Page 346 - Encyclopedia of Philosophy of Language

Formal Semantics
define functions [ • ] which map all expressions of the language to objects of the right types in the model: sentence type expressions to truth values, property type expressions to properties, object-to-property type expressions to functions from objects to properties, and so on. If one assumes for a moment that the basic domain U is the set of natural numbers, and that the predicate letter O stands for being an odd number, then the interpretation of a property expression Ajc.0(x), notation [lx.O(x)], is a function / which yields 1 for every iteU which is an odd number, and 0 for all even numbers. The property expression Ax— O(x) has the characteristic function of all even num- bers as its interpretation. The interpretation of AP.V/* (universal quantification over individuals), notation [AP.VP], is the function Fe(U ->{1,0}) -> {1,0} with F(/) = 1 for the function/in £/->{l,0} with/(u)=l for all i4eU, and F(g)=0 for all g =£f. As a final exam- ple, AP.Vx(—O(x) -»P(x)) has as its interpretation the characteristic function F which maps every func- tion fe U -» {1,0} mapping every even number to 1, to 1, and all other functions in t/->{l,0} to 0. Note
that AP.Vx(—O(x) -* />(*)) would be an appropriate translation for the natural language phrase everynon- odd thing.
are powerful enough to express involvement in a for- mula. F, logically involves F2 if and only if the formula VP.(F.,(P)-»•£2(^))istrueineverymodel.HerePisa variable of the right type for arguments of £, and F2.
Conversely, one may want to impose certain restric- tions on the possible interpretations of the basic vocabulary by stipulating that certain concepts should involve others. For instance, one may want to ensure that the concept of walking involves the concept of moving relative to something. Assuming that these are expressions Ax W(x) for walking and A>»Ax.M(x, y) for moving with respect to, one can express the requirement as: kx.W(x) should involve ix3y.(ob-
ject(y) & M(x,y)). The semantic requirement is then imposed by restricting attention to models in which the first concept does indeed involve the second one. The desired involvements can be expressed as formu- lae. Such formulae, intended to constrain the class of possible models with the purpose of enforcing certain relations between elements in the vocabulary of the language, are called 'meaning postulates.' Given a natural language fragment and a set of meaning pos- tulates for that fragment, a sentence of the fragment is 'analytic' if it is true in any model satisfying all the meaning postulates. A sentence of the fragments is 'synthetic' if it does have counterexamples among the models satisfying all the meaning postulates. Given that the meaning postulates constrain the meanings of the vocabulary in the right way, one may assume that the real world (or some suitable aspect of it) will make all the meaning postulates true, so the synthetic sentences are precisely those that the world could be in disagreement with. The logically valid sentences are
The model theoretic approach to meaning now
equates the intuitive concept of 'meaning' with the
precise model theoretic concept of 'interpretation,'
that is, with values of a function [ • ] generated by an
appropriate model. Note that [ • ] is ultimately defined
in terms of the interpretations of certain basic
expressions in the model (for instance, predicates like
O, universal quantification over individuals). The
interpretation of these basic expressions can be those that are true in any model of the language,
described in terms of truth in the model, modulo appropriate assignments to free variables. The mean- ing of O reduces to the truth or falsity of O(x) in the model, relative to assignments of individuals to x, the meaning of universal quantification over individuals reduces to truth or falsity of Vx<p in the model, relative to assignments of properties to Ax<p, and so on. Thus it can be said that meaning is ultimately defined in terms of truth in a model.
Next, logical validity of inferences is defined in termsoftruth,bysayingthataninferencefromprem- ises <pi through (pn to conclusion \l/ is valid if and only if every model in which all of <p, through <pn evaluate to true will make \f/ true as well. In fact, for typed languages, the concept of logical validity can be extended to arbitrary expressions denoting charac- teristic functions. Let E\ and E2 be expressions for characteristic functions of the same type. If one characteristic function F, involves another one, F2, if FI and F2 have the same types and F2 yields 1 for every argument for which F, yields 1, then E\ logically involvesE2ifandonlyifineverymodel,[£,]involves [£ j|. To give a rather trivial example, Ax—O(x) logi- cally involves Ax(0(x)v—O(x)). Typed languages
irrespective of any meaning postulates.
This overview of models, interpretations, logical
inference, logical involvement, and the analytic/ synthetic distinction makes clear that truth is the tor- toise which carries the whole edifice of semantics on its back.
8. MeaninginContext
The very simple account of meaning given in the pre- vious sections breaks down if one wants to extend the treatment to intensional phenomena. Consider ex- ample (22).
Johnseeksa girlfriend. (22)
Example (22) might mean that John is looking for Sue, who happens to be his girlfriend, but it might also mean that John is answering small ads in the lonely hearts column because he wants to create a situation in which he has a girlfriend.
The setup of the previous sections would only give the first sense of the sentence. A standard way to get thesecondsenseisbymakingadistinctionbetween 'extensional' and 'intensional' interpretations of phrases. The extensional interpretation of a phrase is
324