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Classical & Nonclassical Logics
an introduction to the mathematics of propositions

October 2005 -- by Eric Schechter (Vanderbilt University)
available from Princeton University Press (not from the author!)
0-691-12279-2 (ISBN-10)
978-0-691-12279-3 (ISBN-13)
online list of addenda and known errata

Prospective reviewers and teachers should contact for copies.

This is an introduction to mathematical logic, intended for undergraduate students with no previous knowledge of logic. This treatment replaces the traditional depth-first approach with a breadth-first (but not shallow!) approach which I believe is more intuitive and understandable for the students.

Free excerpt (471 kb PDF): Chapter 1, the introduction for teachers, discusses further how this book differs from other logic books. Chapter 2, the introduction for students, might be useful by itself as a brief introduction to logic (regardless of whether you buy the book).

Also downloadable: The most recent version of a 1-hour introductory talk that I have presented to some undergraduate math clubs. For TeX enthusiasts, here's the source file.

Should we use this book for our "how to do proofs" course?   No, I recommend against using any logic book for that purpose. I'd explain that with this analogy: A book on organic chemistry will tell you "what is really going on" in your kitchen, but a conventional cookbook is more useful if you want to learn to prepare food. Likewise, a course in logic will explain "why proofs work the way they do" -- but for the skills of doing ordinary mathematical proofs, I'd recommend some ordinary math (e.g., general topology), taught using the Moore method.


CAUTION: This book does not fit the traditional syllabus for an introductory logic course. It should only be adopted by teachers who are aware of the differences and are persuaded of this book's advantages.

Before the 20th century, mathematicians saw their work mostly as empirical or descriptive, an uncovering of the laws of the physical universe; each question had only one correct answer. Logic was seen as an explanation of reasoning, and few people could see room for more than one explanation. So-called "classical" logic, developed by Frege, Russell, and others, was the dominant paradigm of logic. Well into the late 20th century, the "one logic only" viewpoint persisted, though a few isolated mathematicians -- Brouwer, Lukasiewicz, Belnap, etc. -- pursued different, nonclassical visions of what "the one true logic" should be.

The formalist revolution began with non-Euclidean geometries in the late 19th century and spread gradually through mathematics. Mathematicians began to see that the physical universe is neither knowable nor necessary to their work. Mathematics came to be viewed as a game of marks on paper, or an investigation of pure thought; different assumptions yield different theories. By the late 20th century, this change of philosophy had spread to logic. Researchers saw different logics merely as different types of abstract structures, no longer requiring philosophical justifications. Gradually, without fanfare, the pluralist (many logics) viewpoint became prevalent in research literature.

It is understandable that textbooks change slowly and lag behind research. Most new research discoveries are too specialized and advanced to be appropriate for beginners. Thus, most of today's introductory logic textbooks still concentrate on just classical logic, focusing mainly on ideas of Gödel (1930's) and earlier. Nonclassical logics can be found only in more advanced books and in research journals, heavily algebraic and inaccessible to beginners. But I believe the pluralist approach to logic is important enough, and can be reformulated to be elementary enough, for beginners; that is the goal of my book. I am attempting to bring to the beginners' classroom a revolution which occurred in the research literature during the last few decades.

A first course in logic should aim at teaching, not just some particular logic or logics, but some deeper ideas that apply to all logics --- interpretation, derivation, completeness, etc. The traditional and pluralist approaches share those deeper goals. But on a more superficial level, the traditional and pluralist approaches have rather little overlap --- see the diagram at right. The pluralist approach gains breadth by sacrificing some depth. Of course, a logician needs both breadth and depth, but both cannot be acquired in the first semester. I am convinced that the breadth-first approach is better for most students, not only for obvious philosophical reasons, but also for pedagogical reasons outlined below.

Thought is so complicated that we may never understand it completely, but some of its ingredients are simple enough to identify and analyze. Ordinary human thought (including that of mathematicians!) is not just classical logic, but a mixture of many logics. Here are some examples:

  •  P   Q  P → Q 

     0   0 
     0   1 
     1   0 
     1   1 
    Classical logic is computationally the simplest of all the major logics. The semantics of classical propositional logic can be described just in terms of tables of 0's and 1's; for instance, the table for classical implication is shown at right. Because the semantics is so simple, the traditional classical-only textbook devotes only a few pages to semantics.
            Classical logic is adequate for the needs of mathematics, but it bears little resemblance to the "implications" of everyday nonmathematical English; it disregards causality, relevance, and other ingredients of everyday reasoning. Presented by itself, the little table at right seems highly arbitrary and artificial to beginners.

  • Relevant logic avoids implications with superfluous hypotheses or other irrelevant clauses -- e.g., as in the formula P → (Q → P) or the statement "If today is Tuesday, then a2 + b2 = c2 is satisfied by right triangles." Such clauses are permitted by classical logic, but they are avoided by most people in everyday conversation. They are also avoided by most mathematicians, who believe themselves to be practicing classical logic plus "good taste." Relevant logic codifies some of that good taste.

  • Constructive logic is known in some of the literature by the less descriptive term "intuitionistic logic." It is the type of reasoning used by constructivists, mathematicians who feel mathematics is (or should be) a collection of procedures, not just statements. For instance, most mathematicians would agree that "Goldbach's conjecture is true or Goldbach's conjecture is false," but constructivists see that statement as nonsense. In their view, what really matters is that it is not yet true that
    We know how to prove Goldbach's conjecture or we know how to disprove Goldbach's conjecture.
    These examples illustrate that the Law of the Excluded Middle, P∨ ¬P (P or not P), is a theorem of classical logic but not of constructive logic. The root of the difference is not in ¬ (negation) or ∨ (disjunction), but in P itself:   Is P a statement or a procedure?

  • Fuzzy logic deals with statements whose values need not be absolutely false (0) or true (1), but may lie somewhere in between. For instance, the statement
    removing one hair from a non-bald man leaves a non-bald man
    is mostly true, though it fails if you iterate it a few million times. Fuzzy logic is used by engineers in the design of thermostats, automobile cruise controls, and other devices to make the best possible use of unavoidably imprecise data. Fuzzy logic (precise thinking about imprecise data) should not be confused with "fuzzy thinking" (i.e., imprecise thinking).

  • Comparative logic can be traced back at least as far as Aristotle. It includes reasoning such as
    If "the coffee is hotter than the milk" is more true than "the tea is hotter than the milk," then in fact the coffee is hotter than the tea.
    That is, [(M → T) → (M → C)] → (T → C). This formula is a theorem of comparative logic but not of classical logic.
           For instance, it fails classically when T=1 and M=C=0. Indeed, with those values, (M → T) → (M → C) is classically true but comparatively false, because M → T is more true than M → C.

When classical logic is presented by itself, it not only seems artificial, but also takes on the aura of "absolute truth," leaving no room for doubt. But reasoning requires doubt; ideally we should teach our students to question everything. In mathematics one must constantly ask:

       What if this condition is not true? What happens then?

But that question is rather abstract and hypothetical. It is difficult for beginning students. Even for advanced students or experienced researchers, such a question by itself is seldom productive. It must be supported by the more concrete question,

       What happens differently in the other examples that we have studied?

It is a basic pedagogical principle — not just in logic, but in any subject — that any abstract idea is best taught with several different examples; one example (classical logic) is hardly enough. Contrasts between different logics make some properties easier to understand. For instance, the properties listed in the table below are explained and proved in CNL; the explanation is made clearer by the fact that the properties take different forms in different logics.

properties →

logics ↓

Excluded Middle: A ∨ ¬A Deduction Principle:
If A B, then
If A, B share no variables, and A → B, then Explosion:
relevant yes B ∨ ¬A can't happen no such result
RM yes B ∨ ¬A B and ¬A (A ∧ ¬A) → (B ∨ ¬B)
constructive No. But if A ∨ B,
then A or B.
A → B B or ¬A (A ∧ ¬A) → B
Wajsberg A ∨ ¬A ∨ (A ↔ ¬A) A → (A → B) no consequence A ∧ ¬A B
classical yes all of the above B or ¬A (A ∧ ¬A) → B

A traditional course on logic may end by covering Gödel's Incompleteness Theorems. But I believe that most of the students in such a course -- though perhaps able to follow the individual steps -- have difficulty understanding the overall concept of incompleteness, chiefly because it is preceded by too few examples of completeness. CNL presents many examples of completeness. By "completeness" I mean a pairing of some semantic system with some syntactic system, such that

{tautologies}   =   {theorems}

-- that is, the collection of always-true formulas of the semantic system is the same as the collection of provable formulas in the syntactic system. (This is known as "weak completeness" in some of the literature.) Also of interest is "soundness" -- i.e., where all the theorms are true (but not necessarily conversely). Below are several examples of such pairings.

Syntactics (axioms) Semantics (interpretation) Completeness?
classical 2-valued; powerset-valued; hexagon proved in book
constructive all topologies; all finite topologies; one infinite topology proved in book
relevant crystal matrix soundness proved; also proved that no finite matrix is adequate
RM Sugihara's Z-valued completeness stated; soundness proved
Rose-Rosser axioms Lukasiewicz [0,1]-valued completeness stated; soundness proved
Wajsberg axioms Lukasiewicz 3-valued completeness proved in book
relevant implication; constructive implication worlds models completeness proved in book
no findable axiomatization arithmetic Gödel's 1st Incompleteness, stated

As we remarked earlier, the semantics of propositional logic is very simple, and gets only a few pages in a traditional, classical-only book. When we turn to nonclassical logics, semantics becomes a much richer and more interesting subject, but it remains fairly concrete and accessible to beginners. Consequently semantics can be used as motivation for the more abstract subject of syntactics. CNL studies semantics for several chapters before even starting syntactic logic.

CNL begins with prerequisite material -- informal set theory and a few basics of topology -- so that it is self-contained, and suitable for a nonhomogeneous class or for self-guided study. The book is intended for beginners, so it keeps algebra to a minimum. For instance, it uses topologies instead of Heyting algebras, because they are more concrete and easier to define. The logics that are developed in the book were selected not for their appealing algebraic structures, but for their philosophical and psychological motivation, their connections to the premathematical experience of students, who may have very little previous mathematical training. That premathematical experience cannot be used as an ingredient in proofs, but it can be used to supply intuition, which is crucial for beginners. Still, after the motivating introductory chapters, CNL is a book of mathematics; all results are rigorously proved.