Page 288 - Encyclopedia of Philosophy of Language

Language and Logic
1. Defining Identity
There are several ways of defining the identity relation. Among the most common are those that call attention to its reflexiveness, on the one hand, and to its obedi- ence to Leibniz's Law on the other. Reflexiveness is the property a relation has if, for everything, the relation holds between that thing and itself; in symbols, Vx (x=x), with' = ' used as the sign for identity. Leibniz's Law is the principle of the indiscernibility of identicals, namely that if x and y are identical, then anything that can be truly predicated of x can also be predicated truly of y. In symbols, there is the scheme (x=y & Fx)->Fy. Reflexiveness and Leibniz's Law define numerical identity, that is sameness of object, and not qualitative identity (sameness of property or quality). In everyday situations, we often say items are identical when we mean they are exactly similar (identical in all their properties, but not numerically one and the same thing).
Given these principles, it is easy to define other standard properties of identity, for example, that it is transitive (if x is identical with y and y is identical with z, then x is identical with z). One principle, however, that can be derived has itself been the subject of controversy (Peirce for example, called it 'all non- sense'). This is the principle of the identity of indis- ceraibles, in symbols, whereFis any predicate, (Fx & Fy)^x=y.
2. Applications in PredicateLogk
Extended with the identity relation, predicate logic can give symbolic versions for numerical claims. To say there are at least two dialects of English, for exam- ple, is to say that there is at least one thing, x, such that x is a dialect of English, and one thing, y, such that y is a dialect of English, and that x is not identical with y. To say there are exactly two truth values is to say
3x3X(*isatruthvalue&yisatruthvalue&~ (x=y))& Vz(zisa truth value -»z=x or z=>0)
in words, there are two nonidentical things, x and y which are truth values, and anything z which is a truth value is the same thing as x or the same thing as y.
The expressive power of the notation also includes capturing the exclusive force of 'else' in sentences like
Jane can run faster than anyone else on the team, which is taken to mean
Jane can run faster than anyone on the team who is not identical with Jane.
3. Russell's Theory of Descriptions
The ability to capture numerical claims is connected to Russell's famous attempt to give contextual defi- nitions of sentences containing definite descriptions. Russell took such sentences to have a truth value whether or not the descriptive phrase in them referred
to anything. Those with nonreferring descriptive phrases come out as uniformly false under his analysis. Russell was thus able to maintain the prin- ciple of bivalence (that each sentence is determinately either true or false).
One supposed advantage of Russell's theory of descriptions was getting round the problem of 'about' in the following form. 'The present King of France is wise,' like 'the golden mountain is made of gold,' seem to be sentences that are about something. Yet there is no present King of France, just as there is no golden mountain. If one accepts that well-formed subject- predicate sentences are about their subjects, then these sentences give rise to deep and challenging puzzles. Since there is no present King of France, is there some nonexistent 'thing' having wisdom predicated of it? Since there is no (actual) golden mountain, is it some possible (but unactualized) golden mountain which is made of gold?
Russell tackled this problem by providing a con- textual paraphrase for whole sentences containing definite descriptions under which the problematic descriptive phrases are supplanted by predicates. The surface, subject-predicate form of sentences with definite descriptions is, according to him, a poor guide to their logical form. The latter is given by treating sentences containing descriptions as conjunctions. Corresponding to:
The golden mountain is made of gold is the conjunction:
at least one thing is a golden mountain and
at most one thing is a golden mountain and
anything which is a golden mountain is made of gold.
Under this analysis, the expression 'golden mountain' occurs only as a fragment of a predicate; thus there is no commitment to the existence of some unique sub- ject of which 'made of gold' is predicated.
In standard notation, the conjunction of three sen- tences giving the Russellian analysis can be expressed as follows (with 'Gx' for 'x is golden (or is made of gold),' 'Mx' for 'x is a mountain'):
3x(Gx&MX)&VxVX((Gx&MX)&(Gy&My) -» Vx((Gx&Mx)-»Gx)
This formula is provably equivalent to the shorter form:
3x((Gx&MX)&VX(G.x&M>>)-»x=y) &Gx)
Although it might seem as if the above sentence should be true, Russell's analysis yields the value false (since the existential claim that there is a golden mountain is false). Russell regarded it as a virtue of the theory that it yielded a uniform result for every sentence containing a vacuous definite description; namely, that the sentence turned out false.
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