 Enthymemes
     5.4 Reduction arguments
     5.5 Limitations
     5.6 Summary
     Exercises 5

6 Validity and arguments
     6.1 Classical validity again
     6.2 Sticking with the classical definition
     6.3 Multi-step arguments again
     6.4 Summary

Interlude: Logic, formal and informal

7 Three propositional connectives
     7.1 \'And\', \'or\' and \'not\'
     7.2 Quirks of the vernacular
     7.3 Formalization
     7.4 The design brief for PL
     7.5 Some simple examples
     7.6 Summary
     Exercises 7

8 The syntax of PL
     8.1 Syntactic rules for PL
     8.2 Construction trees
     8.3 Main connectives
     8.4 Sub formulae and scope
     8.5 Bracketing styles
     8.6 A final brief remark on symbolism
     8.7 Summary
     Exercises 8

9 The semantics of PL
     9.1 Interpreting atomic wffs
     9.2 Interpreting molecular wffs
     9.3 Valuations
     9.4 Evaluating complex wffs
     9.5 Calculating truth-values
     9.6 Three points about valuations
     9.7 Short working
     9.8 Summary
     Exercises 9

10 \'A\'s and \'B\'s, \'P\'s and \'Q\'s
     10.1 Styles of variable: our conventions
     10.2 Basic quotation conventions
     10.3 A more complex convention
     10.4 Summary
     Exercises 10

11 Truth functions
     11.1 Truth-functional vs. other connectives
     11.2 A very brief word about \'functions\'
     11.3 Full truth-tables
     11.4 \'Possible valuations\'
     11.5 Short cuts
     11.6 Truth-functional equivalence
     11. 7 Expressive adequacy
     11.8 \'Disjunctive normal form\'
     11.9 Other adequate sets of connectives.
     11.10 Summary
     Exercises 11

12 Tautologies
     12.1 Tautologies and contradictions
     12.2 Generalizing about tautologies
     12.3 A point about \'form\'
     12.4 Tautologies and necessity
     12.5 A philosophical aside about necessity
     12.6 Summary
     Exercises 12

13 Tautological entailment
     13.1 Two introductory examples
     13.2 Tautological entailment in PL
     13.3 Expressing inferences in PL
     13.4 Truth-table testing in PL
     13.5 Vernacular arguments again
     13.6 \'Validity in virtue of form\'
     13.7 \',,\"\' and \'.\'.\'
     13.8 Some simple metalogical results
     13.9 Summary
     Exercises 13

Interlude: Propositional logic

14 PLC and the material conditional
     14.1 Why look for a truth-functional conditional?
     14.2 Introducing the material conditional
     14.3 \':J\' is conditional-like
     14.4 \'If\', \'only if\', and \'if and only if\'
     14.5 The official syntax and semantics of PLC
     14.6 \'F, \':.\', \'::J\', \'=\', and \'-\'
     14.7 Summary
     Exercises 14

15 More on the material conditional
     15.1 Types of conditional
     15.2 In support of the material conditional
     15.3 Against identifying vernacular and material conditionals
     15.4 Robustness
     15.5 \'Dutchman\' conditionals
     15.6 Summary
     Exercises 15

16 Introducing PL trees
     16.1 \'Working backwards\'
     16.2 Branching cases
     16.3 Signed and unsigned trees
     16.4 More examples
     16.5 Summary
     Exercises 16

17 Rules for PL trees
     17.1 The official rules
     17.2 Tactics for trees
     17.3 More examples
     17.4 Testing for tautologies
     17.5 Comparative efficiency
     17.6 Summary
     Exercises 17

18 PLC trees
     18.1 Rules for PLC trees
     18.2 Examples
     18.3 An invitation to be lazy
     18.4 More translational issues
     18.5 Summary
     Exercises 18

19 PL trees vindicated
     19.1 The tree method is sound
     19.2 The tree method is complete
     19.3 A corollary and a further result
     19.4 Summary
     Exercises 19

20 Trees and proofs
     20.1 Choices, choices ...
     20.2 Trees as arguments in PL
     20.3 More natural deductions?
     2004 What rules for trees?
     20.5 \'P and \'r\', soundness and completeness
     20.6 Summary

Interlude: After propositional logic

21 Quantifiers
     21.1 Quantifiers in arguments
     21.2 Quantifiers in ordinary language
     21.3 Quantifiers and scope
     21.4 Expressing quantification unambiguously
     21.5 Summary

22 QL introduced
     22.1 Names and predicates
     22.2 Connectives in QL
     22.3 Adding the quantifiers
     22.4 Domains, and named vs. nameless things
     22.5 Summary
     Exercises 22

23 QL explored
     23.1 The quantifiers interrelated
     23.2 Expressing restricted quantifications
     23.3 Existential import
     23.4 More on variables
     23.5 \'Revealing logical form\'
     23.6 Summary
     Exercises 23

24 More QL translations
     24.1 Translating English into QL
     24.2 Translating from QL
     24.3 Moving quantifiers
     24.4 Summary
     Exercises 24

25 Introducing QL trees
     25.1 The V-instantiation rule
     25.2 Rules for negated quantifiers
     25.3 The 3-instantiation rule
     25.4 More examples
     25.5 Open and closed trees
     25.6 Summary

     Exercises 25

26 The syntax of QL
     26.1 How not to run out of constants, predicates or variables
     26.2 How to introduce quantifiers
     26.3 The official syntax
     26.4 Some useful definitions
     26.5 Summary
     Exercises 26

27 Q-valuations
     27.1 Q-valuations vs. interpretations
     27.2 Q-valuations defined
     27.3 The semantics of quantifiers: a rough guide
     27.4 The official semantics
     27.5 A toy example
     27.6 Five results about (extended) q-valuations
     27.7 Summary
     Exercises 27

28 Q-validity
     28.1 Q-validity defined
     28.2 Some simple examples of q-validity
     28.3 Thinking about trees again
     28.4 Validity, q-validity, and \'quantification logical form\'
     28.5 The undecidability of q-validity
     28.6 Countermodels and invalidity
     28.7 Summary
     Exercises 28

29 More on QL trees
     29.1 The official rules
     29.2 Further examples of closed trees
     29.3 Extending the (V) rule
     29.4 What can be learnt from open trees?
     29.5 Summary
     Exercises 29

30 QL trees vindicated
     30.1 Soundness
     30.2 Completeness: strateg
     30.3 Consistent, saturated sets are satisfiable
     30.4 Systematic trees
     30.5 Completeness completed
     30.6 Summary

Interlude: Developing predicate logic

31 Extensionality
     31.1 Interpretations vs. valuations
     31.2 Extensional and intensional contexts
     31.3 Quotation
     31.4 Intentional contexts are intensional
     31.5 Modal contexts are intensional
     31.6 Summary

32 Identity
     32.1 Numerical vs. qualitative identity
     32.2 Equivalence relations
     32.3 The \'smallest\' equivalence relation
     32.4 Leibniz\'s Law
     32.5 Leibniz\'s Law and co-referential designators
     32.6 Summary
     Exercises 32

33 The language QL =
     33.1 Adding identity to QL
     33.2 Translating into QL =
     33.3 Numerical quantifiers
     33.4 Summary
     Exercises 33

34 Descriptions and existence
     34.1 Definite descriptions
     34.2 Descriptions and scope
     34.3 More translations
     34.4 Existence statements
     34.5 Summary
     Exercises 34

35 Trees for identity
     35.1 Leibniz\'s Law again
     35.2 Self-identity
     35.3 Descriptions again
     35.4 \'One and one make two\'
     35.5 Soundness and completeness again
     35.6 Summary
     Exercises 35

36 Functions
     36.1 Functions re-introduced
     36.2 Adding functions to QL =
     36.3 Functions and functional relations
     36.4 Partial functions and free logic
     36.5 Definite descriptions again
     36.6 Summary

Further reading
     Matters arising
     Other texts

Index

Back Cover

Better Copy Available: http://libgen.org/book/index.php?md5=c7d58ef8901bb486ff0cf780cbfeb673&open=0