# Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions

###### Eric Schechter
Publisher:
Princeton University Press
Publication Date:
2005
Number of Pages:
507
Format:
Hardcover
Price:
79.50
ISBN:
0-691-12279-2
Category:
General
[Reviewed by
Wesley Calvert
, on
06/14/2006
]

The goal of any logic is to describe the arithmetic of truth and falsity. The most basic sort of logic, propositional logic, allows the operations "AND," "OR," "NOT" and "IF… THEN…" Predicate logic adds the quantifiers "FOR ALL" and "THERE EXISTS," and many more extensions are possible. The present book, and consequently this review, is mostly concerned with propositional logics.

By this point, most readers are probably somewhat disquieted by the reference to "logics" in the plural. A "logic," strictly speaking, is made up of the following:

1. A collection statements, to which truth values can be assigned. They may be either atomic, or may involve logical operations like "AND," etc.
2. A collection of truth values. Classically, there are two options, "TRUE" and "FALSE," although more are sometimes used, on which we will say more later.
3. An interpretation of the logical operations, explaining how, for instance, from the truth values of P and Q, to compute the truth value of "IF P THEN Q."
4. A notion of logical consequence, given as a set of "rules of inference." Perhaps the most classical of these is the Modus Ponens: "From the fact that both 'P' and 'IF P THEN Q' are true, we may conclude that Q is true."

When most mathematicians (indeed, most logicians) talk about logic, they mean Classical Logic. This logic admits only the two familiar truth values. Each statement has exactly one of the two truth values (the Law of the Excluded Middle). Counterfactuals are always true: IF I am writing this review while sitting on the moon, THEN my computer is made of cheese. A second negation exactly negates the first: NOT NOT P is equivalent to P.

While these rules are familiar, and perhaps obvious, to most mathematicians, many alternative logics have been developed over the years for a variety of reasons, some of them quite useful. Traditionally, the applicability in philosophy and, more recently, computer science, have accounted for most of the appeal of the best-known nonclassical logics, but important mathematical work has been done on them.

One standard way to get a nonclassical logic is to change the rules of inference (and, correspondingly, the interpretations of the logical operations). For instance, the constructivists of the early twentieth century showed that a great deal of mathematics could be carried out without proof by contradiction. The restriction of classical logic they used is usually known as "Intuitionistic" logic, although Schechter calls it "Constructive" logic. Other appropriate choices of rules of inference can create logics that are useful, for instance, for software verification.

Another way to get a new logic is to change the set of truth values (and, necessarily, change the interpretations and rules of inference). One example of this is allowing truth values on the interval [0,1], with 0 being completely false, 1 completely true, and a full range in between. This is the beginning of the "Fuzzy" logics often used in engineering.

This wide variety of logics notwithstanding, those of us who have taught introductory logic classes have often found that classical logic is quite enough for students to digest in a semester's time, and so I was skeptical when I saw that Schechter intended to introduce no fewer than thirteen different logics in the book.

My skepticism increased when I saw his warning that the book "may be too elementary for a graduate course" — coupled with an early chapter entitled "Topologies and their interiors." In fairness, Schechter does suggest that the topology could be omitted from the course, and I am in general agreement that it could be. On the other hand, upper level undergraduates could probably learn enough from the topology chapter in a reasonable amount of time to continue with a course that used it.

Still, the book does have a rather serious audience problem. I agree with Schechter's assessment that the book is not nearly deep enough for a graduate course. Very well, then, but what undergraduates should use it? Schechter states that "Regardless of previous background, … any liberal arts undergraduate student might wish to take at least one course in logic." Having taught such a class for first-year liberal arts students while at the University of Notre Dame, I agree. But the mathematical prerequisites of this book seem a bit steep for such an audience. It is difficult to imagine a student understanding much of the book without a background in discrete mathematics at perhaps a sophomore level, and examples from calculus are frequent. To further confuse the issue, the "Introduction for students" explicitly downplays the level of help that the study of logic will give toward either understanding of mathematics or critical thinking. In his explanation of the latter disavowal, his statement that "Any reader who feels emotionally troubled is urged to seek help from a physician, not from this logic book" is especially off-putting, and following an account of Gödel's (rather gruesome) mental illness and death, it seems unlikely to appeal to many of its intended audience.

At the level of an upper-level undergraduate course for math or computer science students, one could teach a very serviceable course from this book. The student would emerge with a superficial but extant understanding of each of several logics (almost certainly not all of them in a semester-long course), and perhaps a strong understanding of how one makes up a logic. The explanation is clear and readable.

There are still a few caveats, though. The organization of the book is perhaps a good deal more logical than psychological, with many chapters beginning with a statement something like, "This chapter should be read simultaneously with the next two chapters — i.e., the reader will have to flip pages back and forth." Although Schechter assures us that "Apparently, this arrangement is unavoidable," I have never seen it before. Also, the terminology is frequently nonstandard, and some explanations cross from clarity and readability into wordiness.

Schechter's "Introduction for teachers" is itself an important document. It contains a learned and appealing argument for "Pluralism," Schechter's name for the inclusion of non-classical logics in early stages of logic education. Most logicians, especially those like me, with a more classical predilection, would find reading this introduction well worth their time.

As a textbook, though, and even more as a reference, this book falls short of the incredibly worthy program Schechter's introduction suggests. In the end, too much is covered too thinly. Very little is said about the applications of any of the logics discussed, and there are so many logics swarming around that I often found myself looking for a scorecard to keep track of which rules of inference were in play and which classical tautologies we were rejecting at any given point.

Still, while there are many good books for an introductory course in classical logic, those who wish to make non-classical logics a part of their course may find this book their best option. Such books are rare, at best, and we may hope that the present book inspires future and more successful attempts to do this.

Wesley Calvert is an assistant professor in the Department of Mathematics & Statistics at Murray State University. His research is in logic and computable algebra.

A Preliminaries 1

1 Introduction for teachers 3

² Purpose and intended audience, 3 ² Topics in the

book, 6 ² Why pluralism?, 13 ² Feedback, 18 ² Acknowledgments, 19

2 Introduction for students 20

² Who should study logic?, 20 ² Formalism and certi¯-

cation, 25 ² Language and levels, 34 ² Semantics and

syntactics, 39 ² Historical perspective, 49 ² Pluralism, 57

² Jarden's example (optional), 63

3 Informal set theory 65

² Sets and their members, 68 ² Russell's paradox, 77 ² Subsets, 79 ² Functions, 84 ² The Axiom of Choice

(optional), 92 ² Operations on sets, 94 ² Venn dia-

grams, 102 ² Syllogisms (optional), 111 ² In¯nite sets

(postponable), 116

4 Topologies and interiors (postponable) 126

² Topologies, 127 ² Interiors, 133 ² Generated topologies

and ¯nite topologies (optional), 139

5 English and informal classical logic 146

² Language and bias, 146 ² Parts of speech, 150 ² Semantic values, 151 ² Disjunction (or), 152 ² Con-

junction (and), 155 ² Negation (not), 156 ² Material

implication, 161 ² Cotenability, fusion, and constants

vi Contents

(postponable), 170 ² Methods of proof, 174 ² Working

backwards, 177 ² Quanti¯ers, 183 ² Induction, 195 ² Induction examples (optional), 199

6 De¯nition of a formal language 206

² The alphabet, 206 ² The grammar, 210 ² Removing

parentheses, 215 ² De¯ned symbols, 219 ² Pre¯x

notation (optional), 220 ² Variable sharing, 221 ² Formula schemes, 222 ² Order preserving or reversing

subformulas (postponable), 228

B Semantics 233

7 De¯nitions for semantics 235

² Interpretations, 235 ² Functional interpretations, 237

² Tautology and truth preservation, 240

8 Numerically valued interpretations 245

² The two-valued interpretation, 245 ² Fuzzy interpre-

tations, 251 ² Two integer-valued interpretations, 258

Sugihara's interpretation, 263

9 Set-valued interpretations 269

² Powerset interpretations, 269 ² Hexagon interpretation

(optional), 272 ² The crystal interpretation, 273 ² Church's diamond (optional), 277

10 Topological semantics (postponable) 281

² Topological interpretations, 281 ² Examples, 282 ² Common tautologies, 285 ² Nonredundancy of sym-

bols, 286 ² Variable sharing, 289 ² Adequacy of ¯nite

topologies (optional), 290 ² Disjunction property (op-

tional), 293

Contents vii

11 More advanced topics in semantics 295

² Common tautologies, 295 ² Images of interpreta-

tions, 301 ² Dugundji formulas, 307

C Basic syntactics 311

12 Inference systems 313

13 Basic implication 318

² Assumptions of basic implication, 319 ² A few easy

derivations, 320 ² Lemmaless expansions, 326 ² De-

tachmental corollaries, 330 ² Iterated implication (post-

ponable), 332

14 Basic logic 336

² Further assumptions, 336 ² Basic positive logic, 339

² Basic negation, 341 ² Substitution principles, 343

D One-formula extensions 349

15 Contraction 351

² Weak contraction, 351 ² Contraction, 355

16 Expansion and positive paradox 357

² Expansion and mingle, 357 ² Positive paradox (strong

expansion), 359 ² Further consequences of positive para-

dox, 362

17 Explosion 365

18 Fusion 369

19 Not-elimination 372

² Not-elimination and contrapositives, 372 ² Interchange-

ability results, 373 ² Miscellaneous consequences of not-

elimination, 375

viii Contents

20 Relativity 377

E Soundness and major logics 381

21 Soundness 383

22 Constructive axioms: avoiding not-elimination 385

² Constructive implication, 386 ² Herbrand-Tarski De-

duction Principle, 387 ² Basic logic revisited, 393 ² Soundness, 397 ² Nonconstructive axioms and classical

logic, 399 ² Glivenko's Principle, 402

23 Relevant axioms: avoiding expansion 405

² Some syntactic results, 405 ² Relevant deduction

principle (optional), 407 ² Soundness, 408 ² Mingle:

slightly irrelevant, 411 ² Positive paradox and classical

logic, 415

24 Fuzzy axioms: avoiding contraction 417

² Axioms, 417 ² Meredith's chain proof, 419 ² Addi-

tional notations, 421 ² Wajsberg logic, 422 ² Deduction

principle for Wajsberg logic, 426

25 Classical logic 430

² Axioms, 430 ² Soundness results, 431 ² Independence

of axioms, 431

26 Abelian logic 437

27 Harrop's principle for constructive logic 443

² Meyer's valuation, 443 ² Harrop's principle, 448 ² The disjunction property, 451 ² Admissibility, 451 ² Results in other logics, 452

Contents ix

28 Multiple worlds for implications 454

² Multiple worlds, 454 ² Implication models, 458 ² Soundness, 460 ² Canonical models, 461 ² Complete-

ness, 464

29 Completeness via maximality 466

² Maximal unproving sets, 466 ² Classical logic, 470

² Wajsberg logic, 477 ² Constructive logic, 479 ² Non-¯nitely-axiomatizable logics, 485

References 487

Symbol list 493

Index 495

A Preliminaries 1

1 Introduction for teachers 3

² Purpose and intended audience, 3 ² Topics in the

book, 6 ² Why pluralism?, 13 ² Feedback, 18 ² Acknowledgments, 19

2 Introduction for students 20

² Who should study logic?, 20 ² Formalism and certi¯-

cation, 25 ² Language and levels, 34 ² Semantics and

syntactics, 39 ² Historical perspective, 49 ² Pluralism, 57

² Jarden's example (optional), 63

3 Informal set theory 65

² Sets and their members, 68 ² Russell's paradox, 77 ² Subsets, 79 ² Functions, 84 ² The Axiom of Choice

(optional), 92 ² Operations on sets, 94 ² Venn dia-

grams, 102 ² Syllogisms (optional), 111 ² In¯nite sets

(postponable), 116

4 Topologies and interiors (postponable) 126

² Topologies, 127 ² Interiors, 133 ² Generated topologies

and ¯nite topologies (optional), 139

5 English and informal classical logic 146

² Language and bias, 146 ² Parts of speech, 150 ² Semantic values, 151 ² Disjunction (or), 152 ² Con-

junction (and), 155 ² Negation (not), 156 ² Material

implication, 161 ² Cotenability, fusion, and constants

vi Contents

(postponable), 170 ² Methods of proof, 174 ² Working

backwards, 177 ² Quanti¯ers, 183 ² Induction, 195 ² Induction examples (optional), 199

6 De¯nition of a formal language 206

² The alphabet, 206 ² The grammar, 210 ² Removing

parentheses, 215 ² De¯ned symbols, 219 ² Pre¯x

notation (optional), 220 ² Variable sharing, 221 ² Formula schemes, 222 ² Order preserving or reversing

subformulas (postponable), 228

B Semantics 233

7 De¯nitions for semantics 235

² Interpretations, 235 ² Functional interpretations, 237

² Tautology and truth preservation, 240

8 Numerically valued interpretations 245

² The two-valued interpretation, 245 ² Fuzzy interpre-

tations, 251 ² Two integer-valued interpretations, 258

Sugihara's interpretation, 263

9 Set-valued interpretations 269

² Powerset interpretations, 269 ² Hexagon interpretation

(optional), 272 ² The crystal interpretation, 273 ² Church's diamond (optional), 277

10 Topological semantics (postponable) 281

² Topological interpretations, 281 ² Examples, 282 ² Common tautologies, 285 ² Nonredundancy of sym-

bols, 286 ² Variable sharing, 289 ² Adequacy of ¯nite

topologies (optional), 290 ² Disjunction property (op-

tional), 293

Contents vii

11 More advanced topics in semantics 295

² Common tautologies, 295 ² Images of interpreta-

tions, 301 ² Dugundji formulas, 307

C Basic syntactics 311

12 Inference systems 313

13 Basic implication 318

² Assumptions of basic implication, 319 ² A few easy

derivations, 320 ² Lemmaless expansions, 326 ² De-

tachmental corollaries, 330 ² Iterated implication (post-

ponable), 332

14 Basic logic 336

² Further assumptions, 336 ² Basic positive logic, 339

² Basic negation, 341 ² Substitution principles, 343

D One-formula extensions 349

15 Contraction 351

² Weak contraction, 351 ² Contraction, 355

16 Expansion and positive paradox 357

² Expansion and mingle, 357 ² Positive paradox (strong

expansion), 359 ² Further consequences of positive para-

dox, 362

17 Explosion 365

18 Fusion 369

19 Not-elimination 372

² Not-elimination and contrapositives, 372 ² Interchange-

ability results, 373 ² Miscellaneous consequences of not-

elimination, 375

viii Contents

20 Relativity 377

E Soundness and major logics 381

21 Soundness 383

22 Constructive axioms: avoiding not-elimination 385

² Constructive implication, 386 ² Herbrand-Tarski De-

duction Principle, 387 ² Basic logic revisited, 393 ² Soundness, 397 ² Nonconstructive axioms and classical

logic, 399 ² Glivenko's Principle, 402

23 Relevant axioms: avoiding expansion 405

² Some syntactic results, 405 ² Relevant deduction

principle (optional), 407 ² Soundness, 408 ² Mingle:

slightly irrelevant, 411 ² Positive paradox and classical

logic, 415

24 Fuzzy axioms: avoiding contraction 417

² Axioms, 417 ² Meredith's chain proof, 419 ² Addi-

tional notations, 421 ² Wajsberg logic, 422 ² Deduction

principle for Wajsberg logic, 426

25 Classical logic 430

² Axioms, 430 ² Soundness results, 431 ² Independence

of axioms, 431

26 Abelian logic 437

27 Harrop's principle for constructive logic 443

² Meyer's valuation, 443 ² Harrop's principle, 448 ² The disjunction property, 451 ² Admissibility, 451 ² Results in other logics, 452

Contents ix

28 Multiple worlds for implications 454

² Multiple worlds, 454 ² Implication models, 458 ² Soundness, 460 ² Canonical models, 461 ² Complete-

ness, 464

29 Completeness via maximality 466

² Maximal unproving sets, 466 ² Classical logic, 470

² Wajsberg logic, 477 ² Constructive logic, 479 ² Non-¯nitely-axiomatizable logics, 485

References 487

Symbol list 493

Index 495

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