Multiple-conclusion logic extends formal logic by allowing arguments to have a set of conclusions instead of a single one, the truth lying somewhere among the conclusions if all the premises are true. The extension opens up interesting possibilities based on the symmetry between premises and conclusions, and can also be used to throw fresh light on the conventional logic and its limitations. This is a sustained study of the subject and is certain to stimulate further research. Part I reworks the fundamental ideas of logic to take account of multiple conclusions, and investigates the connections between multiple - and single - conclusion calculi. Part II draws on graph theory to discuss the form and validity of arguments independently of particular logical systems. Part III contrasts the multiple - and the single - conclusion treatment of one and the same subject, using many-valued logic as the example; and Part IV shows how the methods of 'natural deduction' can be matched by direct proofs using multiple conclusions.
Author(s): D. J. Shoesmith, T. J. Smiley
Publisher: CUP
Year: 1980
Language: English
Pages: 411
Title ......Page 3
Copyright ......Page 4
Contents ......Page 5
Preface ......Page 9
Introduction ......Page 15
PART I. MULTIPLE AND SINGLE CONCLUSIONS ......Page 23
1.1 Consequence ......Page 25
1.2 Rules of inference ......Page 35
1.3 Sequence proofs ......Page 39
2.1 Consequence ......Page 42
2.2 Compactness ......Page 50
2.3 Rules of inference ......Page 53
3.1 Definition ......Page 56
3.2 Extended proofs ......Page 61
3.3 Adequacy ......Page 65
4.1 Multiple-conclusion calculi ......Page 71
4.2 Single-conclusion calculi ......Page 76
4.3 Decidability ......Page 83
5.1 The range of counterparts ......Page 86
5.2 Compactness ......Page 89
5.3 Axiomatisability ......Page 94
5.4 Sign ......Page 96
5.5 Disjunction ......Page 104
6.1 Infinite rules and proofs ......Page 109
6.2 Marking of theorems ......Page 112
PART II. GRAPH PROOFS ......Page 117
7.1 Arguments ......Page 119
7.2 Graphs ......Page 122
7.3 Premisses and conclusions ......Page 125
7.4 Validity and form ......Page 128
7.5 Subproofs ......Page 137
7.6 Symmetry ......Page 143
8.1 Developments ......Page 147
8.2 Validity ......Page 151
8.3 Inadequacy ......Page 154
9.1 Junction and election ......Page 162
9.2 Cross-referenced Kneale proofs ......Page 168
9.3 Cross-referenced circuits ......Page 172
9.4 Other non-abstract proofs ......Page 177
10.1 Cornered-circuit proofs ......Page 181
10.2 Conciseness ......Page 188
10.3 Relevance ......Page 195
10.4 Articulated tree proofs ......Page 201
10.5 Abstract proofs in general ......Page 204
11.1 Single-conclusion arguments ......Page 208
11.2 Proofs with circuits or corners ......Page 212
11.3 Articulated sequence proofs ......Page 216
11.4 Hilbert proofs ......Page 220
12.1 Infinite arguments ......Page 226
12.2 Kneale proofs ......Page 233
12.3 Cornered-circuit proofs ......Page 246
12.4 Marking of theorems ......Page 250
PART III. MANY-VALUED LOGIC ......Page 257
13.1 Definitions ......Page 259
13.2 Examples ......Page 261
13.3 Compactness ......Page 265
14.1 Matrix functions ......Page 270
14.2 Separability ......Page 272
14.3 Equivalence ......Page 276
14.4 Monadicity ......Page 279
15.1 Cancellation ......Page 284
15.2 Compact calculi ......Page 287
15.3 Stability ......Page 291
16.1 Many-valued counterparts ......Page 297
16.2 Principal matrices ......Page 307
17.1 Multiple-conclusion calculi ......Page 311
17.2 Examples ......Page 316
17.3 Single-conclusion calculi ......Page 320
18.1 Axiomatisation ......Page 326
18.2 Duality ......Page 328
18.3 Counterparts ......Page 330
19.1 Finite axiomatisation ......Page 339
19.2 Monadic matrices ......Page 340
19.3 Examples ......Page 344
19.4 Limitations ......Page 348
19.5 The general case ......Page 351
19.6 Further examples ......Page 355
19.7 Single-conclusion calculi ......Page 361
19.8 Rosser and Turquette ......Page 369
PART IV. NATURAL DEDUCTION ......Page 373
20.1 Proof by cases ......Page 375
20.2 Classical predicate calculus ......Page 380
20.3 Intuitionist propositional calculus ......Page 388
Bibliography ......Page 400
Index ......Page 404
