Hyperplans as functions over alternatives

The shortcomings of Gibbard-Yalcin semantics are clear, but they can be overcome. Put simply:

if one wants to understand value in terms of properties of hyperplans, we need to be able to represent a scale of value using hyperplans. This is what we are going to do in this section. To that effect, we propose to change the Gibbard-Yalcin semantics in two ways. First, let us drop the assumption that the arguments of hyperplans are information states. This assumption was necessary to account for deontic modals, but it is not clear that evaluatives rely on epistemic information in the same way that deontics do.

Secondly, the way that we are going to characterize hyperplans, the property of an epistemic state (or of any set of possible worlds) that the hyperplan will “see”, so to speak, are not simply

14A different Kratzerian strategy would be to turn to the distinction between strong and weak necessity modals (Fintel and Iatridou2005; Von Fintel and Iatridou2008), which has been applied to the contrast between extreme and non-extreme deontic adjectives (crucial vs. important, see Portner and Rubinstein 2016). However, that distinction is still too coarse to capture the scalarity of evaluatives.

the possible worlds contained in it, but rather the possible alternatives that are salient at that state. In our characterization, hyperplans are going to be functions from sets of sets of worlds to subsets thereof containing the chosen, or preferable, alternatives at the original set. This second tweak will allow us to model degrees of positive and negative value as degrees of priority: a proposition has a high degree of positive value just in case one’s plan is to give priority to that proposition over certain alternatives.

Besides the technical problem that was pointed out in the previous subsection—that everything that is known is normatively required—my main reason for turning to a different definition of hyperplans is that the Gibbard-Yalcin proposal is too coarse-grained. Intuitively, hyperplans select desirable, permissible or ideal states of affairs among different options. But the worlds that make those states of affairs true might also make a lot of other things true, things that we do not want a hyperplan to “care for”, so to speak.

Recall our previous example: we want Camila’s planning state to be represented by hyperplans that require her to start packing. But given that it is late, the worlds in which she starts packing are also worlds in which it is late. However, we do not want to say that hyperplans that require her to start packing also require it to be late. Being late is not part of what is required; rather, it is part of the circumstances relative to which the hyperplan requires the agent to start packing.

More generally, whatever is required by a hyperplan should not include circumstances such as it being late. To capture this, rather than have hyperplans select sets of possible worlds, we can make them select sets of propositions. In the case of Camila, a hyperplan would be a function that, given her situation, selects the singleton set containing the proposition that Camila starts packing, rather than the set of worlds that make that proposition true. By making a hyperplan select sets of propositions, we are not wedded anymore to the view that whatever is already true across the selected worlds comes out required by a hyperplan. In order to be required by a hyperplan, it is not enough for a proposition to be true across worlds that are selected as preferable. Rather, the relevant proposition has to be selected by the hyperplan among a set of possible alternatives.

For further illustration, let us consider the example that we previewed at the end of §3.3.1(and which will be a running example in this and the following chapter). Suppose that Kanye West has released a new album, and we are about to learn what Grammy nominations he got. Since he is an ambitious, global artist, he only cares for two categories: Song of the Year and Album of the Year; but since he is African American and the Grammys are famously racist, it is not guaranteed that he will be nominated, let alone awarded—however good the new album is.¹⁵ We can think of Kanye’s predicament in terms of his epistemic state, defined by the proposition that he releases a new album. In Yalcin’s view, a hyperplan takes you from an information state, understood as something like what you see on the left column of Table3.2, to a subset of that state where the impermissible options are ruled out. Alternatively however, we can also think of that very same situation in terms of the set of possible alternatives or developments ahead of him: namely, that his new album gets 0, 1 or 2 nominations (out of the categories that he cares about). This is represented on the right column of Table3.2:

Consider again information state e: that state is characterized by a set of worlds, namely worlds at which Kanye releases a new album. We will say that a hyperplan, when fed that set of

15 ‘ “[My Beautiful] Dark [Twisted] Fantasy” and “Watch the Throne”: neither was nominated for Album of the Year, and I made both of those in one year. I don’t know if this is statistically right, but I’m assuming I have the most Grammys of anyone my age, but I haven’t won one against a white person.’ (K. West, NY Times, 2013).

Epistemic state e Set U of alternatives at e

Kanye’s new album receives 0 nomination Kanye’s new album receives 1 nomination Kanye releases a new album

Kanye’s new album receives 2 nominations

Table 3.2: Kanye’s epistemic state e as a set of worlds and as a set of alternatives.

possible worlds, distinguishes the different alternatives at that information state and selects one or more. That is, hyperplans see the right, rather than the left column of Table3.2. In order to model such fine-grainedness, we will say that hyperplans do not take sets of possible worlds as arguments and return subsets of those sets as their value. Rather, we will say that their arguments are sets of propositions that represent the different alternatives that are open in a given situation. Their values will be subsets of those sets of alternatives, which represent the preferable or chosen alternatives relative to the former set of alternatives.

Hyperplans are defined as follows (an illustration is given in Table3.3):¹⁶

Definition 10 (Hyperplans) A hyperplan is a total function h from 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 case V are the chosen alternatives at U .

Set U of alternatives Set U of alternatives

Kanye’s new album receives 0 nomination Kanye’s new album receives 0 nomination Kanye’s new album receives 1 nomination ↦ Kanye’s new album receives 1 nomination Kanye’s new album receives 2 nominations Kanye’s new album receives 2 nominations

Table 3.3: A hyperplan as a function from sets of alternatives to sets of alternatives.

Note that we are not too far from Gibbard’s characterization. In our view, a hyperplan is a function from sets of alternatives to subsets thereof. But we can formulate this as a set of conditional imperatives, with an alternative set in the antecedent and the preferred options in the consequent. For instance, the conditional imperative version of the hyperplan characterized by Table3.3would be:

(3.17) If given a choice between 0, 1 and 2 nominations, let it be 2 nominations!

16The notion of alternatives comes mainly from question and focus semantics, but the details of that literature are not important for our present concerns (see e.g. Aloni2007, 2018; Alonso-Ovalle 2006; Hamblin 1973;

Karttunen2008; Rooth1992 for classic and recent references). All we need to do is keep in mind that a set of alternatives is a set of propositions, which in turn are sets of possible worlds. This notion should also not be confused with the notion of (relevant) alternatives often used in (meta-)epistemology (D. Lewis1996; Stine1976).

Similarly, the expression alternative should not make the reader think that hyperplans can select only one. We might as well speak of sets of options (where it seems more intuitive to say that one can choose more than one) or outcomes—although the latter seems to suggest that these are outcomes of something, which need not be the case (thanks to Carla Umbach for suggesting this).

Closer to our present interests, a similar notion of alternatives to the one employed here is used by Starr to model imperatives (Starr2018). It bears mentioning that hyperplans, in this characterization, are formally very similar to choice functions from economics, which select the most preferable among a set of given options (see Hansson and Grüne-Yanoff2018for an introductions, esp. §5 and ff.). Indeed, Silk (2015) has defended a preference semantics for expressivists which recruits and deploys such concepts from economics. His proposal is scalar and has much in common with the present proposal, although it is not aimed specifically at capturing the scalarity of particular natural language items.

Note, however, that we are describing a model, but we haven’t said yet what sentences or for-mulae this model could represent. In the next section of this chapter, we will use this model to describe a formal language, and in Chapter4we will use a slightly more sophisticated ver-sion of this model to represent the meaning of evaluative sentences of natural language. At this stage, we are simply considering some formal properties of a possible plan of action, one that has as its outcome that Kanye obtains two out of two Grammy nominations. Eventually, this plan will figure in the semantics of sentences that evaluate positively that situation, that is, sentences like it’d be great if Kanye got 2/2 nominations or it’s good that he got 2/2 nom-inations. Those sentences will be true relative to a point of evaluation that contains a plan according to which, when faced with a choice between 0, 1 and 2 nominations, 2 nominations are preferred. But recall that, even if it is Kanye who is evaluating those sentences, the plan according to which he gets 2/2 nominations will not be such that Kanye himself chooses for him to obtain 2/2 nominations. This would be obviously bad for reasons given in §3.3.1: first, Kanye is not the one making the choice. And secondly, no single person decides the outcome of the Grammy nominations. As we said, we need to understand these sentences as endorsing a plan for an agent to choose whatever action leads to that outcome (in the case of a vote, this should most reasonably be understood as a plan for each of the people taking part in the vote to vote appropriately).

The general picture will be that evaluative sentences represent properties of hyperplans; and to make an evaluative judgment is, as we will see in later chapters, to propose to adopt hy-perplans that share a certain property. More specifically, positive evaluative adjectives will be characterized by hyperplans that prefer, or choose, the things under evaluation over cer-tain alternatives; while negative evaluative adjectives will be characterized by hyperplans that dis-prefer, or avoid, the object of evaluation. That is, in a nutshell, our approach.

It is important to stress another parallelism with possible worlds and propositions that was already mentioned: just as possible worlds might be thought of as maximally opinionated states of belief, hyperplans can be thought of as maximally specific states of planning. In our terms, this means that hyperplans are defined for absolutely any set of possible alternatives (as said in the definition, hyperplans are total functions). That is, for any set of propositions U , a hyperplan returns a subset of U consisting of the chosen alternatives at U . But just as actual states of belief are never maximally opinionated, real-life plans (not hyperplans) are never defined for absolutely any contingency. Therefore, we can define PLANS as partial functions from sets of propositions to preferred alternatives; or correspondingly as sets of hyperplans that agree on some of their instructions but not on others.

Hyperplans can avoid or dis-prefer alternatives as well, which will be crucial to the semantics of evaluative adjectives of negative polarity, such as bad, cruel or ugly. The easiest (if not the most elegant) way to model negative preference is to assign a negative function to each hyperplan h, call it h

, that selects the alternative(s) that h avoids or dis-prefers out of every set of alternatives:

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 hyperplanh and sets of propositions U, V such that V ⊆ U ∖ h(U ), h

⟨h, U ⟩ = V just in case V are the alternatives atU that h dis-prefers.

The function h

behaves like a hyperplan in that it is a total function on sets of alternatives, and it can select only some or all alternatives as undesirable. In addition, note that, since the range

of a negative hyperplan relative to a hyperplan h and a set of alternatives X is a subset of X minus the result of applying h to X, the positive and negative choices of a hyperplan never overlap.

To sum up: in this section, we introduced Gibbard’s notion of hyperplan as a view about the content of normative judgments. In Gibbard’s view, hyperplans are semantic primitives, just like possible worlds. Then, we saw via Yalcin’s 2012proposal how to implement Gibbard’s ideas in a compositional, intensional semantics for deontic ought that, in addition, no longer treats hyperplans as primitives, but rather as functions over sets of possible worlds. However, we argued that, in order to apply a hyperplan semantics to evaluative sentences, we needed to move beyond Yalcin’s characterization of hyperplans. By conceiving of hyperplans as functions that select preferred and dis-preferred sets of alternatives, we have seen how to overcome the problems that Yalcin’s implementation had, while preserving much of the spirit of that proposal.

In the next section, we make all of the preceding observations more formal by defining a simple propositional language that introduces scalar evaluative operators whose semantics make use of hyperplans.