Truth (Princeton Foundations of Contemporary Philosophy) Read online

  (Evidently designating a proposition by a that-clause is not quite the same as denoting one by a noun phrase beginning with “the proposition that.”) But despite doubts and difficulties, it seems that propositionalists outnumber sententialists.

  It also seems that, though many other kinds of things are spoken of as true and false—from assertions and beliefs and conjectures and declarations to remarks and statements and thoughts and utterances—there are at present no serious candidates for the role of fundamental truthbearers beyond propositions and sentences (the latter in the sense of semantic tokens).

  In particular, while the traditional theories (3)-(5) were formulated in terms of beliefs, few today regard beliefs as the primary bearers of truth and falsehood, for the following sort of reason, among others: It is possible to believe a disjunction (for instance, that it was either Professor Plum or Colonel Mustard who killed Mr. Boddy) without believing either disjunct, and such a belief may be true; but a disjunction cannot be true unless one disjunct is; so there must be truths that are not believed. Similar considerations apply to assertions. In any case, the very notions of «assertion» and “belief” are ambiguous, since we must distinguish the act or state or event or process or whatever of asserting or believing from the content thereof, from what is asserted or believed. The most common view today seems to be that the content is a proposition, and that if the act or state or event or process can be called “true” at all, it is only in a derivative sense.

  Items of yet other sorts are also spoken of as true or false. One speaks, for instance, of false teeth or false friends. But here we are clearly dealing with different senses of the key words “true” and “false,” roughly synonymous with “genuine” and “fake,” as can be seen from the fact that while a false sentence or proposition is as much a sentence or proposition as a true one, false teeth or friends are no teeth or friends at all. (What is false about them is the proposition that they are teeth or friends.) In sum, other truthbearers beyond sentences and propositions can be and will be more or less ignored for the remainder of this book. But the division between sententialists and propositionalists, which cuts across the division of theorists into realists and antirealists and deflationists, must be borne in mind as we turn to the examination of the views of particular authors, beginning with Tarski.

  CHAPTER TWO

  Tarski

  TARSKI WAS, BY HIS OWN ACCOUNT, primarily a mathematician, though also “perhaps a philosopher of a sort.” He foresaw important applications for a notion of truth in mathematics, but also saw that mathematicians were suspicious of that notion, and rightly so given the state of understanding of it circa 1930. In a series of papers in Polish, German, French, and English from the 1930s on he attempted to rehabilitate the notion for use in mathematics, and his efforts had by the 1950s resulted in the creation of a branch of mathematical logic known as model theory. This fact alone makes Tarski's work an enduring achievement, even if its philosophical bearing remains in dispute. What follows is a simplified account (telescoping earlier and later developments) of the most basic features of Tarski's influential work.

  2.1 “SEMANTIC” TRUTH

  Tarski held that almost all words of ordinary language, “true” included, are ambiguous, and his first task was to distinguish the sense of “true” that concerned him, and to give it a label. To give a somewhat facetious example, a follower of Williams James might hold that “Snow is white” is true on the grounds that it is useful for people to believe so, while a follower of William Jennings Bryan might hold that “Snow is white” is true on the grounds that the Bible often uses the phrase “white as snow.” Tarski considered himself a follower of Aristotle, because he held that “Snow is white” is true on the grounds that snow is white (and that “to say of what is that it is, is true”).

  He gave his notion of truth the label “semantic” truth, which could mislead readers familiar only with the present-day usage of “semantics” for the branch of linguistics concerned with meaning. Tarski was following a usage current among positivistically inclined philosophers in the 1930s, on which “semantic” was one of a trio of terms. A notion was called syntactic if it pertains to relations among words, semantic if it pertains to relations between words and things, and pragmatic if it pertains to relations among words and things and people. Thus a notion involving only words and compilations of words (such as the Bible) would count as syntactic, while a notion that brought in people (as believers, for instance) would count as pragmatic. By contrast, Tarski's notion of truth was “semantic” simply because his principle

  (1) “Snow is white” is true iff snow is white.

  involves words and a thing, snow, but not people. No more should be read into the “semantic” label than that.

  Anything of the same form as (1) is called a T-biconditional. The general pattern

  (2) “_______” is true iff_______.

  where the same sentence goes into each blank, is called the T-scheme. Tarski is famous for the emphasis he places on the T-scheme, but against a background of classical logic, which Tarski assumes without question, the scheme is equivalent to a pair of rules, T-introduction and T-elimination:

  (3a) from “_______” to infer “‘_______’ is true”

  (3b) from “‘_______’ is true” to infer “_______”

  Tarski's whole discussion could be recast in terms of these rules. Nonetheless we will stick to formulations in terms of the scheme in what follows.

  Another notion Tarski counted as “semantic” was that of the satisfaction of a condition by a thing or things. Parallel to (1) Tarski held, for instance,

  (4a) “x is white” is satisfied by snow iff snow is white.

  (4b) “x is the same color as y” is satisfied by snow and milk iff snow is the same color as milk.

  Yet another notion Tarski counted as “semantic” was that of denotation, as illustrated by

  (5) “snow” denotes snow.

  Tarski's ambition was to rehabilitate for use in rigorous mathematics the whole list of what he called “semantic” notions. (Note, however that these are more or less what we called “alethic” notions in §1.3, and meaning is not on Tarski's list, though it would be at the head of any list of “semantic” notions in present-day linguistics.)

  2.2 OBJECT LANGUAGE VS METALANGUAGE

  Before going further we need to look at how quotation marks are being used in (1). The sequence of letters es, en, oh, double-yu, and so on, is an orthographic type that is used in English to express the truth that snow is white, but that could be used in some other language to express the falsehood that grass is red. Setting aside complications owing to ambiguities within a single language, there are two ways we could acknowledge this fact in a reading of (1). One way would be to take the quotation marks with what they enclose to designate the semantic type consisting of the orthographic type together with its English meaning. The other way would be to take it to denote the orthographic type, but take “true” to be elliptical for “true in English.” The latter is Tarski's way. For him quotation marks are used to designate orthographic types, and what he is concerned with under the label “truth” is truth-in-English, or truth-in-L for some other language L.

  Tarski calls the language L for which truth is being discussed the “object language,” and the language L* in which truth is being discussed the “metalanguage.” So far we have been writing as if both languages L and L* were just English. But Tarski insists that paradox will be inevitable if one takes both L and L* to be the whole of some natural language, including the word “true” itself.

  For then one can write down something like this:

  (6) The sentence numbered (6) is not true.

  One then has

  (7a) The sentence numbered (6) is “The sentence numbered (6) is not true.”

  which together with the T-biconditional

  (7b) “The sentence numbered (6) is not true” is true iff the sentence numbered (6) is not true.

&
nbsp; yields the contradiction

  (7c) The sentence numbered (6) is true iff the sentence numbered (6) is not true.

  If, however, one distinguishes the object language L from the metalanguage L*, and takes “true sentence of L” to belong only to the latter and not the former, then while one can still write down this:

  (6') The sentence numbered (6') is not a true sentence of L.

  it will be only a sentence of L*, and there will be no paradox in the conclusion that it is neither a sentence of L that is true nor a sentence of L that is not, since it will not be a sentence of L at all. Thus for Tarski, though it is sentences of one language L that go into the blanks in (2), what results from filling the blanks is a sentence of another language L*.

  Three points should be noted about the relationship between L and L* as described so far. First, since for any instance of (2) the same sentence of L that is mentioned on the left side is used on the right side as a clause in a biconditional sentence of L*, the metalanguage must contain the object language. Second, in order to be able to mention sentences of L, the metalanguage must contain the device of quotation. Third, the metalanguage must contain something more besides, namely, the truth predicate “is true-in-L.” But the first two requirements can be relaxed.

  First, L* need not literally contain L, so long as one can translate L into L*. In subsequent mathematical applications of Tar-ski's work, often L and L* are each a fragment of some natural language, today most often English, but with L being abbreviated and written in symbols rather than words, while L* is not. Thus items of nonlogical vocabulary that may be written “less than” or “perpendicular” in L* might be abbreviated to “<” or “” in L, while the logical vocabulary of negation, conjunction, disjunction, conditional, biconditional, and universal and existential quantification that in L* would be written in words “not, and, or, if, iff, all, some” would in L be written in symbols “” In such a case, L is not literally contained in L* and one does needs to translate L into L*, even if the “translation” involved amounts to no more than undoing abbreviations.

  Second, L* must be able to mention sentences of L somehow, but it need not be by quoting them. Thus one might designate “Snow is white” as the sentence consisting of the 19th letter of the alphabet followed by the 14th followed by the 15th, and so on, or the sentence “~x~(x = x)” as the negation of the existential quantification with respect to the variable ex of the negation of the sentence consisting of the identity symbol flanked by two occurrences of the variable ex. Tarski speaks of designating a sentence of L by a “structural description,” a label broad enough to cover any of these possibilities.

  With the first requirement relaxed the T-scheme becomes

  (2’) “_______” is true in L iff _ _ _ _ _ _ _ _.

  where an object language sentence goes into the first blank, and its translation into the metalanguage into the second blank. With the second relaxed it becomes

  (2”) ........... is true in L iff _ _ _ _ _ _ _ _.

  where a “structural description” of a sentence of the object language goes into in the first blank, and its translation into the metalanguage into the second blank. (2”) is Tarski's ultimate “official” version of the T-scheme.

  Tarski calls a proposed definition of the truth predicate formally correct if it is mathematically rigorous, and materially adequate if it implies every instance of the T-scheme for the object language. Tarski's aim is to make the notion of truth respectable by giving a formally correct, materially adequate definition. Tarski's aim is achieved by using some set-theoretic apparatus available in the metalanguage but not the object language. Invocation of set-theoretic apparatus makes it utterly unlikely that his mathematical definition could be achieving what a lexicographer's definition aims to do, namely, to give an account of what ordinary people have meant all along by the word. It is no part of Tarski's aim to do so.

  To put the matter another way, traditional logic distinguished a predicate's extension, the things to which the predicate correctly applies, from its intension, the information about a thing conveyed by applying the predicate to it; and Tarski is only aiming at a definition that will have the right extension, without worrying about the intension. For some object languages he finds two or more formally correct, materially adequate definitions, differing in meaning or intension from each other. The smaller the object language, the more likely it is that one can find two definitions thus coextensive but nonsynonymous. With such a pair, though each definition would tell us something all truths have in common, at most one could be telling us what “truth” means. Yet in such a case Tarski is equally prepared to adopt either definition.

  But why bother with definition at all? In mathematics, new terms are indeed often introduced by defining them in terms of old ones and deriving basic results about them by proof from older results plus the new definitions. But new terms sometimes are (and as Aristotle already pointed out, on pain of circularity or infinite regress, sometimes must be) introduced simply as undefined primitives, with basic facts about them assumed as unproved axioms. Why not, then, simply introduce the truth predicate as a primitive, and the T-biconditionals as axioms?

  Tarski finds a philosophical reason for not taking the notion of truth as a primitive in a view called “physicalism” current in his day among the positivistically inclined. According to this view, the only scientifically admissible primitives are logical and physical. (Arithmetical and geometrical notions were assimilated to logical and physical, respectively.) The traditional literature on truth, however, certainly makes the notion seem, not logical or physical, but psychological or metaphysical.

  2.3 RECURSIVE DEFINITION

  Tarski also has another reason why, even if one did take truth as a primitive, one shouldn't take the T-biconditionals as one's axioms for it. That procedure won't suffice for establishing various theorems about truth.

  One of the first theorems one might want to establish is that consistency is a necessary condition for truth, that deductions starting from true premises can never lead to contradictory conclusions. To establish this one might hope to start from the analyses that logicians such as Frege and Russell had given of deduction, showing that every deduction can be broken down into a series of steps proceeding according to very simple rules, such as inference to a disjunction from either of its disjuncts, or from a conjunction to either of its conjuncts. The program would then be to show, on the one hand, that such rules are truth-preserving, always carrying one from true premises to true conclusions, and on the other hand, that contradictory conclusions, one the negation of the other, cannot both be true. To carry out this program, one would need such principles as the following:

  (8a) A disjunction is true iff at least one of its disjuncts is true.

  (8b) A conjunction is true iff both of its conjuncts are true.

  (8c) A negation is true iff what it negates is not true.

  Now T-biconditionals, if taken as axioms, would enable one to deduce each instance of any of these principles. For example, from the T-biconditionals

  (9a) “Coal is black” is true iff coal is black.

  (9b) “Blood is green” is true iff blood is green.

  (9c) “Coal is black or blood is green” is true iff coal is black or blood is green.

  one can deduce this instance of (8a):

  (10) “Coal is black or blood is green” is true iff “Coal is black” is true or “Blood is green” is true.

  But there is no hope of getting any universal principle on the order of (8abc) just from the T-biconditionals.

  One can, however, for a suitably restricted object language, hope to do the reverse, and obtain all T-biconditionals from a shortish list of principles including (8abc) and a few more. To avoid technicalities, let us consider just a toy example. Consider a language L whose only nonlogical vocabulary consists of the numerals “0” and “1.” The only atomic, or logically simple, sentences of the language are equations “m = n” between numerals
. The only sentences of the language are those that can be obtained from the atomic sentences by repeated use of negation, conjunction, and disjunction (writing “m ≠ n” for the negation of “m = n”).

  We can supplement the principles (8abc) by a few more principles bringing in the auxiliary alethic notion of denotation, thus:

  (8d) An equation is true iff the denotations of the numerals on the two sides are identical.

  (8e) The denotation of the numeral “0” is the number zero.

  (8f) The denotation of the numeral “1” is the number one.

  Then (8abcdef) together give what is called a recursive definition of the truth predicate, enabling us to deduce a T-biconditional for every one of the (infinitely many) sentences of the language, as required by Tarski's criterion of material adequacy.

  We can deduce the T-biconditional for any sentence in a canonical way, following the structure of the sentence itself. If our target sentence is “0 = 1 or 1 ≠ 1,” we note it is the disjunction of one atomic sentence with the negation of another, and so start with the atomic sentences and work our way up, thus: