Ambiguities and Types

If the Propositional Functions Account turns out to be the correct semantic account of first-order modal languages, then these languages happen to be ambiguous in unexpected ways. Consider the formulaP a. Taken on its own, this expression has as its semantic value a proposition, namely, the proposition that Jordan is tall. However, in the context of the formula∀ˆx(P a), the expressionP a

has as its semantic value a1st-level propositional function, namely, that propositional function which necessarily, for everyx, mapsxto the proposition that Jordan is tall.

Moreover, in the context of the formula∀ˆx(∃ˆy(P a∧Qxy))the semantic value ofP aturns out to be a2nd-level propositional function. Namely, it is that2nd-level propositional functionf which necessarily, for everyy, mapsyto that propositional functiongwhich necessarily, for everyx, mapsx

to the proposition that Jordan is tall.

A second kind of ambiguity is illustrated by considering the open formulaQxywhen embedded in, respectively, the formulas∃ˆy(∃ˆx(Qxy))and∃ˆx(∃ˆy(Qxy)). In the context of the first closed formula,

∃ˆy(∃ˆx(Qxy)), the formulaQxy has as its semantic value that2nd-level propositional functionf

which necessarily, for everyx, mapsxto that propositional functiongwhich necessarily, for every

y, mapsyto the proposition thatyis a father ofx. In the context of the second closed formula,

∃ˆx(∃yˆ(Qxy)), the formulaQxy has as its semantic value that2nd-level propositional functionf

mapsy, to the proposition thatx(noty) is a father ofy(not ofx).

Thus, in an extended sense of ‘open formula’, according to which open formulas are those expres- sions of the language that have as semantic values propositional functions, the formula ‘P a’ may itself be considered an open formula. When in the context of a formula such as ‘∀ˆx(P a)’, the formula ‘P a’ does not have as its semantic value a proposition. Instead, its semantic value is a1st-level propositional function.

On Stalnaker’s own account, the ambiguities just noted are resolvedin situ. The same expression has different semantic values depending on the larger linguistic context in which it occurs. Here, instead of adopting thein situstrategy, the Propositional Functions Account is given for languages stripped of ambiguities of the kind in question. By focusing on languages without these ambiguities the consequences of the Propositional Functions Account become clearer.8 _{To the languages for which}

the Propositional Functions Account is here given I will call TMˆL-languages. The ambiguities noted in ML-languages are resolved in two ways.ˆ

In order to account for the fact that expressions such asQxycan express different propositional functions, the ordering of the variables of the language will be exploited. The two propositional functions that were the possible semantic values ofQxyin the context of, respectively, the formulas

∃ˆy(∃ˆx(Qxy))and∃xˆ(∃yˆ(Qxy)), are the semantic values of, respectively, the formulasQx1x2and

Qx2x1.

In order to account for the fact that an expression such asP amay express propositional functions of different levels, types are added to the language. The type-hierarchy adopted is the one presented in (Gallin,1975, p. 68). The variables of the language are typed withe, the type of individuals. The variables of the language are thusx1_e, x2_e, . . .. Similarly, the individual constants of the language are typed with the typee, and then-ary predicates of the language are typed with the typehe1, . . . , eni, the type ofn-ary relations. So, whereas beforeawas an individual constant, nowaeis an individual

constant, and whereas beforeP andQwere, respectively, a unary predicate and a binary predicate, nowPheiandQhe,eiare, respectively, a unary predicate and a binary predicate.

LetU andSbe the following subsets of the setP of all types:

• U is the smallest set such thathibelongs toU, and ifτ belongs toU, thenhe, τibelongs toU; • Sis the smallest set such thatheibelongs toS, and ifτ belongs toS, thenhe, τibelongs toS. The setsU andS are, respectively, the sets of types of propositional functions and of property functions. Thus,hiis the type of propositions,he,hiiis the type of1st-level propositional functions,

he,he,hiiiis the type of2nd-level propositional functions, and so on. Moreover,heiis the type of properties,he,heiiis the type of1st-level property functions,he,he,heiiiis the type of2nd-level property functions, and so on.

Each one of the types in, respectively,U andS, is abbreviated as follows:

1. For each natural numbern,hn,0iis an abbreviation of the type of nth-level propositional 8_{This is not to say that there are no reasons to prefer the in situ strategy. On the contrary, the in situ strategy is} conservative with respect to the present usage of first-order modal languages.

functions;

2. For each natural numbern,hn,1iis an abbreviation of the type ofnth_{-level property functions.}

This means thath0,0iis the type of propositions,h0,1i, and in general,h0, niis the type ofn-ary relations. Furthermore,hn,0iis the type ofnth-level propositional functions, andhn,1iis the type ofnth-level property functions.

The set of complex expressions of TML-languages is now defined:ˆ 1. Every variable of typeτ is a term of typeτ;

2. Every constant of typeτ is a term of typeτ;

3. Ifsis a term of typehi, ni,s1, . . . snare terms of typee, andkis the highest index of all the vari- ables occurring free ins, s1, . . . , sn, then, for each natural numberm,(ss1. . . sn)hmax(i,k)+m,0i is a term of typehmax(i, k) +m,0i;

4. ϕis a term of typehn,0i, then(¬ϕ)hn,0i,(@ϕ)hn,0i,(2ϕ)hn,0iare terms of typehn,0i; 5. Ifϕ, ψare terms of typehn,0i, then(ϕ∧ψ)hn,0iis a term of typehn,0i;

6. Ifϕis a term of typehn+ 1,0i, thenxˆ_en+1(ϕ)hn,1iis a term of typehn,1i; 7. Ifsis a term of typehn,1i, then(∀s)hn,0iis a term of typehn,0i.

For instance:

• (Qh0,2ix1eae)h1,0iis a formula whose type is that of a1st-level propositional function; • (Qh0,2ix1eae)h2,0iis a formula whose type is that of a2nd-level propositional function; • xˆ2_e((Qh0,2ix1exe2)h2,0i)h1,1iis a term with the type of a1st-level property function;

• (∃ˆx2_e((Qh0,2ix1ex2e)h2,0i)h1,1i)h1,0i is a term whose type is that of a 1st-level propositional function.

Note that there is no formula corresponding to the string(Qh0,2ix2eae)h1,0i. That is, if the variable

with the highest index occurring in the formula is the variablexn_e, then the formula has at least the type of anth-level propositional function. The string(Qh0,2ix2eae)h1,0iviolates this constraint, since the variable with the highest index occurring in it isx2_e, whereas this string is labelled with the type

h1,0i. In order to count as a formula, the string would have to be labelled with a typehn,0i, for

n >1. Note also thatxˆ2

e((Qh0,2ix1exe3)h3,0i)h2,1iis not a term, since the variable being bound is not

x3_e, contrary to what is required by clause 7.

To those interested in the minutiae of the Propositional Functions Account, in the remainder of §2.3the Propositional Functions Account is presented in more detail and it is shown how the Kripkean model-theory may be used to model the account. Others may want to skip ahead to §2.4.