Good, bad, better & worse

We turn now to the semantics of evaluative sentences of natural language with thin evaluative adjectives taking sentential clauses as their arguments.

It is good that

It is good that will have roughly the same semantics as the operator ⇑ ofL_V, except for the fact that the hyperplan’s argument will be supplied by the alternative-generating function of the index.

In addition, we need to ensure that the proposition under evaluation—or more exactly, its

“worldly” component, the set of worlds that make it true—is among the alternatives deter-mined by the index. We can think of this as a kind of semantically hardwired “ought implies can” principle; in order to evaluate anything, it has to be an option. We write it ‘a_i[ϕ](w_i)’;

which reads: the set of alternatives a_i including (the worldly component of) ϕthat are deter-mined at worldw_i.

Thus,it is good thatis a sentential operator: its arguments are propositions, and when it receives an argument, it returns a set of world-hyperplan-alternative set triplets. The truth-conditions of a sentence likeit is good thatϕare as follows:

(4.17) [[it is good thatϕ]]_⟨w

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

A propositionϕis good, relative to an index of evaluationidetermining a worldw_i, a hyperplan hiand a set of alternativesaijust in case the set of worlds that constitute theworldlycomponent ofϕ(or simply ϕ, for short) is among the set of alternativesa_i (includingϕ) that are chosen by h_i. In other words: the index supplies a set of alternatives, which includes ϕ; and then the hyperplan selects a subset of those alternatives. If and only if ϕ is among the selected propositions, it is good thatϕ.

Turning to our previous example (1Nstands for the sentenceKanye gets 1 nomination):

(4.18) [[it is good that Kanye gets 1 nomination]]_⟨w

i,hi,ai⟩=1iff {w^′∶ [[1N]]_⟨w′,hi,ai⟩=1} ∈h_i(a_i[1N](w_i))

In words: relative to an index⟨w_i, h_i, a_i⟩, the sentenceit is good that Kanye gets 1 nomination will be true just in case the worldly component of the sentenceKanye gets 1 nomination(the set of worlds at which Kanye gets 1 nomination) is among the alternatives that are chosen by the hyperplanh_i relative to the set of alternativesa_i(which includes Kanye getting 1 nomination).

Suppose that the alternatives at some index of evaluation are Kanye getting 0, 1 or 2 nomi-nations. Then, this sentence will be true just in case the hyperplan of the index is such that, when it takes as argument the set of propositions {Kanye gets 0 nominations, Kanye gets 1 nomination,Kanye gets 2 nominations}, it returns a subset of that set that at least contains the propositionthat Kanye gets 1 nomination.³

At least as good as, better than

Similarly to the operator > of L_V, the meaning of the comparative operator it is better that results from applyingit is good thatacross subsets of the alternatives determined by the index.

But let us first look at the more basic it is at least as good that. This corresponds to the ≥ operator ofL_V, and so will have the following denotation:

(4.19) [[thatϕis at least as good as thatψ]]_⟨w

i,hi,ai⟩=1iff∀a^′⊆ai, if[[it is good thatψ]]_⟨w

i,hi,a^′⟩=1, then[[it is good thatϕ]]_⟨w

i,hi,a^′⟩=1

(4.19) says that, relative to an index of evaluationi, a propositionϕis at least as good as another propositionψjust in case every subseta^′of the seta_iof alternatives determined by the index is such that, ifψ isgoodrelative tohi(a^′), thenϕisgood too relative tohi(a^′). In other words, there is no subset ofa_i relative to whichψ comes outgoodbutϕdoes not.

In turn, this means the following: for every subset a^′ of a_i, the following must hold: if ψ is good is true relative to⟨w_i, h_i, a^′⟩, thenϕis goodis true relative to⟨w_i, h_i, a^′⟩. This requires evaluating those sentences along the model ofit is good that, which we have just seen. Each of those sentences has the following truth-conditions:

3We are purposefully abstracting away from a number of features of evaluative adjectives in their use as sen-tential operators. For example, sentence mood and whether the complement clause is anif- orthat-clause alter what we may call the “epistemic status” of the complement clause in ait is good that-sentence. Consider the following generalization from Sode2018, p. 407:

(i) It is good that the cat is fat ↝the cat is fat

(ii) It is good if the cat is fat ↝the cat might be fat

(iii) It would be good if the cat were fat ↝the cat is not fat

See Sode2018for discussion of this feature ofgoodand its obvious connection to the semantics of conditionals and mood.

Another general feature of thin evaluatives likegoodandbadthat we are bracketing for now is the fact that they have beneficiary arguments:

(iv) It is good for you that the cat is fat.

The prepositional phrasefor youin that sentence does not denote neither an experiencer (as similar PPs do when they appear with predicates of personal taste Glanzberg2007; Stephenson2007b; Stojanovic2007) nor an opinion holder. We come back to beneficiary arguments in Chapter5.

(4.20) a. [[it is good thatψ]]_⟨w

i,hi,a^′⟩=1iff{w^′∶ [[ψ]]_⟨w′,hi,a^′⟩=1} ∈h_i(a^′(w_i)) b. [[it is good thatϕ]]_⟨w

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

And each of them is true just in case the proposition in its scope is among the alternatives selected by the hyperplan of the index, relative to the alternatives of the index. For (4.19) to be false, one must be able to find a subset of alternatives that makes (4.20a) true but (4.20b) false.

That subset of alternatives must include bothψ andϕand it must be such that the hyperplan of the index selectsψbut notϕ.

Let us look at an example:

(4.21) That Kanye gets 2 nominations is at least as good as him getting 1 nomination.

This sentence has the following truth conditions (2Nand1Nstand forKanye gets 2 nominations andKanye gets 1 nomination, respectively):

(4.22) [[That 2N is at least as good as that 1N]]_⟨w

i,hi,ai⟩=1iff∀a^′⊆a_i, if[[it is good that 1N]]_⟨w

i,hi,a^′⟩=1, then[[it is good that 2N]]_⟨w

i,hi,a^′⟩=1

(4.21) is true at an indexijust in case every subseta^′of the alternatives of the index is such that, if the proposition that Kanye gets 1 nomination is selected by the hyperplan ofiata^′(wherea^′ contains the proposition that Kanye gets 1 nomination), then the proposition that Kanye gets 2 nominations is equally selected by the hyperplan ofiat a^′ (wherea^′ contains the proposition that Kanye gets 2 nominations).

Suppose that the alternatives of the indexa_iare Kanye getting 0, 1 or 2 nominations (a_i={2N, 1N, 0N}). And suppose that the hyperplan of the index is as follows:

h_i,({2N,1N,0N}) = {2N,1N} hi,({2N,1N}) = {2N,1N} h_i,({2N,0N}) = {2N} h_i,({1N,0N}) = {1N}

Relative to this index, (4.21) comes out true, because there is no subset ofa_i that contains both 2N and1N as alternatives and such thath_ichooses1N but not2N.

Now suppose that we evaluate (4.21) relative to the same set of alternatives, but relative to a different hyperplanh^′:

h^′,({2N,1N,0N}) = {2N,1N} h^′,({2N,1N}) = {1N}

h^′,({2N,0N}) = {2N} h^′,({1N,0N}) = {1N}

Relative to this hyperplan, (4.21) comes out false, because there is a subset ofa_i at which h^′ chooses1N but not2N, namely the set{2N,1N}.

As we saw in the semantics ofL_V, it is straightforward to define the comparative it is better thatfrom the equativeit is at least as good as:

(4.23) [[it is better thatϕthan thatψ]]_⟨w

i,hi,ai⟩=1iff [[thatϕis at least as good as thatψ]]_⟨w

i,hi,ai⟩=1and [[thatψis at least as good as thatϕ]]_⟨w

i,hi,ai⟩=0.

For (4.23) to be true, we need to be able to find a subset of the alternatives of the index including bothrelatarelative to which the hyperplan of the index selectsϕbut notψ.

(4.23) [[it is better thatϕthan thatψ]]_⟨w

i,hi,ai⟩=1iff∃a^′⊆a_i such that [[it is good thatϕ]]_⟨w

i,hi,a^′⟩=1and[[it is good thatψ]]_⟨w

i,hi,a^′⟩=0 Now consider the following sentence:

(4.24) It is better that Kanye gets 1 nominations than that he gets 2.

Relative to our previous set of alternativesa_i and hyperplanh_i, (4.24) comes out false because every subset ofa_icontaining2N and1Nrelative to whichh_iselects1Nis also a subset relative to which h_i selects2N. In order to make (4.24) true, we would need to find one such subset selecting 1N but not 2N. Hyperplan h^′, on the other hand, is like that: relative to the set {2N,1N},h^′selects1N but not2N. Therefore, (4.24) is true relative toh^′(anda_i).

Summing up, in this semantics,being goodjust means being chosen among the relevant set of alternatives. And for something to be better than another thing is just for there to exist a set of alternatives including both options relative to which the former option is chosen but the second is not. Comparative “betterness” is derived from absolute “goodness”, following a Delineation approach.

It is bad that, it is worse that

As the reader probably expects, we assign to it is bad that a denotation based on that of the operator⇓ofL_V:

(4.25) [[it is bad thatϕ]]_⟨w

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

⟨hi, ai[ϕ](wi)⟩

A propositionϕis bad relative to an indexithat determines a hyperplan h_i and a set of alter-nativesa_ijust in caseϕis a proposition thath_i dis-prefers ata_i (wherea_i includesϕ).

The meaning of the negative polarity comparativeworsederives from the equative at least as bad as, which in turn is derived fromit is bad that(just asbetterwas derived fromat least as good as, which was derived fromit is good that). The equative would mean the following:

(4.26) [[thatϕis at least as bad as thatψ]]_⟨w

i,hi,ai⟩=1iff∀a^′⊆a_i, if[[it is bad thatψ]]_⟨w

i,hi,a^′⟩=1, then[[it is bad thatϕ]]_⟨w

i,hi,a^′⟩=1

(4.26) says that, relative to an index of evaluationi, a propositionϕis at least as bad as another propositionψ just in case every subseta^′ of the seta_i of alternatives determined by the index is such that, ifψ isbadrelative toh_i(a^′), thenϕisbadtoo relative toh_i(a^′). In other words, there is no subset ofai relative to which ψ comes outbad butϕ does not. The comparative worsewould be defined as follows:

(4.27) [[it is worse thatϕthan thatψ]]_⟨w

i,hi,ai⟩=1iff [[thatϕis at least as bad as thatψ]]_⟨w

i,hi,ai⟩=1and [[thatψis at least as bad as thatϕ]]_⟨w

i,hi,ai⟩=0

For (4.27) to be true, we need to be able to find a subset of the alternatives of the index including bothrelataand relative to which the hyperplan of the index dis-prefersϕbut notψ.

(4.27) [[it is worse thatϕthan thatψ]]_⟨w

i,hi,ai⟩=1iff∃a^′⊆a_i s.t.

[[it is bad thatϕ]]_⟨w

i,hi,a^′⟩=1and[[it is bad thatψ]]_⟨w

i,hi,a^′⟩=0

Before moving on, a complication for this proposal should be highlighted (Carla Umbach,p.

c.). Note that, as was pointed out in §2.5.1, positive form equatives invite an inference to the positive form, while negative form equatives do not:

(2.59) The first movie was as good as the second. ↝̸Both movies were good (2.60) The first movie was as bad as the second. ↝Both movies were bad Our semantics does not predict this, as the equativeas bad as does not involve a reference to the positive formbadrelative to the alternatives defined at the index. A simple solution would be to include that in the meaning of the equative, perhaps as a presupposition.

(4.28) [[thatϕis at least as bad as thatψ]]_⟨w

But since the comparativeworseis derived from the equative, this simple solution would have the bad prediction that the comparative worse would also involve a reference to the positive form.

Rett (2007, p. 217), albeit with different motivations, offers the following alternative path:

assume, first, that the equative means something stronger than at least as, namely exactly as.

Then, equatives formed with antonym adjectives have the same truth-conditions: ϕis as good asψis equivalent toϕis as bad asψ. Secondly, assume that equatives are ambiguous between a “positive-entailing” (c.f., (4.28)) and a “positive-neutral” interpretation (c.f., (4.26)). That is, as good/bad aswould each have the following interpretations:

(4.29) ϕis as good asψ

Third, negative antonyms are in general marked with respect to their positive counterparts. For adjectives liketallandshort, this is shown by e.g., the fact that only the positive antonym takes measure phrases (she is 1.63m tall {# short}). Thus, assume thatbad is equally marked with respect to good. Finally, assume a principled dis-preference to marked forms when these are truth-conditionally equivalent. The result of all this is that the “positive-neutral” interpretation ofbadis ruled out, and only the “positive-entailing” interpretation remains:

(4.29) ϕis as good asψ

On the other hand, this view predicts that positive form equatives are ambiguous between a

“positive-neutral” and “positive-entailing” interpretations. We remain unsure as to whether this is the right prediction, but we will leave this issue for future work.

Relation betweengood, better, worseandbad

In Chapter3, we assumed that the binary relations≥and>were inter-definable using⇑and⇓. Now, by contrast, we have two linguistic items, betterandworse, each of which respectively defined using the simple predicatesgood andbad. What is, then, the relation between them?

As we said, good and bad should not be duals: all that is not good isn’t bad, and all that is not bad isn’t good. We want to allow for certain things to be neither good nor bad. On the other hand, what we do want is thatbetter andworsemirror each other. The semantics does not establish this, so we need to establish by brute force:

Axiom 7 (Better≈worse) for any descriptive and atomic (i.e. non-Boolean) sentences of En-glishϕandψand any index of evaluationi,ϕis better thanψis trueiif and only ifψis worse thanϕis true ati.

Defining pairs of antonyms in this way has been observed to incur into problems (Kennedy 1999, 2013a), in particular when dealing with cross-polar anomalies, but we will ignore that issue for the time being.

The non-intersective character of the extensions of positive and negative hyperplans (let’s call it non-intersectionality; see the definition of negative hyperplans in §3.3.3), the axiom of No Reversaldefined in Chapter3and and the axiomBetter≈worsedefined here result in a number of welcome predictions in this semantics, which we present informally here.

First, a basic antonymity pattern holds between good and bad. In virtue of non-intersectionality, if it is good that ϕ is true, then it is bad that ϕ is false. And conversely, ifit is bad thatϕis true, thenit is good thatϕis false.

Secondly, note however that the previous entailment does not hold in the opposite direction: if it is good thatϕis false, it does not follow thatit is bad thatϕis true. And conversely, ifit is bad thatϕis false, it does not follow thatit is good thatϕis true. This means two things: first, goodandbadare not duals: all that is not good isn’t bad, and all that is not bad isn’t good. And second, some things are neither good nor bad. Both of these are welcome.⁴

Thirdly,ϕis better thanψis compatible with any application ofgoodto any ofϕandψ, except it is good thatψ andit is not good thatϕ. This holds in virtue ofNo Reversal. Similarly,ψ is worse thanϕis compatible with any application ofbadto any ofϕandψ exceptit is bad that ϕandit is not bad thatψ, also in virtue ofNo Reversal. But now we can go further. In virtue ofBetter≈worse, these consequences of better/ worseare interchangeable: ϕis better than ψ rules out the conjunction ofit is bad that ϕandit is not bad that ψ; andψ is worse thanϕ rules out the conjunction ofit is good thatψandit is not good thatϕ. And finally, in virtue of non-intersectionality, the conjunction of it is bad thatϕandit is good thatψ is ruled out too.

In sum, ifϕis better thanψ, then it cannot be the case thatψ is good andϕis bad.

This concludes our initial application of the language ofL_V to evaluative adjectives of natural

4Perhaps we would like as well to predict intra-adjectival gaps, that is, some things being neither good nor not good, and neither bad nor not bad. Since hyperplans are maximally decided, they will be defined for each and every possible alternative. To circumvent this, a procedure of supervaluation over hyperplans could be introduced such thatϕis good relative to a set of hyperplans iff it is good relative to all hyperplans in the set, and so on.

This might be independently desirable: recall that hyperplans are maximally decided planning states, but real world planners are hardly (if ever) maximally decided. Quantifiying over hyperplans would be the obvious way of formalizing undecidedness.

language. Remember that we have focused on a relatively small fragment of English, namely the two simplest evaluative adjectives,goodandbad, in positive and comparative form and tak-ing sentences as arguments. It is time to expand this story, and we’ll do that in three directions:

first, we’ll look at argumentsotherthan sentences, namely action types and individuals. Then, we will see how to incorporate increasing degrees of “thickness” into the picture, that is, of descriptive elements.

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