Monday, August 20, 2012

Fifth Salvo: Tarski's Snow Job

Virtually a sacred text in the purlieus of analytical philosophy, (1) can hardly fail to be boringly familiar.

(1)     Snow is white, if and only if the sentence “snow is white” is true.

Whoops, more than a mere parody of (2), (1) figures as the evil twin of (2) who figures in turn as the good brother already elected early on by Alfred Tarski himself to serve as the public face of his quite technical biconditional.

(2)     The sentence “snow is white” is true iff snow is white.

Thanks to the fact that (2) is a mere platitude and featuring (2) over (1) as a pneumonic device abbreviating his biconditional, Tarski could hoodwink even hardened professionals into crediting his technical biconditional with being “more or less analytic” (Beall p. 1) as well.  How’s that for a snow job!

As the evil twin who is quite on a par with (2) when it comes to constituting Tarski’s biconditional, (1) lends itself much less readily to being viewed as a platitude if only because we expect snow to remain white even in the absence of all sentences, trading indeed on the thesis that conditionals in English are characteristically infected with modal import.

Invited by my use of the word “infect” here to protest that (1) should be allowed to be as unproblematic as (2) in so far as both alike are to be glossed in terms of “material implication”, one may need to be reminded that early in the last century one spoke of “the paradoxes of material implication” among which (1) might well be registered.
Actually, translating “p materially implies q” by “~ p v q” can produce “funny” business of its own.  Thus (1) supplies “Either snow is not white or the sentence ‘Snow is white’ is true.  Hardly your everyday idiomatic disjunction!  Call it a material disjunction evaluating which it will suffice to ignore the rubbish on the left, while plumping for the truth on the right, thereby extending Tarski’s snow job beyond the conditional to disjunction.

Emerging here one encounters at least a minor puzzle as to how serious a glitch figures in this informal privileging of (2) over (1) in current discussions about . . . truth.  For it may be urged that privileging (2) over (1) over wide stretches of these discussions is just the sort of simplification that advances serious research in one discipline after another.  Fair enough; an immediate case in point lies in (2)’s tacitly recognizing a truth that (1) brazenly flouts, namely that it is because snow is white that the corresponding sentence has the semantic value it does, and not the reverse. Alas, this “advance” has “all the benefits of theft over honest toil” that Bertrand Russell satirized years ago, the theft being of course at the expense of the traditional Correspondence Theory of truth, with Tarski’s own official even-handed biconditional failing to respect the critical asymmetry that is highlighted by (1) being juxtaposed with (2).

Deconstructing Tarski’s snow job in terms of how his famous biconditional combines a platitudinous (2) with a highly non-platitudinous, and even freaky-deaky, (1) that trades on worries about material implication, one comes to grasp how Tarski could have vacillated in his attitude toward the Correspondence Theory of truth.  Viewing (2) rightly as being all of a piece with it, he could hardly fail to appreciate that his originality lay rather with his biconditional taken as a whole.  And here we have learnt that what counts is how (3) semantically relates to four.

(3)     Snow is green.

(4)     The sentence “snow is green” is true.

Starting from a ‘true’ free language from which (4) is absent, the noise “true” enters in terms of a rule that takes (4) and (3) to be alike in truth value.  That (4) inherits its truth value from (3) in accordance with the Correspondence Theory one quickly, and risibly, finds to be altogether occluded in the recent discussions collected in Deflationism and Paradox, edited by J. C. Beall and Bradley Armour-Garb.

          Not that the introduction (and elimination) rules for “true” aka T-in and T-out are not expressly defined by way of first adding to ‘Snow is green’ the two words ‘is’ and ‘green’, thereby yielding “‘Snow is green’ is true’” and later eliminating them.  This second operation is especially instructive.  Knowing nothing about snow, you are given the sentence “the sentence ‘Snow is green’ is true” without, however, any hint as to its true value.  No matter.  Invoking the rule T-out, you validly deduce that snow is green, again quite aware of your ignorance as to its truth value.

          By parity of reasoning then the false sentence “Snow is green” inherits or quasi-inherits its pseudo truth value from its false premise, namely (4).  Teasing apart these two senses of “inherit”, the one naturalistically oriented to the Correspondence Theory of truth, the other semantically prompted by Tarski’s snow job, one comes to grasp how one might continue, with Tarski, to sympathize with an inflationary Correspondence Theory even while deflating it on behalf of a more fundamental semantic approach to truth.

          Which is by no means to concede that fans of this fifth salvo could be expected to rest satisfied with this result.  To the contrary, they are more likely to rally around (5) even as they are prepared to relish the keen tension that sizzles when (5) is juxtaposed with (6).

(5)     It is because snow is not green that the
sentence “Snow is green” is false.

(6)     The sentence “’Snow is green’ is true’” and the sentence “Snow is green” have the same truth value.

Resolving this tension, the dominant deflationary theory of truth as sponsored by Tarski’s snow job may be expected to relax its fetishistic preoccupation with (6) in its fiercely tendentious disregard of (5).  Accordingly we may expect a mature theory of truth to plump for (5) very nearly as much as for (6).