Semantics

To give a semantics forL_V, first we define a semantic structure, a semantic model and a notion of truth-at-an-index of evaluation. Then, we give compositional semantic rules for the sentences ofL_V.

First, a structureS is a tuple formed by a setW of possible worlds and a set of hyperplansH.

Definition 12 (Semantic structure forL_V) S= ⟨W, H⟩

These parameters play a role in evaluating the truth of sentences ofL_V at an index of evaluation.

All sentences ofL_V have uniform semantic values, but whereas sentential constants will impose a certain condition on the set of possible worldsW, formulae containing the operators of L_V will impose conditions on the set of hyperplansH. His a set of hyperplans, which are functions from sets of propositions to sets of propositions. Propositions are sets of possible worlds, and since the domain and range of any hyperplan inHare sets of propositions, the domain and range of any hyperplan are subsets of the power set ofW,P (W), that is, the set of all propositions.

In §3.3.3we defined hyperplans as follows:

Definition 10 (Hyperplans) A hyperplan is a total functionhfrom non-empty sets of proposi-tions to non-empty sets of proposiproposi-tions such that, for every non-empty set of proposiproposi-tionsU, V such thatV ⊆U,h(U) =V just in caseV are the chosen alternatives atU.

And as an auxiliary notion, we have a negative preference function on hyperplans, h

. For any set of propositions U and any hyperplan h, h

selects the propositions among U that are dis-preferred. Recall our previous definition:

Definition 11 (Negative hyperplans) A negative hyperplan is a total function h

from hyper-plans and non-empty sets of propositions to non-empty sets of propositions such that, for every hyperplanhand sets of propositionsU, V such thatV ⊆U∖h(U), h

⟨h, U⟩ =V just in caseV are the alternatives atU thathdis-prefers.

A model is a tupleM consisting of a structure S and a valuation function V mapping every sentential constant ofL_V to a subset ofW×H, that is, to a set of world-hyperplan pairs.

Definition 13 (Model forL_V) M = ⟨S, V⟩

Sentences ofL_V are evaluated for truth at indices of evaluation, which are triplets formed by a possible world, a hyperplan and the set of all propositions P (W). P (W) is held constant across all sentences ofL_V, but it will be useful to have it in the index in order to show how the binary operators ofL_V work.

Definition 14 (Index forL_V) ⟨w_i, h_i,P (W)⟩

That is, the index provides a world a hyperplan and the set of all propositions (we will often useias variable over indices).

Sentences are evaluated for truth at indices of evaluation. As we saw in §3.3.2, we want the sentential constants of L_V to be descriptive. Ignoring P (W), which is held constant, this means that sentential constants should impose a condition on the world but not on the hyperplan parameter of the index. And to do that, we need the valuation function V to assign to the sentential constants sets ofw, h,P (W)triplets formed by pairwise combining each of a subset of worlds with every possible hyperplan andP (W). That way, sentential constants will come out world-, but not hyperplan-sensitive.¹⁷

(3.18) [[p]]^M

⟨wi,hi,P(W)⟩ = 1 iff ⟨w_i, h_i,P (W)⟩ ∈ V(p)(we skip superscript M in what fol-lows)

The semantic rules of composition for the Boolean connectives are the following:

(3.19) [[∼ϕ]]_⟨w Let us turn now to the evaluative operators introduced byL_V.

The positive form:⇑,⇓

The unary operators⇑,⇓have the following semantic entries:

(3.23) [[⇑ϕ]]_⟨w

i,hi,P(W)⟩=1iff{w^′∶ [[ϕ]]_⟨w′,hi,P(W)⟩=1} ∈h_i(P (W))

In words: ‘⇑ ϕ’ is true at an index i just in case the set of possible worlds in the worldly component of ϕ’s denotation is among the alternatives chosen by the hyperplan of the index, relative to the set of all propositions.

(3.24) [[⇓ϕ]]_⟨w

i,hi,P(W)⟩=1iff{w^′∶ [[ϕ]]_⟨w′,hi,P(W)⟩=1} ∈ h

⟨h_i,P (W)⟩

In words: ‘⇓ϕ’ is true at an indexijust in case the hyperplan ofiis such that the set of worlds in the worldly component ofϕis among the dis-preferred outcomes.

Let us highlight three aspects that can be potentially confusing at this stage: first, note that the only relationship between ⇑ and ⇓ that falls out from the semantics so far is that their extensions are non-intersective (this is a consequence of the definition of negative hyperplans given in §3.3.3). This establishes that if⇑ϕis true, then⇓ϕis false; and if⇓ϕis true, then⇑ϕ is false. But that does not hold in the opposite direction: if⇑ϕis false, it does not follow that

⇓ϕis true, andviceversa. This is how it should be, since⇑and⇓are not duals. This will be clearer in Chapter4when we map⇑ontogoodand⇓ontobad, asgoodandbadare not duals:

all that is not good isn’t bad, and all that is not bad isn’t good. As we will see shortly though,

⇑and⇓are further relatedviathe binary operators ofL_V.

17Remember that lettingV assign sets of possible worlds (rather than sets ofw, h,P (W)triplets) to sentential constants while reserving the more “complex” denotations for evaluative sentences gets us into trouble: if we did that, we would not be assigning uniform semantic values to all formulae. In turn, this would make it problematic to account for Boolean combinations of descriptive and non-descriptive sentences. See §3.3.2.

Secondly, recall that we uniformly assign to any sentence of L_V a set of world-hyperplan-P (W)triplets. This means that we cannot simply say that an indeximakes ‘⇑ϕ’ true if[[ϕ]]

is one of the chosen alternatives at h_i... because [[ϕ]] is not (only) a set of worlds—it is a set of world-hyperplan-P (W)triplets. In order to make a sentence likeϕthe argument of an operator like ⇑, we need to abstract away from its hyperplan and alternative components and retain its “wordly” component, the set of worlds in the denotation of ϕ. To do that, we take {w^′ ∶ [[ϕ]]_⟨w′,hi,P(W)⟩ =1}, that is, the “worldly” component of ϕ, and we say that that set of worlds is among the chosen alternatives. So for instance, ifϕis the sentenceMora runs, then [[Mora runs]]is the set of worldw, hyperplan handP (W)triplets such that Mora runs atw.

Now ifMora runsis embedded under ⇑, we get the formula ‘⇑Mora runs’. But, according to our semantics, the semantic argument of⇑is not a set of world-hyperplan-P (W)triplets, but only the worldly component of that set—the worlds at which Mora runs. And then, ‘⇑Mora runs’ is true at an index of evaluationijust in case the set formed by those worlds belongs in the set of preferred alternatives of the hyperplan ofirelative to the set of alternativesP (W). Finally, note that, in the truth conditions for⇑and⇓, the hyperplan of the index is invariably fed the set of all sets of worlds,P (W), which is also a constant parameter in the index of evaluation.

In practice, this means that the hyperplan of the index does not behave like a function—because it receives a constant argument. This is a formal oddity, but it is an idealization that for now will make our lives simpler. When we move fromL_V to natural language evaluatives, we will see how hyperplans can and do take as arguments varying sets of alternatives that are also supplied by the index of evaluation (analogously to the way in which, in Yalcin’s proposal, hyperplans were fed epistemic states).

Equatives & comparatives: ≥,>

The binary operators are defined with the aid of the semantics of either⇑or⇓. Let’s start with

≥:

(3.25) [[ϕ≥ψ]]_⟨w

i,hi,P(W)⟩=1iff∀X⊆ P (W)such that{w^′∶ [[ϕ]]_⟨w′,hi,P(W)⟩=1} ∈X&

{w^′∶ [[ψ]]_⟨w′,hi,P(W)⟩=1} ∈X, if[[⇑ψ]]_⟨w

i,hi,X⟩=1, then[[⇑ϕ]]_⟨w

i,hi,X⟩=1.

In words, a proposition ϕis at least as valuable as a propositionψ just in case, relative to an index i, every subset of alternatives containing ϕ and ψ among which hi chooses ψ is one among whichϕis also chosen. In other words, there is no set of alternatives containingϕand ψ relative to whichh_ichoosesψbut notϕ.

Alternatively, we can define≥with⇓: (3.26) [[ϕ≥ψ]]_⟨w

i,hi,P(W)⟩=1iff∀X⊆ P (W)such that{w^′∶ [[ϕ]]_⟨w′,hi,P(W)⟩=1} ∈X&

{w^′∶ [[ψ]]_⟨w′,hi,P(W)⟩=1} ∈X, if[[⇓ϕ]]_⟨w

i,hi,X⟩=1, then[[⇓ψ]]_⟨w

i,hi,X⟩=1.

In words, a proposition ϕis at least as valuable as a propositionψ just in case, relative to an indexi, every subset of alternatives containing ϕ andψ among which h_i disprefers ϕis one among whichψ is also dis-preferred. In other words, there is no set of alternatives containing ϕandψ relative to whichh_i dis-prefersϕbut notψ.

From this we can easily define the comparative connective>:

(3.27) [[ϕ>ψ]]_⟨w

i,hi,P(W)⟩=1iff[[ϕ≥ψ]]_⟨w

i,hi,P(W)⟩=1and[[ψ≥ϕ]]_⟨w

i,hi,P(W)⟩=0 In words: relative to an indexi, a proposition ϕis more valuable than a propositionψ just in caseϕis at least as valuable asψbut not the other way around. Again, this can be defined with

⇑or⇓. That is, one cannot find a set of alternatives (a subset ofP (W)containingϕandψ) at whichψ[ϕ] but notϕ[ψ] is chosen [dis-preferred], although one can find a set of alternatives at whichϕ [ψ] but not ψ [ϕ] is chosen [dis-preferred]. In other words, the hyperplan of the index is such that there is at least one subset of P (W)containingϕandψ at which ϕ[ψ] is

Defining the binary operators (≥,>) ofL_V using either⇑or⇓has the consequence that a relation is established between ⇑ and ⇓. Simply put, that relation is that the orderings induced by recursive applications of either⇑or⇓mirror each other. If we use⇑to order two propositions, sayϕandψ, it follows that an ordering has been established involving⇓(and viceversa).¹⁸ This can be seen more clearly if we suppose that, where ⇑is mapped onto good and⇓ is mapped ontobad, >is mapped onto betterif read left to right and worseif read right to left. We will discuss this further in the next chapter.

By purposefully holding fixed the hyperplan’s argument in the semantic entry for the unary operators, we can see the difference between the unary and the binary operators of L_V: at an index of evaluationi, the unary operators⇑and⇓impose certain conditions on the hyperplan ofirelative to a fixed set of alternatives, namelyP (W). By contrast, relative to the same index of evaluation, when evaluating formulae containing the binary operators, the hyperplan ofiis not fedP (W). Rather, the hyperplan is fed subsets ofP (W); and the various binary relations are defined by quantifying over those subsets.

The equal-value connective≈& defining a degree system

The equative connective≥also lets us define an equal-value relation≈as one would expect:

(3.30) [[ϕ≈ψ]]_⟨w

i,hi,P(W)⟩=1iff[[ϕ≥ψ]]_⟨w

i,hi,P(W)⟩=1and[[ψ≥ϕ]]_⟨w

i,hi,P(W)⟩=1 Additionally, using≈we can define a degree system by identifying a propositionϕ’s degree of value, writtenµ_V(ϕ), with the set of propositions that stand in the≈relation toϕrelative to an indexi.

(3.31) [[µV(ϕ)]]_⟨w

i,hi,P(W)⟩= {ψ∶ [[ϕ≈ψ]]_⟨w

i,hi,P(W)⟩}

Note, however, thatµ_V is not part of the vocabulary ofL_V. The sole purpose of definingµ_V is to show that we can conceive of degrees as equivalence classes of propositions (Cresswell 1976; Klein 1980, a.o.), and use the formal apparatus of degree semantics (that is, measure

18(Informal) proof: if⇑ϕand∼⇑ψ, thenϕ>ψ. By the inter-definability of>via⇑or⇓, it follows that there is a subsetX of alternatives ofP (W)includingϕandψsuch that⇓ψand∼⇓ϕrelative toX.

functions likeµ) to represent the meaning of certain constructions. This will be convenient in

§4.6.

To proceed in this way is to use a Delineation approach to the semantics of the operators of L_V: we start by giving a non-scalar semantics for the unary operators ⇑and ⇓, and we then give a scalar semantics for the binary operators ≥, >and ≈by quantifying over subsets of the set of alternatives relative to which the unary operators are evaluated, namely P (W). The procedure is similar to the one we presented at the end of Chapter2fortall: the positive form tall is assigned a non-scalar extension relative to a comparison class. Any individual in that comparison class will fall in the extension, anti-extension or extension gap of tall. And in order to define the comparativetaller, we introduce quantification over comparison classes: an individualxis taller than an individualyjust in case there exists a comparison class relative to whichxis tall butyis not (introducing quantification over comparison classes, in turn, makes comparatives appropriatelyinsensitiveto the comparison class supplied by context).

IteratedL_V operators?

Before moving on, let us observe that it is not straightforward to say what, if anything, the meaning of formulae containing iterated instances of the operators ofL_V should be. What is the meaning of ‘⇑⇑ϕ’, ‘⇓∼⇑ϕ’ or ‘⇑ϕ>ψ’?

At the outset, let me point out that it is not clear how to interpret the corresponding English sentences. Assume, as we will, that⇑represents the positive form of an evaluative adjective, e.g. good, and that > represents the comparative, e.g. better. Then, a formula like ‘⇑⇑ ϕ’

would be equivalent to the English sentence it is good that it is good that ϕ. And a formula like ‘⇑ϕ>ψ’ would be equivalent toIt is good that it is better that ϕthan thatψ. These are sentences whose meaning is intuitively difficult to spell out.

Regardless, it is important to say what our semantics predicts the corresponding formulae to de-note. In general, the meaning of any evaluative formula that is under the scope of an evaluative operator is equivalent to the “total” proposition, i.e. W, for the following reason: evaluative operators impose a condition on, and only on, the worldly component of the formulae that they take as arguments. But the worldly component of any evaluative formula is the set of all worlds. Therefore, any evaluative operator that takes an evaluative formula as its argument will only “see” the worldly component of that formula, namely the set of all possible worldsW. Consider this example:

(3.32) [[⇑⇑ϕ]]_⟨w

i,hi,P(W)⟩=1iff{w^′∶ [[⇑ϕ]]_⟨w′,hi,P(W)⟩=1} ∈h_i(P (W))

Since ‘⇑ϕ’ imposes no condition on the world parameter, the set of worlds at which⇑ϕis true is the set of all worlds. Thus,

(3.33) [[⇑⇑ϕ]]_⟨w

i,hi,P(W)⟩=1iffW ∈hi(P (W))

Admittedly, this is a little strange: ‘⇑⇑ ϕ’ is true just in case W is among the chosen set of alternatives relative to the hyperplan of the index. However, the set of all worlds makes no proposition false, and therefore, it does not seem like a valid choice for a hyperplan.

Furthermore, consider Boolean combinations including evaluative formulae under the scope of other evaluative operators, such as ‘⇑∼⇓ϕ’. If our semantics predicts that evaluative formulae in the scope of evaluative operators contribute the total proposition, and since the embedded formula is negated, the meaning of ‘∼⇓ϕ’ under⇑turns out to be the empty set.

(3.34) [[⇑∼⇓ϕ]]_⟨w

i,hi,P(W)⟩=1iff

{w^′∶ [[∼⇓ϕ]]_⟨w′,hi,P(W)⟩=1} ∈h_i(P (W)) =1iff {w^′∶ [[⇓ϕ]]_⟨w′,hi,P(W)⟩=0} ∈h_i(P (W)) =1iff

∅ ∈h_i(P (W))

And given that the empty set is included in every set whatsoever, it is included in the set of preferred alternatives, thereby making ‘⇑∼⇓ϕ’ trivially true at any index of evaluation.

There are solutions that one might try to apply to make these results less awkward. Syntac-tically, one could ban evaluative operators from embedding evaluative formulae and Boolean combinations thereof. Semantically, it might make sense to preclude the total proposition W from being a possible value of h(P (W)), for any h. That would make evaluative formulae embedded under evaluative operators uninterpretable. Either move could be welcome, in view of the observation that the corresponding natural language sentence are hard to interpret. But given that they are hard to interpret, it is not clear either that the predictions that we obtain from our semantics are actually bad in the first place. We leave this issue for future consideration.