In formal logic, sentences and arguments are translated into mathematical languages with well-defined properties. If all goes well, properties of the argument that were hard to discern become clearer.

This text describes two formal languages which have been of special importance to philosophers: truth-functional sentential logic and quantified predicate logic. The book covers translation, formal semantics, and proof theory for both languages.

This can be used as the textbook for a semester long course in logic, for a unit on logic, or for self-directed study. Each chapter contains practice exercises; solutions to selected exercises appear in an appendix.

The author is an assistant professor of philosophy at the University at Albany, SUNY. (lulu.com)

Chapter 1 What is logic?
Logic is the business of evaluating arguments, sorting good ones from bad ones. In everyday language, we sometimes use the word ‘argument’ to refer to belligerent shouting matches. If you and a friend have an argument in this sense, things are not going well between the two of you.

In logic, we are not interested in the teeth-gnashing, hair-pulling kind of argument. A logical argument is structured to give someone a reason to believe some conclusion. ...

Download forall x: An Introduction to Formal Logic

PDF format, 590KB, 160Pages.

P.D. Magnus
University at Albany, State University of New York

Contents
1 What is logic? 5
1.1 Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Two ways that arguments can go wrong . . . . . . . . . . . . . . 7
1.4 Deductive validity . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Other logical notions . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Sentential logic 17
2.1 Sentence letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Other symbolization . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Sentences of SL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Truth tables 37
3.1 Truth-functional connnectives . . . . . . . . . . . . . . . . . . . . 37
3.2 Complete truth tables . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Using truth tables . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Partial truth tables . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Quantified logic 48
4.1 From sentences to predicates . . . . . . . . . . . . . . . . . . . . 48
4.2 Building blocks of QL . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Translating to QL . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5 Sentences of QL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.6 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Formal semantics 83
5.1 Semantics for SL . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2 Interpretations and models in QL . . . . . . . . . . . . . . . . . . 88
5.3 Semantics for identity . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4 Working with models . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.5 Truth in QL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6 Proofs 107
6.1 Basic rules for SL . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2 Derived rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Rules of replacement . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.4 Rules for quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.5 Rules for identity . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.6 Proof strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.7 Proof-theoretic concepts . . . . . . . . . . . . . . . . . . . . . . . 129
6.8 Proofs and models . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.9 Soundness and completeness . . . . . . . . . . . . . . . . . . . . . 132
Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A Other symbolic notation 140
B Solutions to selected exercises 143
C Quick Reference 156

Visit forall x: An Introduction to Formal Logic Website

forall x is an introduction to sentential logic and first-order predicate logic with identity, logical systems that significantly influenced twentieth-century analytic philosophy. After working through the material in this book, a student should be able to understand most quantified expressions that arise in their philosophical reading.

In the Introduction to his volume Symbolic Logic, Charles Lutwidge Dodson advised: “When you come to any passage you don’t understand, read it again: if you still don’t understand it, read it again: if you fail, even after three readings, very likely your brain is getting a little tired. In that case, put the book away, and take to other occupations, and next day, when you come to it fresh, you will very likely find that it is quite easy.”

The same might be said for this volume, although the reader is forgiven if they take a break for snacks after two readings.

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