symbolic logic.pdf

An Accessible Introduction to Serious Mathematical Logic
Tony Roy
version 7.1
pages - 1003
March 19, 2016

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Preface.................................................. i
Contents .................................................v
Named Definitions ........................................ix
Quick Reference Guides ..................................xvii
I The Elements: Four Notions of Validity ...................1
1 Logical Validity and Soundness ............................4
1.1 Consistent Stories . . . . . . . . . . . . . . . . . . . 5
1.2 The Definitions . . . . . . . . . . . . . . . . . . . . 10
1.3 Some Consequences . . . . . . . . . . . . . . . . . . . 22
2 Formal Languages......................................... 30
2.1 Sentential Languages . . . .. . . . . . . . . . . . . . 31
2.2 Quantificational Languages . . . . . . . . . . . . . . 46
3 Axiomatic Deduction...................................... 67
3.1 General . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2 Sentential . . . . . . . . . . . . . . . . . . . . . . 72
3.3 Quantificational . . . . . . . . . . . . . . . . . . . 80
4 Semantics ................................................96
4.1 Sentential . . . . . . . .. . . . . . . . . . . . . . . 96
4.2 Quantificational . . . . . . . . . .... . . . . . . . .113
5 Translation .............................................138

5.1 General . . . . . . . . . . . . . . . . . . . . . . . 138
5.2 Sentential . . . . . . . . . . . . . . . . . . . . . . 140
5.3 Quantificational . . . . . . . . . . . . . . . . . . . 170
6 Natural Deduction....................................... 208
6.1 General . . . . . . .... . . . . . . . . . . . . . . . 208
6.2 Sentential . . . . . . . . . . . . . . . . . . . . . . 218
6.3 Quantificational . . . . . ..... . . . . . . . . . . . 270
6.4 The system ND+ . . . . . . . . . ... . . . . . . . . . 318
II Transition: Reasoning About Logic ......................329
7 Direct Semantic Reasoning ...............................331
7.1 General . . . . . . . . . . . . . . . . . . . . . . . 332
7.2 Sentential . . . . . . . . . . . . . . . . . . . . . . 335
7.3 Quantificational . . . . . . . . . . . . . . . . . . . 350
8 Mathematical Induction ..................................378
8.1 General Characterization . . . . . . . . . . . . . . . 378
8.2 Preliminary Examples . . . . . . . . . . . . . . . . . 384
8.3 Further Examples (for Part III) . . .. . . . . . . . . 397
8.4 Additional Examples (for Part IV) . . . . . . . . . . 407
III Classical Metalogic: Soundness and Adequacy ...........423
9 Preliminary Results .....................................426
9.1 Semantic Validity Implies Logical Validity . . . . . . 426
9.2 Validity in AD Implies Validity in ND .. . . . . . . . 431
9.3 Validity in ND Implies Validity in AD . . . . . . . . 438
9.4 Extending to ND+ . . . . . . ... . . . . . . . . . . . 459
10 Main Results........................................... 464
10.1 Soundness . . . . . . . . . . . . . . . . . . . . . .465
10.2 Sentential Adequacy . . . . . . . . . . . . . . . . . 472
10.3 Quantificational Adequacy: Basic Version . . . . . . 483
10.4 Quantificational Adequacy: Full Version . . . . . . . 498
11 More Main Results ......................................514