Azzouni, Jody. Tracking Reason: Proof, Consequence, and Truth.

Cain, James

AZZOUNI, Jody. Tracking Reason: Proof, Consequence, and Truth. Oxford: Oxford University Press, 2006. vi + 248 pp. Cloth, $49.95--Azzouni's aim is to articulate philosophically adequate accounts of truth, proof, and logical consequence without falling into (what he perceives to be) the pitfalls of the following: Platonism, the truthmaker assumption (if a sentence, statement or belief is true then it is, at least in part, made true by how the world is), treating existential affirmations as entailing ontological commitments, and holding that normative rules or principles of reason are introspectively accessible. The reader is often referred to Azzouni's earlier work for more detailed explanations of why these are supposed to be pitfalls. The book divides into three parts: I. Truth, II. Mathematical Proof, and III. Semantics and the Notion of Consequence.

According to Azzouni, the truth predicate renders natural languages inconsistent. This is shown by the reasoning behind the liar paradox. (Fortunately common practices keep the inconsistency at bay.) Azzouni's approach to truth is to focus upon what he takes to be the essential functional role played by the truth predicate and to show how to regiment language so that it contains a device playing that role (presumably without introducing inconsistency). Azzouni accepts "the Quinean dicta that the functional role of the truth predicate is semantic ascent and descent--when coupled with an appropriate quantifier" (p. 109); this allows one to make "blind truth-endorsements" [that is, "uses of the truth predicate where the sentences endorsed or denied don't themselves appear" (p. 14)], and ineliminable blind truth-endorsement "is the raison d'etre for the presence of the truth idiom in the vernacular" (pp. 15-16). Azzouni argues that affirmation of a set of T-biconditionals of the form "S" is true iff S will not (contrary to what many think) license a full range of blind truth-endorsements--it will not enable one to endorse statements whose expression transcends one's own linguistic resources.

Azzouni develops a regimentation in terms of anaphorically unrestricted quantifiers. To give a rough idea of this notion, which is formalized in Chapter Three, think of a crude attempt to symbolize "Something Jones says is true" in first order logic as "[there exists]x(Sx & x)," where "Sx" symbolizes "Jones says x." This would treat "x" as playing the role of an ordinary variable and the role of a wff. Though this violates the syntax of normal first order languages, Azzouni argues we can develop intelligible languages in which variables play such a dual role. Imagine someone saying, "For any blah-blah-blah, if Jones says blah-blah-blah then blah-blah-blah." Though this is not standard English, it seems to be intelligible, and we can think of it as saying that everything Jones says is true, or, alternatively, as making a blind truth endorsement of everything Jones says.

Part II deals with mathematical proof. How can mathematical proofs lead to results that, historically speaking, are so resilient? Euclid's proof that there is no greatest prime is as convincing today as it was in ancient times. Since Azzouni rejects mathematical Platonism his answer cannot appeal to the idea that proofs provide enduring insights into mathematical reality. On the other hand Azzouni does not think sociological explanation in terms of group conformity (which may nicely account for synchronic uniformity of opinion within a social group) can account for mathematical agreement across cultures and over long stretches of history. Azzouni's solution is that mathematical proofs are what he calls "derivation indicators." A derivation in a given system is mechanically checkable by algorithms, and "... the algorithmic systems that codify which rules may be applied to produce derivations in a given system" are at least implicitly recognized by mathematicians (p. 143). This (implicit or explicit) grasp of algorithms accounts for mathematical agreement. Strangely, as Azzouni recognizes, over the course of history mathematicians generally have not consciously realized that derivations could be formulated from their proofs. Typically mathematicians feel that they are dealing with mathematical objects (and imagining that they are doing so may be psychologically crucial to their mathematical productivity), but this should not be taken to show that they are actually describing mathematical reality. Azzouni compares mathematicians to novelists who think about their characters as if they actually exist, when of course they do not.

Part III concerns the notion of consequence. After critically discussing accounts of consequence that have descended from Tarski's work, Azzouni tries to show how insights gained from the Loweinheim-Skolem theorem and Henkin-style constructions of models can be used in support of a Tarskian approach that avoids the pitfalls enumerated at the beginning of this review. And just as psychological aspects of mathematical reasoning (with its apparent reference to mathematical objects) can be misleading, so too psychological aspects of drawing consequences (with its apparent appeal to what is or is not imaginable) can be misleading.--James Cain, Oklahoma State University.