Truth (Princeton Foundations of Contemporary Philosophy) Read online

  (11a) “0 = 1” is true iff zero is identical with one.

  (11b) “1 = 1” is true iff one is identical with one.

  (11c) “1 ≠ 1” is true iff one is not identical with one.

  (11d) “0 = 1 or 1 ≠ 1” is true iff zero is identical with one or one is not identical with one.

  Given that (8abcdef) provide a recursive definition of truth, if one were to take the truth predicate as a primitive governed by certain axioms, (8abcdef) would be the axioms to take, rather than the infinite list of instances of the T-scheme. But only a direct definition, which is to say, something of the form “A sentence is true iff… ,” with no occurrence of the truth predicate on the right side, would permit us to regard the truth predicate as merely an abbreviation for some compound of notions previously understood. And “physicalist” scruples lead Tarski to demand such a direct definition. Fortunately for him, the nineteenth-century algebraist Richard Dedekind had decades earlier devised a method by which a recursive definition can always be converted into a direct definition, making use of set-theoretic apparatus (specifically, the mathematical notion of function).

  But one can understand the allusions to Tarski in our discussion of the deflationism-realism-antirealism debate without having to know any of the technical details of Dedekind's method and Tarski's exploitation of it, or of the various mathematical uses to which the notion of truth, once defined, was put by Tarski and later model theorists. The reader who wishes may proceed at this point straight to the next chapter and post-Tarskian philosophical debates. (Some of the more technical material that occupies the remaining sections of this chapter would be needed to follow the correspondingly technical portions of our discussion of Kripke's work on the paradoxes in later sections of chapter 7, but those sections may themselves be treated as optional reading.)

  2.4 * DIRECT DEFINITION

  To illustrate some of the finer points, we need a couple of examples more substantial than the toy language of the previous section. The first will be the formal language of arithmetic. This language contains variables v1 and v2 and v3 and so on for natural numbers. (We do not write variables “v1” and “v2” and “v3” and so on because the custom when writing in unformalized English about formal languages is to let expressions of the latter name themselves, omitting quotation marks.) The language contains besides the numerals or constants 0 and 1 also operators + and •. This is all the nonlogical vocabulary.

  We call something an atomic term if it is either a variable or a constant. Then more terms such as (v1+ 1) • (1 + 1) are built up from atomic terms using the operators and parentheses for punctuation. A term is called open if it is or contains a variable or variables, and closed otherwise. It does not make sense to ask for the numerical value of an open term such as v1 + 1, since that would depend on the value of the variable; only closed terms such as 1 + 1 can be said to have a numerical value or denotation. Tarski's first task is to give a rigorous definition of this auxiliary notion of the denotation | t | of a closed term t.

  A recursive definition sufficient to determine, step by step, the denotations of more and more complex closed terms, is provided by the following clauses:

  (12a) |0 | is zero.

  (12b) |1 | is one.

  (12c) | s + t| is the sum of | s | and | t |.

  (12d) | s • t| is the product of | s | and | t |.

  The first step in applying Dedekind's method to convert the recursive definition into a direct one is to prove an existence and uniqueness lemma, telling us that there is one and only one function f taking closed terms as inputs and giving natural numbers as outputs, that satisfies the following conditions:

  (13a) f(0) is zero.

  (13b) f(1) is one.

  (13c) f(s + t) is the sum of f(s) and f(t).

  (13d) f(s • t) is the product of f(s) and f(t).

  All the hard work goes into the proof of this lemma, which we omit. When the proof is complete, | t | may be defined to be f(t) where f is the function of the lemma, and (12abcd) follow immediately as theorems.

  The atomic formulas of the language consist of two terms flanking the identity symbol. Then other formulas are built up from atomic formulas by the logical operations. An occurrence of a variable in a formula is called bound if it is “caught” by a quantifier, and free otherwise. For example in

  (14) v2(v1 = v2 • v2)

  which intuitively says that v1 is a perfect square, v1 is free and v2 is bound. A formula is called closed if all occurrences of variables in it are bound, and otherwise is called open. Open formulas may also be called conditions, and closed formulas sentences. It does not make sense to ask whether a condition such as (14) is true, since that would depend on the value of the free variable. Tarski's next task is to give a rigorous definition of the notion of truth for sentences.

  A recursive definition sufficient to determine, step by step, the truth values of more and more complex sentences is given by the following clauses:

  (15a) s = t is true iff | s | is identical with | t |.

  (15b) ~A is true iff A is not true.

  (15c) A B is true iff A is true and B is true.

  (15d) A B is true iff A is true or B is true.

  (15e) vA(v) is true iff for every closed term t, A(t) is true.

  (15f) vA(v) is true iff for some closed term t, A(t) is true.

  In (15ef), A(t) represents the result of substituting the term t for every free occurrence of the variable v in A(v).

  The lemma needed for conversion to a direct definition by Dedekind's method is that there is one and only one function g taking closed formulas as inputs and giving as output either the number one (representing truth) or the number zero (representing falsehood), and satisfying the conditions

  (16a) g(s = t) is one iff | s | is identical with | t |.

  (16b) g(~A) is one iff g(A) is not one.

  and four more related to (15cdef) as (16ab) are related to (15ab). Then A is defined to be true iff g(A) is one, where g is the function of the lemma, and (15abcdef) become theorems given this definition.

  The definition becomes more complicated for an object language with variables ranging over objects not all of which are denoted by closed terms of the language. Such is the case with the formal language of geometry, which has variables for points of the Euclidean plane, but no constants or operators. The only items of nonlogical vocabulary are two predicates B and C, for betweenness and congruence, abbreviating the following:

  (17a) ______ lies on the line segment between ______ and ______ .

  (17b) ______ is as exactly as far from ______ as ______ is from ______.

  There are three kinds of atomic formulas, from which to build up other formulas: B followed by three variables, C followed by four variables, and = flanked by a variable on each side.

  For geometry we cannot begin a recursive definition of truth for closed formulas with a clause like (15a) about truth for closed atomic formulas, since there are no closed atomic formulas. Nor can we handle quantification by clauses like (15ef), since there are no closed terms. Tarski's solution to these difficulties is to bring in the auxiliary alethic notion of satisfaction. He gives a recursive definition (convertible to a direct definition by Dedekind's method) of what it is for an open formula to be satisfied by an assignment of a sequence of points to its free variables. For example, the atomic formula C v1v2v1v3 is satisfied by the assignment of p, q, r to v1, v2, v3 iff p is exactly as far from q as it is from r. For another example, the existentially quantified open formula v1Cv1v2v1v3 is satisfied by the assignment of q, r to v2, v3 iff for some point p the formula C v1v2v1v3 is satisfied by the assignment of p, q, r to v1, v2, v3. Truth for closed formulas is simply the special case of satisfaction by the “empty sequence” of points. We will not give the details, since Tarski's approach or some close variant can be found expounded in all good logic textbooks today.

  2.5* SELF-REFERENCE

  The language of geometry is a case where there can be found coextensive but no
nsynonymous truth definitions. For a major result of Tarski is that for the language of geometry truth under his “semantic” definition coincides with the “syntactic” notion of deducibility from a list of axioms for Euclidean geometry. By contrast with this result, Kurt Gödel's famous incompleteness theorems tell us that there is no system of axioms from which all truths of arithmetic can be deduced.

  This difference between geometry and arithmetic is closely connected with the fact that we have in arithmetic, though not geometry, a way of referring in the formal language to expressions of the formal language, comparable to the ability of natural language to refer to its own expressions by quoting them. To be sure, arithmetic is about numbers, not expressions; but a code number #E may be assigned to every expression E, as in effect happens today when expressions are transmitted electronically by encoding them as sequences of zeros and ones, the binary digits of a number. The formal language L of arithmetic has numerals 0 and 1 for zero and one and closed terms 1 + 1 and (1 + 1) + 1 and so on that can in effect serve as numerals 2 and 3 and so on for other numbers. The numeral e for the code number e = #E of an expression E amounts to something like a quotation of the expression, and we may suggestively write “E” for it.

  Many important notions pertaining to expressions of the language can then be expressed in the language. One notion we can express is

  (18) x is the code number of an equation whose two sides have the same denotation.

  For a fixed system of axioms, giving us a fixed notion of provability, or deducibility from the axioms, another notion we can express is

  (19) x is the code number of a provable sentence.

  Sentences can be constructed that in effect refer to themselves. (So can pairs of sentences that refer to each other, and sequences of sentences in which each refers to the next.) If “x is the code number of a sentence with property II” is expressible by a formula P(x) of L, a sentence ∑ of L can be constructed that in effect says of itself that it has the property II. What ∑ literally says is that for any number x, if S(x), then P(x), where S(x) is a formula that provably holds of one and only one number, namely, the code number of the sentence ∑ itself. Thus the sense in which ∑ says of itself that it has the property is that the following is a theorem:

  (20) ∑ ↔ P(“∑”)

  Such self-referential sentences are the analogues of natural language sentences of the type

  (21) The sentence numbered (21) has the property .

  Gödel exploited self-reference to construct a sentence that says of itself that it is not provable, which sentence he showed is indeed not provable, hence is true: an unprovable truth. Tarski exploited self-reference to show that no formula T(x) of the language L of arithmetic can express “x is the code number of a true sentence of L,” because for any formula T(x) one can construct a sentence having ∑ ↔ ~ T(“∑”) rather than ∑ ↔ T(“∑”). The details are given in textbooks of intermediate-level logic.

  2.6 * MODEL THEORY

  Languages like those of formal arithmetic and formal geometry are called first-order languages. They are the main object of study in model theory. To specify such a language one must do two things. On the “syntactic” side, one must specify a vocabulary of nonlogical symbols. One must indicate what constants, operators, and predicates occur in the language (as for arithmetic we have two constants 0 and 1 and two two-place operators + and • , while for geometry we have one three-place predicate B and one four-place predicate C). Application of the usual logical apparatus to a vocabulary of nonlogical symbols yields an “uninterpreted language.”

  On the “semantic” side, one must specify an interpretation. What this means is that one must indicate what domain (as numbers for arithmetic, or points for geometry) the variables are supposed to range over. Also, one must associate objects of appropriate kinds to each item of nonlogical vocabulary. That means that for each constant, one must say what element of the domain it denotes (as 0 denotes zero). For each operator, one must say what function on the domain it indicates (as + indicates addition). For each predicate, one must specify its extension, that is, one must say which sequences of elements of the domain satisfy it (as triples of points with the first between the other two satisfy B). Providing an interpretation turns an “uninterpreted language” into an “interpreted language.”

  Note that an “interpretation” in the technical sense just indicated need not assign any meaning to the nonlogical vocabulary. Predicates need to be assigned an extension, but not an intension. In describing the language of geometry, for instance, we read C as “______ is exactly as far from as ______ is from , ______,” and used this reading to specify which quadruples satisfy an atomic formula beginning with C. But all that really matters for Tarski's definition of truth is the set of quadruples. This we could have arrived at a different way, by reading C as “______ is exactly as far from ______ as ______ is from ______ and some swans are black” For with this alternate reading, though the intension would be different, still (since some swans are black) the extension would be the same.

  For a language containing, say, an operator meaning “Euclid knew that…,” the difference between the two readings of C might make a difference. Indeed, black swans being native to Australia, not Egypt, Euclid knew the first but not the second of the following:

  (22a) For any two points, the first is exactly as far from the second as the second is from the first.

  (22b) For any two points, the first is exactly as far from the second as the second is from the first, and some swans are black.

  Hence v1v2 Cv1v2v2v1 would be true with our original swanless reading of C, but not with the alternate swanful reading. But such operators as are not present in first-order languages. Their presence would obstruct Tarski's whole procedure, which depends on the truth values of compounds depending only on the truth values of their components, as the truth value of ~A depends only on the truth value of A. With an operator like in the language (and with Euclid having been less than omniscient), this feature would be lost, since A would be true for some true A and not for others.

  If we change not only the intension but the extension of our basic predicates, truth values can change even in first-order languages. We can obtain from the geometric language another geometric language by keeping the vocabulary but changing the interpretation. Instead of taking the variables to range over points of the Euclidean plane or pairs of real numbers (x, y), take them to range over points in the interior of the unit circle, or pairs of real numbers (x, y) with x2 + y2 < 1. Instead of reading the nonlogical predicate B as lying between on a straight line, read it as lying between on an arc of a circle orthogonal to the unit circle. Instead of reading the nonlogical predicate C as equal in distance according to the usual Cartesian distance formula, read it as equal in distance according to the alternative Poincaré distance formula. Then where the sentences that came out true in the old language were the theorems of Euclidean geometry, the sentences that come out true in the new language will be the theorems of what is known as hyperbolic geometry. (Such is the so-called Poincaré disk model of non-Euclidean geometry, famously illustrated by Escher's Circle Limit graphics; the technical details are immaterial for present purposes.)

  Model theory generally treats truth for whole batches of different interpreted languages having the same nonlogical vocabulary, or what is the same thing, truth for an uninterpreted language relative to whole batches of different interpretations. Now though the two geometries in our example do not have the same set of theorems, they do have some theorems in common. These include all logical truths, such as the law v1(v1 = v1) that everything is self-identical. Tarski suggested that logical truths may indeed be characterized as precisely those sentences that remain true for all interpretations. This characterization is adopted in logic textbooks today. Indeed, most of the basic material on truth and logical truth in modern logic texts derives from Tarski. (A few basic results do antedate Tarski, notably the Löwenheim-Skolem and Gödel compl
eteness theorems. Such results were originally stated in terms of a notion of truth not reduced to more ordinary mathematical notions, leaving the impression that “metamathematics” was a subject outside mathematics. Tarski's definition brought it inside.)

  The subject of model theory today extends far beyond such basic material. It has grown into a flourishing enterprise with applications across mathematics and allied fields. Tarski's work has an ongoing influence in philosophy as well, and a greater one than will be evident just from the scattered mentions of his name in the chapters to follow.

  CHAPTER THREE

  Deflationism

  IN TURNING FROM TARSKI'S WORK to the deflationist-realistantirealist debate we are turning from work largely motivated by concerns about the paradoxes to work more directly motivated by questions about the nature of truth. About this work there is a great deal to be said even ignoring the bearing of the paradoxes, so let us set them aside for the time being, and return to the question with which we began this book, “What do the different truths about different topics all have in common, to make them all truths?”