2.2 Compositional Semantics and Thorough Contingentism
2.3.4 Modelling the Account
Recall Stalnaker’s views of what it takes for a model to be intended. The intended model does not consist of ‘modal reality’. Instead, it is a representation of certain features of reality that the theorist aims to capture. For the purposes of the Propositional Functions Account, the relevant features are the semantic values of the different expressions of the language, and the relationships between these. This representational use of the model-theoretic semantics requires that a particular class of set-theoretic entities be singled out to do the job of representing the semantic values of the different expressions of the language.
elements ind(). The admissible semantic values of expressions of typehe1, . . . , eni, for each natural numbern, consist of elements of the set of functionsfwith domainW and such that for everyw∈W,
f(w)⊆(d(w))n. Contrary to what was the case in the model-theoretic semantics specified in §2.2, only elements in this set are considered, since otherwise certain formulas contradicting Thorough Serious Actualism would be true in the models for the language. The sets of entities that represent the semantic values of the remaining expressions of typeshn,0iandhn,1iare defined in a similar fashion, as we shall see.
The relevant class of models is now defined in more detail.
Definition 1(PF-Models). A PF-model based on an inhabited model structureIS=hW,, Diis a pairM =hIS, Vi, whereV is a valuation function assigning a value to each individual constant and
n-ary relation letter in the following way:
1. For every (atomic) expressionsof typee,V(s)∈D()
2. For every atomic expressionsof typeh0, ni, for every natural numbern,V(s), is a function with domainW and such that, for everyw∈W,V(s)(w)⊆(d(w))n.
The next step is to extend the definition of value to the remaining expressions of the language. In order to do so, it is useful to first define a hierarchy of ‘domains’ ofn-ary relation functions:
Definition 2(Domains ofn-ary Relation Functions).
• Dh0,ni={f ∈(( S w∈W D(w))n)W :f(w)⊆(D(w))n} • Dhm+1,ni={f ∈( S w∈W D(w)×Dhm,ni)W :f(w) ={ho, gi:o∈D(w)}}
The values of expressions of typehn,0iandhn,1i— that is, of expressions whose type is, respectively, that ofn-ary propositional functions and that ofn-ary property functions — belong, respectively, to the setsDhn,0iandDhn,1i.
Now, let~onbe shorthand for the sequenceo1, . . . , onof meta-variables. Also, letV(o1/o1,...,on/on)
extend the original valuationV by assigning, for each1≤i≤n, the individual constantoito the
elementoi∈
S
w∈W
S(w), whereoiis not in language. Let~onbe shorthand for the sequenceo1, . . . , on
andV(~on/~on)be shorthand forV(o1/o1,...,on/on). Finally, letst
t0 be the result of substitutingt0fortin
s, ands~tn
~ t0
nbe the result of substitutingt 0
1fort1,. . .,t0nfortnin terms.
The value of the typed formulas and complex predicate of the language is defined as follows — note that the definition of valueis notthe usual one, since it does not appeal to variable-assignments:
Definition 3(Value of a typed formula and complex predicate).
1. V((ss1. . . sn)hu,0i) =f ∈Dhu,0isuch that, for everywj ∈W, oj ∈d(wj),1≤j ≤u:f(w1)(o1). . .(wu)(ou) =h such that: h={w∈W :hV(~ou/~ou)((s1)~xu ~ou), . . . , V (~ou/~ou)((sn)~xu ~ou)i ∈V(s)(w1)(o1). . .(wi)(oi)(w)}
2. V((¬ϕ)hn,0i) =f ∈Dhn,0isuch that, for everywj ∈W, oj ∈d(wj),1≤j≤n:f(w1)(o1). . .(wn)(on) =h such that: h=W −V(ϕ)(w1)(o1). . .(wn)(on) 3. V((2ϕ)hn,0i) =f ∈Dhn,0isuch that, for everywj ∈W, oj ∈d(wj),1≤j≤n:f(w1)(o1). . .(wn)(on) =h such that, h={w∈W :V(ϕ)(w1)(o1). . .(wn)(on) =W} 4. V((@ϕ)hn,0i) =f ∈Dhn,0isuch that, for everywj ∈W, oj ∈d(wj),1≤j≤n:f(w1)(o1). . .(wn)(on) =h such that h={w∈W : ∈V(ϕ)(w1)(o1). . .(wn)(on)} 5. V((ϕ∧ψ)hn,0i) =f ∈Dhn,0isuch that, for everywj ∈W, oj ∈d(wj),1≤j≤n:f(w1)(o1). . .(wn)(on) =h such that: h=V(ϕ)(w1)(o1). . .(wn)(on)∩V(ψ)(w1)(o1). . .(wn)(on) 6. V(ˆxne+1(ϕ)hn,1i) =f ∈Dhn,1isuch that, for everywj ∈W, oj ∈D(wj),1≤j≤n:f(w1)(o1). . .(wn)(on) =h
such that, for everyw∈W:
h(w) ={o∈D(w) :w∈V(ϕ)(w1)(o1). . .(wn)(on)(w)(o)}
7. V((∀s)hn,0i) =f ∈Dhn,0isuch that,
for everywj ∈W, oj ∈D(wj),1≤j≤n:f(w1)(o1). . .(wn)(on) =h
such that:
h={w∈W :V(s)(w1)(o1). . .(wn)(on)(w) =d(w)}
Finally, a termϕof typehiistrue in a modelif and only if ∈V(ϕ).
This concludes the exposition of the Propositional Functions Account. We are now in a position to show why the account is not austere, pace Stalnaker, instead favouring higher-order necessitism.
2.4 Overgeneration of the Propositional Functions Account
Before turning to the case for the Overgeneration of the Propositional Functions Account, I will briefly offer some comments on its virtues. The Propositional Functions Account is an elegant account of the semantics of first-order modal languages. Not only is the account consistent with the conjunction of Aliens and Contingentism — contrary to what was the case with the Literal Account — it also avoids certain somewhat puzzling features of the Haecceities Account. Whereas according to the Haecceities Account the semantic value of an individual constant consists of an haecceity, according to the Propositional Functions Account the semantic value of an individual constant consists of an individual (that is actually something). The latter is, arguably, a more natural view.
These are advantages of the Propositional Functions Account from the standpoint of Thorough Contingentists committed to Thorough Actualism. There is yet another advantage of the Propositional Functions Account over the classic accounts that is orthogonal to the question whether any of these theses is true. Contrary to the other accounts, the Propositional Functions Account does not require an appeal to a notion of semantic value relativised to variable-assignments. Variable-assignments turn out to be, on the Propositional Functions Account, relics of the model-theoretic formalism used to model the semantics of quantified expressions. These relics should not be reflected in an account of the real semantics of quantified expressions. Arguably, these features of the Propositional Functions Account make it more attractive in comparison to the classic accounts. Arguably, the availability of the Propositional Functions Account reveals that the classical accounts confuse the elements of models with the things that they represent.
In this section it will be shown that, despite the advantages of the Propositional Functions Account over the classic accounts, the Propositional Functions Accountovergeneratesfrom the standpoint of Higher-Order Contingentists committed to Thorough Serious Actualism. To explain what precisely is meant with the overgeneration claim, let me introduce some notions and theses. Say that a proposition is anattribution of being tox just in case it is the proposition thatx is something, and that it is anattribution of being(simpliciter) just in case it is possible that there is somexsuch that it is an attribution of being tox. Consider the following theses:
Necessity of Being. Necessarily, there is some individual.
Haecceity Necessitism. Necessarily, every haecceity is necessarily something.
Attributions of Being–Necessitism. Necessarily, every attribution of being is necessarily some- thing.
The Propositional Functions Account overgenerates from the standpoint of proponents of Higher- Order Contingentism committed to Thorough Serious Actualism in the following sense:
Overgeneration of the Propositional Functions Account. The Propositional Functions Account, together with Thorough Serious Actualism and Necessity of Being, implies both i) Haecceity Necessitism, and ii) Attributions of Being–Necessitism.
Let me start by showing that the Propositional Functions Account, Thorough Serious Actualism and Necessity of Being together imply that Jordan’s haecceity is necessarily something, and that the attribution of being to Jordan is necessarily something.
Consider the following expressions: (3) yˆe(a=y)h1,1i
(4) (∃yˆe(a=y))h1,0i.
Note that the expressionsyˆe(a=y)h1,1iand(∃ˆye(a=y))h1,0iare used, for instance, in formulating the claim that(2∀ˆxe(2(Qax→(∃yˆe(a=y)∧ ∃ˆye(x=y)))))h0,0i, i.e., the claim that necessarily, for every individualx, necessarily, if Michael Jordan is a father ofx, then Michael Jordan is something andxis something. Here,xis being used for the variablex1e,yis being used for the expressionx2e, andzforx3e.
According to the Propositional Functions Account the semantic values of(3)and(4)are, respec- tively, i) a first-level property function which necessarily, for every individualy, mapsyto the property of being Jordan, and ii) a propositional function which necessarily, for every individualy, mapsyto the proposition that Jordan is something. From the thesis of Necessary Being and i) it follows that a) necessarily, some individual is mapped to the property of being Jordan — and so, necessarily, some individual is related to the property of being Jordan; and from the thesis of Necessary being and ii) it follows that b) Necessarily, some individual is mapped to the proposition that Jordan is something — and so necessarily, some individual is related to the proposition that Jordan is something.
Finally Thorough Serious Serious Actualism and a) together imply that necessarily, the property of being Jordan is something. Moreover, Thorough Serious Actualism and b) together imply that the proposition that Jordan is something is something.
These consequences are generalisable. Consider the following expression: (5) zˆe(z=xe)h2,1i.
According to the Propositional Functions Account, the semantic value ofzˆe(z = xe)h2,1i is that
2nd-level property functionfwhich necessarily, for everyx, mapsxto that1st-level property function which necessarily, for everyy, mapsyto the property of beingx.
From the thesis of Thorough Serious Actualism it follows that necessarily, for everyx, there is a
1st-level property functiongwhich necessarily, for everyy, mapsyto the property of beingx. From
the thesis of Necessity of Being it follows that i) necessarily, for everyx, there is a1st-level property functiongwhich necessarily, maps someyto the property of beingx. Thorough Serious Actualism and i) together imply that necessarily, for everyx, necessarily, the property of beingxis something. That is, Thorough Serious Actualism and i) together imply Haecceity Necessitism.
Similarly, consider the expression (6) (∃ˆze(z=xe))h2,0i
According to the Propositional Functions Account, the semantic value of(∃ˆze(z=xe))h2,0iis that
2nd-level propositional functionfwhich necessarily, for everyx, mapsxto that1st-level propositional
function which necessarily, for everyy, mapsyto the proposition thatxis something.
From the thesis of Thorough Serious Actualism it follows that necessarily, for everyx, there is a
1st-level propositional function which necessarily, for everyy, mapsyto the proposition thatxis something. From the thesis of Necessity of Being it follows that ii) necessarily, for everyx, there is a1st-level propositional function which necessarily, maps somey to the proposition thatx is something. Thorough Serious Actualism and ii) together imply that necessarily, for everyx, necessarily, the proposition thatxis something is itself something. That is, Thorough Actualism and ii) together imply Attributions of Being–Necessitism.
Hence, the Propositional Functions Account overgenerates from the standpoint of proponents of Higher-Order Contingentism.
How significant is this result? To begin with, Stalnaker’s own higher-order modal theory is committed to Thorough Serious Actualism, as well as to the negation of Haecceity Necessitism and of Attributions of Being–Necessitism. Arguably, Stalnaker is also committed to the necessary being of at least some entities, such as mathematical entities and other abstract objects. Thus, the Overgeneration of the Propositional Functions Account reveals that Stalnaker’s own higher-order modal theory is inconsistent with the Propositional Functions Account. Thus, he cannot hope to appeal to it in order to address the challenge of offering a satisfactory account of the semantics of first-order modal languages consistent with his higher-order modal theory.
The significance of the overgeneration of the Propositional Functions Account goes beyond Stalnaker’s own higher-order modal theory. First, note that typical higher-order contingentists should be at least as opposed to the truth of Haecceity Necessitism as they are to the truth of the negation of No Actual Haecceity, since Haecceity Necessitism implies the falsehood of No Actual Haecceity.
Indeed, higher-order contingentists such as Adams, Fine, Prior and Stalnaker all reject the truth of the conjunction of Haecceity Necessitism and Attributions of Being–Necessitism. Moreover, it is difficult to see how some higher-order entities may fail to be something, while at the same time it is necessary that all haecceities are necessarily something, and that all attributions of being are necessarily something. Arguably, the conjunction of Haecceity Necessitism and Attributions of Being–Necessitism is true only if Higher-Order Necessitism is itself true.
Second, the thesis of Necessity of Being is rather plausible. For instance, the thesis is a direct consequence of the view that there is at least one necessary being. But according to many, things such as the empty set, the number one, and other mathematical entities are all necessary beings.
Finally, in chapter3a defence of Thorough Serious Actualism is offered, a defence that I will assume here to be successful.
Given this information, the plausible higher-order contingentist theories are committed to the Necessity of Being and to Thorough Serious Actualism, and to the falsehood of Haecceity Necessitism and Attributions of Being–Necessitism. The Overgeneration of the Propositional Functions Account
shows that proponents of plausible higher-order contingentist theories will not find in the Propositional Functions Account an account of the semantics of first-order modal languages consistent with their commitments.
Now, the Propositional Functions Account appears to be independently attractive, as was previously shown, in the first paragraphs of the present section. Assuming the truth of Thorough Serious Actualism and of Necessity of Being, the independent attractiveness of the Propositional Functions Account constitutes a pro tanto reason in favour of Haecceity Necessitism and Attributions of Being–Necessitism. So, the attractiveness of the Propositional Functions Account constitutes a pro tanto reason in favour of Higher-Order Necessitism.10