The Propositional Functions Account

The semantic value of an expression of typeeis an individual (that is actually something), and the semantic value of a constant of typeh0, niis an-ary relation. In particular, the semantic value of

=h0,2iis, as expected, the identity relation. LetJ·Khz1,...,znidenote a function which, when applied to an expressionϕof a TMˆL-language, mapsϕto its semantic value, except that ifϕis a variablexi_e,

1≤i≤n,_Jϕ_Khz1,...,zniiszi, wherenis any natural number. Here is a specification of the semantic values of the remaining terms of the language (except for the variables, which have no semantic value whatsoever):9

1. J(ss

1_{. . . s}n₎

hmax(i,k)+m,0iK = f0, wheref0 is that(max(i, k) +m)

th_{-level propositional}

function which is such that, necessarily, for everyy1, for every((max(i, k) +m)−1)th-level

propositional functionf1 and every(i−1)th-level property functionf1∗,f0(y1) =f1if and

only if_Js_K(y1) =f1∗and, necessarily, for everyy2, for every((max(i, k) +m)−2)th-level

propositional functionf2and every(i−2)th-level property functionf2∗,f1(y2) =f2if and only

iff₁∗(y2) =f2∗and, necessarily,. . .and, necessarily, for everyyi, for every((max(i, k) +m)−

i)th-level propositional functionfi, for every0th-level property functionfi∗,f(i−1)(yi) =fi

if and only if f₍∗_i−1)(yi) = fi∗ and, necessarily, fory(i+1), for every ((max(i, k) +m) −

(i+ 1))th-level propositional functionf(i+1),fi(y(i+1)) =f(i+1) if and only if necessarily,

for everyy_(i+2), for every((max(i, k) +m)−(i+ 2))th-level propositional functionfi+2,

f(i+1)(y(i+2)) = f(i+2)and. . .and necessarily, for everyy(max(i,k)+m), for every0th-level

propositional functionf_(max(i,k)+m),f_(max(i,k)+m)obtains if and only iff_i∗holds between Js

1

Khy1,...,y(max(i,k)+m)iand. . .andJs

n

Khy1,...,y(max(i,k)+m)i.

2. _J(¬ϕ)hn,0iK=f0, wheref0is thatn

th_{-level propositional function which is such that, neces-}

sarily, for everyy1, for every(n−1)th-level propositional functionf1, for every(n−1)th-level

propositional functionf₁∗,f0(y1) =f1if and only ifJϕK(y1) =f ∗

1 and, necessarily,. . .and,

necessarily, for everyyn, necessarily, for every0th-level propositional functionfn, necessar-

ily, for every0th-level propositional functionf_n∗, necessarily,f(n−1)(yn) =fnif and only if

f_n∗₋₁(yn) =fn∗and, necessarily,fnobtains if and only if it is not the case thatfn∗obtains;

3. J(2ϕ)hn,0iK=f0, wheref0is thatn

th_{-level propositional function which is such that, neces-}

sarily, for everyy1, for every(n−1)th-level propositional functionf1, for every(n−1)th-level

propositional functionf₁∗,f0(y1) =f1if and only ifJϕK(y1) =f ∗

1 and. . .and, necessarily,

for everyyn, for every0th-level propositional functionfn, for every0th-level propositional

functionf_n∗,f(n−1)(yn) =fnif and only iffn∗−1(yn) =fn∗ and, necessarily,fnobtains if and

only if necessarily,f_n∗obtains; 4. _J(@ϕ)hn,0iK=f0, wheref0is thatn

th_{-level propositional function which is such that, neces-}

sarily, for everyy1, for every(n−1)th-level propositional functionf1, for every(n−1)th-level

propositional functionf₁∗,f0(y1) =f1if and only ifJϕK(y1) =f ∗

1 and. . .and, necessarily,

for everyyn, for every0th-level propositional functionfn, for every0th-level propositional

functionf_n∗,f_(n−1)(yn) =fnif and only iffn∗−1(yn) =fn∗ and, necessarily,fnobtains if and

only if actually,f_n∗obtains;

5. _J(ϕ∧ψ)hn,0iK=f0, wheref0is thatn

th_{-level propositional function which is such that, neces-}

9_{Hopefully, it will be clear that the Propositional Functions Account could have been offered in the higher-order modal} language MLP(enriched with some extra primitives) presented in chapter1, even if it was there given in what might be called ‘logical English’.

sarily, for everyy1, for every(n−1)th-level propositional functionf1, for every(n−1)th-level

propositional functionf₁∗, for every(n−1)th_{-level propositional functionf}0

1,f0(y1) =f1if

and only if_Jϕ_K(y1) = f1∗ andJψK(y1) =f

0

1 and, necessarily,. . .and, necessarily, for every

yn, necessarily, for every0th-level propositional functionfn, necessarily, for every0th-level

propositional functionf_n∗, necessarily, for every0th-level propositional functionf_n0, necessarily,

f(n−1)(yn) = fnif and only iff₍∗_n−1)(yn) = f ∗ n andf 0 (n−1)(yn) = f 0 nand, necessarily,fn

obtains if and only iff_n∗andf_n0 both obtain; 6. _Jxˆn_e+1(ϕ)hn,1iK=f0, wheref0is thatn

th_{-level property function such that, necessarily, for}

every y1, for every (n−1)th-level property functionf1, for everynth-level propositional

function f₁∗, f0(y1) = f1 if and only ifJϕK(y1) = f ∗

1 and. . .and, necessarily, for every

yn, for every 0th-level property functionfn, for every1st-level propositional function fn∗,

fn−1(yn) =fnif and only iffn∗−1(yn) =fn∗and, necessarily, for everyyn+1, for every0th-level

propositional functionf_n∗₊₁, necessarily,fnholds ofyn+1if and only iffn∗(yn+1) =fn∗+1and

f_n∗₊₁obtains.

7. J(∀s)hn,0iK=f0, wheref0is thatn

th_{-level propositional function such that, necessarily, for}

everyy1, for every(n−1)th-level propositional functionf1, for every(n−1)th-level property

functionf₁∗,f0(x1) =f1if and only ifJsK(y1) =f ∗

1 and,. . .and, necessarily, for everyyn, for

every0th-level propositional functionfn, for every0th-level property functionfn∗, necessarily,

f_(n−1)(yn) =fnandf₍∗_n−1)(yn) =f

∗

nand, necessarily,fnobtains if and only iffn∗holds of

everything.

Contrary to the Literal account, the Propositional Functions Account is consistent with the conjunction of Aliens and Actualism. Prima facie, the account is also consistent with the conjunction of No Actual Haecceity and Thorough Actualism. However, this is not so. As shall be seen, the Propositional Functions Account is committed to claims that imply the falsehood of No Actual Haecceity, for instance, the claim that necessarily every haecceity is necessarily something. Before presenting these problematic consequences of the Propositional Functions Account, it will be shown how the Kripkean model-theory may be used to provide a model (i.e., a representation) of what is, according to the account, the semantics of first-order modal languages.