ANTECEDENTES INVESTIGATIVOS

A different route available to thorough contingentists consists in thinking that the mistake with the Propositional Functions Account has been that of thinking that propositional functions must betotal. An alternative option is to take propositional functions to be partial, defined only for some individuals. For instance, according to this proposal the second-level propositional functionf that is the semantic value of(a=x)h1,0iis a relation that necessarily, for every individualx, obtains betweenx and a propositionhif and only ifhis the proposition thatxis identical toa. Thus, if the proposition thatxis identical toadoes not exist, then the propositional functionf does not relatexto any proposition whatsoever. The property that is the semantic value ofxˆ(a=x)h0,1iis determined in the same way, as a function of the propositional function that is the semantic value of(a=x)h1,0i. Iff does not relate an individualxto any proposition, thenxis not in the extension of the property that is the semantic value ofxˆ(a=x)h0,1i. Call this proposal thepartial functions proposal.

The partial functions proposal comes with its own problems. As will be shown, the fact that it is possible that there are some individuals for which a propositional function is undefined has the consequence that the recursive clauses of the account of semantic value do not assign semantic values

to expressions that ought to have a semantic value.

Higher-order contingentists are sympathetic to the view that Attributions of Being–Necessitism is false, and in particular that there are some individualsxsuch that the proposition thatxis something is itself something only contingently. As in the previous section, letEh0,1ibe a1-ary predicate letter whose semantic value is the property of being something. Consider the expression(Ea)h1,0i, and

letf be the propositional function that is the semantic value of this expression. Letw be some counterfactual possibility such that the proposition that Michael Jordan is something is nothing atw, and at which the empty set is something. Since the proposition that Jordan is something is nothing atw, it is not the case that the propositional functionf relates the empty set to a proposition at the relevant counterfactual possibility.

How is the property that is the semantic value ofxˆ(Ea)h0,1idetermined in terms of the propo- sitional functionf? The two natural options available are: i) necessarily, for every individualx,x

has the property if and only iff mapsxto a proposition and that proposition is true; ii) necessarily, for every individualx,xhas the property if and only if eitherf mapsxto a proposition and that proposition is true, orfmapsxto no proposition whatsoever.

If option i) is adopted, then it is not the case that the empty set has the property that is the semantic value ofxˆ(Ea)h0,1iatw, since the functionf that is the semantic value of(Ea)h1,0idoes not relate the empty set to any proposition whatsoever atw. If option ii) is adopted, then the empty set does have, atw, the property that is the semantic value ofxˆ(Ea)h0,1i. Option i) is the one that

delivers the right result in the present case. The intended semantic value ofxˆ(Ea)h0,1iis the property of being such that Michael Jordan is something, that property that necessarily, for every individual holds of that individual if and only if Michael Jordan is something. Since Jordan is nothing atw, the empty set does not have the property of being such that Jordan is something.

Let us thus adopt option i). Consider now the propositional function (¬(Ea))h1,0i. By the

recursive clause for negated expressions, the propositional functiongthat is the semantic value of the expression(¬Ea)h1,0ialso does not relate the empty set to any proposition whatsoever. But then the empty set also does not have, atw, the property that is the semantic value ofxˆ(¬Ea)h0,1i. This,

however, is the wrong result. The intended semantic value ofxˆ(¬Ea)h0,1iis the property of being such that Jordan is nothing. Insofar as Jordan is nothing atw, the empty set has, atw, the property of being such that Jordan is nothing atw. Since the empty set does not have, atw, the property that is the semantic value ofxˆ(¬Ea)h0,1iaccording to the the partial functions proposal, the semantic value ofxˆ(¬Ea)h0,1iaccording to the partial functions proposal is not its real semantic value.

Note also that if option i) is adopted then the semantic values ofxˆ(Ea)h0,1iandxˆ(¬Ea)h0,1i turn out not to be exhaustive properties. In addition, the adoption of option i) forces the rejection of certain plausible principles of first-order modal logic. Letbhave as its semantic value the empty set. For instance, the adoption of option i) requires the rejection of the following claim:

Even though it is the case that the empty set is something atwand Michael Jordan is nothing atw(let us assume, since the proposition that Michael Jordan is something is itself nothing atw), it is not the case that, atw, the empty set has the property that is the semantic value ofxˆ(¬(Ea). But(9)is a principle valid in fairly minimal first-order modal logics, such as the one offered inStalnaker(1994).

Before proceeding, it is important to make it clear that this result is not intended to show that the partial functions proposal is contradictory. Instead, the argument shows that the property that is delivered by the partial functions account as the semantic value ofxˆ(¬Ea)h0,1iis not the property that in fact is the semantic value of this expression. Since the partial functions account is unable to deliver the right semantic values of some of the expressions of the language, it is unsatisfactory.