Modality, Weights, andInconsistent Premise Sets*

Alex Silka.silk@bham.ac.uk

Dra of May

Abstract

It is a commonplace that some of our desires are stronger than others;that certain values, norms, and rules are more important than others; andthat states of affairs can be more likely or typical than others. Some authorshave claimed that standard analyses of modals cannot capture how the truthconditions of modal sentences can be sensitive to such differences in strengthand priority. I develop an interpretation of a standard premise semantics formodals that captures various ways in which weights and priorities can gurein the interpretation of modal sentences. I then extend the proposed analysesin considering certain inference patterns involving modals and comparatives,and modals and conditionals. e resulting framework provides an attractiveaccount of the relation between the conventional meanings of modals and thelogic of weights and priorities.

*anks to Fabrizio Cariani, Kai von Fintel, IreneHeim, Ezra Keshet, Dan Lassiter, Bob Stalnaker,Eric Swanson, and participants at SALT for helpful discussion, and to two anonymous reviewersfor DEON for their valuable comments. A preliminary version of the ideas in this paper can befound in Anca Chereches (Ed.), Proceedings of Semantics and Linguistic eory (SALT) , –, §.

IntroductionIt is a commonplace that some of our desires are stronger than others; that certainvalues, norms, and rules are more important than others; and that states of affairscan be more likely or typical than others. ese differences in strength and prior-ity can affect which modal claims are true. Suppose you promised Alice that youwould meet her for lunch, but you also promised your ailing mother that you woulddrive her to the hospital for a critical operation. You realize that you can’t keep bothpromises, but it ismuchmore important that you keep your promise to yourmother.Suppose also that there are no other normatively relevant factors. Intuitively, () istrue and () is false. (Here and throughout, assume that the modals are given auniform type of normative reading.)

() You must keep your promise to your mother.() You must keep your promise to Alice.

Call this case ‘ ’.Cases like raise a prima facie challenge for the classic premise

semantic framework formodals. Modals are treated as receiving their reading or in-terpretation from a contextually supplied set of premises. Since modals can them-selves occur in intensional contexts, it is standard to index premise sets to a world ofevaluation (written: ‘Pw’). To a rst approximation, given a consistent set of premisesPw, ‘Must ϕ’ says that the prejacent ϕ follows from Pw. Making room for non-trivialinterpretations of modal claims given inconsistent premise sets, ‘Must ϕ’ says thatϕ follows from every maximally consistent subset of Pw. Slightly less roughly, givena premise set Fw (a “modal base”) that describes some set of relevant backgroundfacts in w, and given a further premise set Gw (an “ordering source”) that representsthe content of some ideal (morality, your goals, etc.) in w, ‘Must ϕ’ says that ϕ fol-lows from every maximally consistent subset of Fw ∪ Gw that includes Fw —or, interms of the simplifying notation in De nition , that ϕ follows from every set in (Fw,Gw).

Equivalently, a preorder (L ). See especially K , a, ; also F , L , V .

K a, ; cf. G : –. is premise semantic implementation isequivalent to the perhaps more familiar implementation in K a, which uses theordering source to induce a preorder on the worlds compatible with the modal base (L ).For simplicity I make the limit assumption (L : –) and assume that ordering the con-sistent subsets of Fw ∪ Gw (that include Fw as a subset) by set inclusion ⊆ yields a set of subsets thatare ⊆-maximal. For semantics without the limit assumption, see L , K a, ,

De nition . (Fw,Gw) ∶= {P ∶ cons(P) ∧ ∀P′ [P ⊂ P′ ⊆ Fw ∪Gw → ¬cons(P′)]∧Fw ⊆ P ⊆ Fw ∪Gw}, where, for a set of propositions S, cons(S) iff ⋂ S ≠ ∅

De nition . ‘Must ϕ’ is true at w iff ∀P ∈ (Fw,Gw) ∶⋂P ⊆ ϕ

What premise sets are called for in a case like ? As David Lewiscounsels us, “Wemust be selective in the choice of premises…By judicious selection,we can accomplish the same sort of discrimination as would result from unequaltreatment of premises” (: –; cf. K b: ). e question iswhether we can do so in a way that captures the full range of data and re ects ourintuitive views about the norms that are relevant in the context.

e ‘must’s in ()–() are interpreted with respect to the relevant norms. enormative force of your promise to Alice should be represented, even if this promiseis ultimately outweighed by the promise to your mother. Intuitively, the norms thatgure in the interpretation of ‘must’ in ()–() are the same as the norms that gure

in the interpretation of ‘have to’ in ().

() If you didn’t visit your mother, you would have to meet Alice for lunch.

So, we might treat as calling for premise sets like the ones in() that describe the relevant circumstances and the contents of the relevant prac-tical norms, where l is the proposition that you meet Alice for lunch, and h is theproposition that you take your mother to the hospital.

() Fw = {¬(h ∩ l)}Gw = {h, l}P = Fw ∪Gw = {¬(h ∩ l),h, l}

But this won’t do. P is inconsistent. Holding xed the relevant circumstance that youcan’t keep both promises, P has maximally consistent subsets P = {¬(h∩ l),h} andP = {¬(h∩ l), l}. Since neither h nor l follows from both P and P, this incorrectlypredicts that each of () and () is false (though it correctly predicts that ‘You mustkeep your promise to Alice or yourmother’ is true). e problem is that there seems

S . I assume that the prejacents of modals and the elements of premise sets are proposi-tions, conceived as sets of possible worlds. For expository purposes I focus only on strong necessitymodals, like ‘must’ and ‘have to’; weak necessity modals, like ‘ought’ and ‘should’, raise complicationsorthogonal to our discussion here (see S for my preferred account).

I will use ¬ϕ as an abbreviation for ϕ′ = W/ϕ, and ϕ ⊃ ψ as an abbreviation for ϕ′ ∪ ψ =W/ϕ∪ψ. I will oen use unitalicized capital letters or ‘w’ in referring to worlds, italicized lowercase(English or Greek) letters in referring to propositions, and italicized capital letters in referring to setsof propositions.

to be no room for representing how your promise to your mother is stronger thanyour promise to Alice, or how the premise h has priority to the premise l. ereseems to be no mechanism for breaking the tie between the maximally consistentsubsets of P.

It is rare to nd explicit articulations of the premises that gure in the inter-pretations of modals. Not unsurprisingly, problems concerning weights and prior-ities among premises have received little attention in the formal semantics litera-ture. Daniel Lassiter () is one of the few to address these issues; his assessmentis not optimistic: “e problem is fundamentally that the theory makes no roomfor one [premise’s] being stronger than another; instead any con ict of [premises]leads to incomparability”; “the theory doesn’t leave any room for [one premise’s] be-ing stronger or weaker than another… [Premises] are all-or-nothing” (: –,–; see also –, ). Considering an essentially equivalent semantics tothe one in De nition , Lou Goble comes to a similar conclusion: “it fails to takethe relative weight or signi cance of [premises] into account” (: –).

ere are various ways we might respond to the problem posed by cases like . ere are rich literatures in deontic logic on capturing pri-orities among default rules and the logic of prima facie obligations. One responsewould be to revise the classic semantics by importing additional apparatus fromthese theories into our semantics for modals. ere may be reasons in the end fordoing so. However, I will argue that with a more nuanced characterization of thepremises that gure in the interpretation of modals, we can capture a wide range oflinguistic data concerning weights and priorities within the classic premise seman-tic framework. Roadmap: First, I develop an interpretation of a standard premise

In context Lassiter is considering the implications of this alleged failure to represent weights forKratzer’s account of (epistemic and deontic) comparatives.

For instance, abstracting away from details of implementation, we might introduce a ranking⪯Gw on the propositions in a given premise set Gw. We could then introduce a function Gw thattakes Fw and ⪯Gw as argument and returns a preorder on the elements of (Fw,Gw). Making ananalogue of the limit assumption, we could then say that ‘Must ϕ’ is true iff ϕ follows from every setin (Fw,Gw) that ranks among the highest in the preorder Gw(Fw,⪯Gw). In ,Gw and ⪯Gw would privilege the maximally consistent subset P over P; it would only be P fromwhich the modal’s prejacent would be said to follow. is would correctly predict that () is true but() is false. is is essentially the route taken by Goble (: –), following work by H() and H (, ); cf., e.g., A & B , B , B & L, T & T , H . See K . for a related account thatrepresents cascades of priorities via an operation of orderedmerging of ordering sources thatmirrorsthe lexicographical product for posets. is account can be seen as developingKratzer’s (b, ,) strategy of “lumping” propositions into additional premises to capture the relative importanceof facts in the interpretation of counterfactuals.

semantics that captures various ways in which weights and priorities can gure inthe interpretation of modal sentences (§§–). I then extend the proposed analysisto the case of graded modal expressions in considering certain inference patternsinvolving modals and comparatives (§). e resulting framework provides an at-tractive account of the relation between the conventional meanings of modals andthe logic of weights and priorities (§).

Weights, priorities, and applicability conditionsWe need a more nuanced enough understanding of the considerations that gurein the interpretation of modals. Norms— like values, goals, desires, etc.— typicallydon’t come in the form of blunt categorical imperatives or commands. ey aren’tusually of the form No matter what, ϕ!. Rather they oen come with what I will callapplicability conditions (“ACs”), or conditions under which they apply. If I want to gofor a run, my desire needn’t be that I go for a run, come what may. More plausibly, itis that I go for a run given that it’s nice outside, among other things. Our preferencesare oen conditional, preferences for certain circumstances. Similarly with moralnorms. A norm against lying needn’t take the form of a categorical prohibition. Itmight be something to the effect that you don’t lie unless not lying would lead to aninnocent person’s death, or would put your family in grave danger, or …. Normscan thus be understood on the model of conditional imperatives, imperatives thatenjoin an action or state of affairs given that certain circumstances obtain. iscaptures the intuitive idea that depending on the circumstances— i.e., dependingon which applicability conditions are satis ed—only certain norms may apply, orbe “in force.” Fixing terminology, call the content of a conditional norm, goal, etc. aconsideration. Given a consideration ‘If C, ϕ’, let C be the consideration’s applicabilitycondition, and ϕ be the consideration’s premise, or what the consideration enjoinsgiven C. (Categorical considerations can be treated as conditional on the tautology.)

ere are a number of ways applicability conditions might be integrated into thesemantics. One option, suggested by the few explicit remarks that there are on thecontents of ordering sources in concrete examples, would be to build them into thepropositions that are the elements of ordering sources. e elements of orderingsources would be identi ed with considerations, construed as material conditionalsC ⊃ ϕ. Simplifying quite a bit, the relevant deontic premise set in

For example: “one should take the propositions that make up the relevant deontic backgroundcontext l(w) to consist just of such conditional propositions,” like that “If someone owns a car and isnot handicapped, he must pay taxes” (F : ; see also –, –, P : ,

might be something like in (), where pa is the proposition that you promise tomeetAlice for lunch; pm is the proposition that you promise to take your mother to thehospital; l is the proposition that you meet Alice for lunch; and h is the propositionthat you take your mother to the hospital.

() Gw = {(pa ∩ ¬pm) ⊃ l, pm ⊃ h}

One might worry that this approach will require the elements of ordering sources tobe extraordinarily complex. In our simplistic example there were only two promisesat play. But in more realistic cases there may be many competing norms, and theremay be no simple way to describe their interaction and the conditions under whichthey apply. Even given the sorts of idealizations common in semantics, it is hardto see contexts as supplying such bodies of premises. A more serious problem withthis way of capturing the role of applicability conditions is that it makes incorrectpredictions. It treats all normative considerations as categorical commands of con-ditionals rather than as conditional commands (and all bouletic considerations aspreferences that a conditional be satis ed rather than as conditional preferences,and so on). is raises problems familiar from discussions of material conditionalanalyses of conditionals.

Suppose Little Timmy has some free time aer school, followed by a piano recitaland a basketball game. Timmy’sMom andDad both tell him to nish his homeworkwhen he gets home from school before he does anything else. Finishing his home-work rst is most important. Timmy’s Mom tells him to practice piano too if he n-ishes his homework in time, but Timmy’s Dad tells him to practice his free throws ifhe nishes his homework in time. Even if he nishes his homework, Timmy won’tbe able to practice both piano and basketball before needing to leave. When Timmygets home from school, his parents are out, unable to be contacted, and they won’tbe returning until they have to leave for the recital and game. Given the currentway of capturing applicability conditions, this case would seem to call for a deonticpremise set like in () that describes Timmy’s parents’ commands, where h is theproposition that Timmy nishes his homework, p is the proposition that he prac-tices piano, and b is the proposition that he practices basketball. is yields the ele-ments of (Fw,Gw)—the maximally consistent subsets of Fw ∪ Gw that includeFw —in () (see n. ).

F : – (following unpublished notes by Kratzer on information-sensitivity), S: –, ).

Cf. H /: –; A-H : ; N : ; H: –;also the “quali cation problem” in AI (MC & H , MC ).

() Fw = {¬(b ∩ p)}Gw = {h,h ⊃ b,h ⊃ p}

() P = {¬(b ∩ p),h,h ⊃ b}P = {¬(b ∩ p),h,h ⊃ p}P = {¬(b ∩ p),h ⊃ b,h ⊃ p}

Intuitively, in light of his parent’s commands, Timmy must work on his home-work rst. However, since ¬h is compatible with P, the deontic premise set in ()incorrectly predicts that the sentences in () are true, assuming that ‘may’ and ‘ispermissible’ are duals of ‘must’.

De nition . ‘May ϕ’ is true at w iff ∃P ∈ (Fw,Gw) ∶⋂ (P ∪ {ϕ}) ≠ ∅

() a. It’s permissible for Timmy not to work on his homework rst.b. Timmy may refrain from working on his homework rst.c. Timmy doesn’t have to work on his homework rst.

If Timmy doesn’t do his homework rst when he gets home, he can’t defend him-self by saying that he complied with his parents’ conditional norms by making theirantecedents false. Little Timmy would never be so cray.

e case of Little Timmy highlights that we must only consider considerationswhose applicability conditions are satis ed when evaluating modal claims. I notedabove that it is standard in linguistic semantics to index premise sets to a world ofevaluation. is feature of the classic semantics makes available an alternative wayof distinguishing considerations whose applicability conditions are satis ed.

Which premise sets are relevant for the evaluation of a given modal sentencecan depend on how things happen to be in the actual world, or on how things couldbe but aren’t, or could have been but weren’t. What Little Timmy’s parents com-mand might change from one world to the next. ey could have told Timmy topractice piano rst thing aer school. us the meaning of a phrase like ‘what Little

Another strategywould be to revise the classic semantic framework by treating ordering sourcesnot as sets of propositions, but as sets of pairs of applicability conditions (propositions) and premises(propositions). (See the input/output logic of M & T , for relatedrepresentations of conditional norms.) A consideration ‘IfC,ϕ’ would be represented by a premise setthat includes the pair ⟨C, ϕ⟩. Only those considerations whose applicability conditions are entailedby the relevant background facts would gure in the interpretation of the modal: ‘Must ϕ’ could betreated as saying that ϕ follows from every set in (Fw,G′w), where G′w is the set of premises ψsuch that there is a pair ⟨C, ψ⟩ in Gw such that ⋂Fw ⊆ C. Since my aim here is to capture the role ofweights and priorities within the classic semantic framework, I won’t pursue this approach. We cancapture the intuitions driving it utilizing resources already present in the classic semantics.

Timmy’s parents command’ that determines the intended reading of ‘must’ in ()can be treated as a function that assigns to every possible world the set of proposi-tions describing the house rules in that world.

() In light of what Little Timmy’s parents command, hemust do his homeworkrst.

Similar remarks hold for the meanings of phrases like ‘in view of the relevant cir-cumstances’, ‘according to U.S. law’, and so on. It is these functions that contextsupplies for the interpretation of modals. Call these functions unsaturated premisesets (written ‘S’). Call the value of an unsaturated premise set given a world of eval-uation a saturated premise set, or simply a premise set (written ‘Sw’).

We can capture the role of applicability conditions in terms of variability in thevalues of unsaturated premise sets at different worlds. Suppose we have a consid-eration ‘If C, ϕ’ which enjoins ϕ given that conditions C obtain. We can representthe content of this consideration with an unsaturated premise set S that assigns toevery relevant C-world a premise set that includes ϕ. For example, the import ofyour desire to go for a run, mentioned above, would be re ected in S’s assigning apremise set that includes the proposition that you go for a run to worlds in which theweather is nice (among other things). e premises in a saturated premise set thusre ect what follows from a body of considerations—what is enjoined by a body ofconditional norms, what is preferred in light of a body of conditional preferences,what is expected in light of a body of evidential relations, etc.— given the circum-stances that obtain in the evaluation world.

Objection: “e characterization of applicability conditions will be problemati-cally circular in certain cases. For example, the applicability condition for the normenjoining you to keep your promise to Alice in won’t just bethat you haven’t promised to take your mother to the hospital. It will presumablybe some more general condition, like that you haven’t made any more importantpromises that are incompatible with your keeping your promise to Alice. But thismakes reference to weights and the relative importance of various norms, which isprecisely what needs to be explained.”

Translated to Kratzer’s terminology, my unsaturated premise sets correspond to “conversa-tional backgrounds,” and saturated premise sets correspond to “modal base” and “ordering source.”I invoke new terminology because of occasional variation in how terms like ‘modal base’, ‘orderingsource’, and ‘conversational background’ are used in the literature— e.g., whether they refer to sets ofpropositions, sets of worlds, functions from worlds to sets of propositions, or functions from worldsto sets of worlds (which is not to say that Kratzer herself was unclear about this).

anks to Dan Lassiter and Paul Portner for pressing me on this issue.

Reply: We as theorists might use modal notions and talk of weights and priori-ties in describing the intuitively relevant considerations in a given circumstance, andin explaining what makes it the case about a concrete situation that such-and-suchpremises are enjoined. But it is the ways things are in the world of evaluation— thefeatures of the world on which the facts about priorities and applicability conditionssupervene— that determines what premises are enjoined. Information about appli-cability conditions, weights, and priorities is encoded in, but not itself recoverablefrom, an unsaturated premise set. It is not explicitly delineated in the semantics. Anintuitive body of considerations and relative priorities determines an unsaturatedpremise set, but not vice versa. (More on this in §.)

Capturing the role of applicability conditions in this way may also stave off thecomplexity worry above with treating the elements of premise sets as material con-ditionals. e complexities in the material conditional antecedents are re ected inthe current framework in terms of how certain distinctions among worlds affect thevalue of an unsaturated premise set at those worlds. Descriptions of the elementsof premise sets need only be as complex as the descriptions of what the relevantconsiderations enjoin.

ApplicationsIn this section I will apply this interpretation of the classic premise semantic frame-work to several examples.

. Little TimmyStart with the case of Little Timmy. Little Timmy’s parents, recall, both requiredhim to work on his homework rst thing aer school. His Mom told him to prac-tice piano next if (and only if) he nishes his homework, whereas his Dad told himto practice basketball. For simplicity, assume that Timmy will in fact either work onhis homework, practice basketball, or practice piano. As before, let h be the proposi-tion that Timmy works on his homework rst, p be the proposition that he practicespiano, and b be the proposition that he practices basketball. Let wC be a world inwhich Timmy completes his homework with time to spare before needing to leavefor the game and recital, and let wC be a world in which Timmy doesn’t complete hishomework. (ese can be treated as representatives of suitable equivalence classesof worlds.) To a rst approximation (see below), we can capture the relevant cir-cumstances and Timmy’s parents’ commands with unsaturated premise sets with

the following properties:

() FwC = FwC= {¬(b ∩ p),h ∪ b ∪ p}

GwC = {h, b, p}GwC= {h,¬(b ∪ p)}

is correctly predicts that () is true and () is false both at wC and at wC.

() Timmy must work on his homework rst.() Timmy may refrain from working on his homework rst.

Even if it isn’t settled whether Timmywill complete his homework before needing toleave— even if wC and wC are both live possibilities— it can be known that he mustwork on his homework.

. Weights and contrary-to-duty imperativesReturn to . What we need to capture is that () is true and ()is false; your promise to your mother is more important than your promise to Alice.

() You must keep your promise to your mother.() You must keep your promise to Alice.

Consider the following worlds, AM, AM, AM, AM, characterized with respect towhether or not you promise to meet Alice and whether or not you promise to helpyour Mother in them. We can capture the normative import of your promises asfollows, again where h is the proposition that you take your mother to the hospitaland l is the proposition that you meet Alice for lunch.

() GAM = {h}GAM = {l}

What matters for present purposes is simply that we capture how Timmy must work on hishomework before practicing piano or basketball. What one says about Timmy’s obligations withrespect to practicing piano and practicing basketball if he does complete his homework will dependon one’s views about the possibility of irresolvable practical dilemmas. I leave open this issue here,since the aim in this paper is to examine cases where the relevant norms are comparable to oneanother, and so the apparent dilemmas can be resolved. See my for discussion of how theframework developed in this paper can be applied to data involving irresolvable dilemmas expressedwith weak vs. strong necessity modals.

GAM = {h}GAM = ∅

enormative import of your promise to Alice is re ected inG’s assigning a premiseset that includes l to some world in which you make this promise, namely AM. enormative import of your promise to your mother is re ected in G’s assigning apremise set that includes h to some world in which you make this promise, e.g.AM (or AM). And the priority of keeping your promise to your mother over yourpromise to Alice is re ected in G’s assigning a premise set that includes h to someworld in which you make both promises, namely AM. (Had your promises been ofequal importance, this would be re ected in a premise set that contains the disjunc-tion h ∨ l.) is unsaturated premise set G correctly predicts that () is true and() is false in the given context, i.e. at AM: h, but not l, follows from every set in (FAM,GAM).

() (FAM,GAM) = {{¬(h ∩ l),h}}

Prima facie con icts among norms needn’t lead to incomparabilities.One might worry that this treatment of makes incorrect

predictions concerning contrary-to-duty imperatives. Consider ()–().

() You have to take your mother to the hospital.() If you don’t take your mother to the hospital, you have to meet Alice for

lunch.

Assuming a standard Kratzerian () treatment of conditionals (De nition ),the antecedent of () adds the proposition ¬h to FAM, and the modal ‘have to’ isinterpreted with respect to this updated modal base F+AM = FAM ∪ {¬h}. Given theproposed premise sets for FAM and GAM, this seems to incorrectly predict that ()is false, as re ected in ()–():

De nition . ‘If ψ, ϕ’ is true at w iff ∀P ∈ (F+w,Gw) ∶⋂P ⊆ ϕ, where F+w =Fw ∪ {ψ}

() F+AM = {¬(h ∩ l),¬h}GAM = {h}

I use ‘have to’ instead of ‘must’ to avoid potential complications from the entailingness of ‘must’.(I.e., many speakers nd ‘Must ϕ, but ¬ϕ’ and ‘Must ϕ, but might ¬ϕ’ to be marked, even for deonticreadings of ‘must’. For such speakers, accepting ‘Must ϕ’ has the potential to violate the presupposi-tion of conditionals ‘If ¬ϕ…’ that ¬ϕ is a live possibility.)

(F+AM,GAM) = {F+AM}() () is true at AM iff

∀P ∈ (F+AM,GAM) ∶⋂P ⊆ l iff{¬(h ∩ l),¬h} ⊆ l

ere are subtle issues concerning time, andwhat information is taken for granted,which may complicate the interpretation of pairs of claims like ()–(). Intu-itively, in interpreting the modal in () one assumes that the acts of keeping yourpromise to yourmother and keeping your promise to Alice are both available to you.In interpreting the modal in the consequent of the conditional in (), by contrast,one assumes that the act of keeping your promise to your mother is no longer anoption. One way of capturing this is to re ne our indices of evaluation. e normsencoded in G must be conditional not only what promises you have made but alsoon what acts are available to you. Let AM be a circumstance in which the optionof keeping your promise to your mother and the option of keeping your promiseto Alice are both available, and let AM be a circumstance in which only the op-tion of keeping your promise to Alice is available. A more ne-grained deonticunsaturated premise set G∗ can be given as follows:

() G∗AM = {h}G∗AM = {l}⋮

Assuming that () is evaluated at a circumstance like AM at which keeping yourpromise to your mother is still available, we continue to predict that () is true.Adopting a double modal analysis of deontic conditionals delivers the correct inter-pretation for (). Roughly, on such an analysis, in interpreting a deontic condi-tional one checks whether themodalized consequent clause is veri ed at all relevant(circumstantially accessible, epistemically accessible, closest) worlds in which theantecedent holds. is delivers the following simpli ed truth conditions for ():

() () is true atw iff for all relevant¬h-worldsw′∶ ∀P ∈ (Fw′ ,Gw′) ∶⋂P ⊆ lAs before, AM and AM can be treated as representatives of suitable equivalence classes of

worlds. e sensitivity to time or available acts may also be captured by treating indices as world-time pairs or situations. I remain neutral on these alternative implementations. For discussion oftimeless contrary-to-duty cases, see, e.g., P & S , .

See F , G , F & I , L , S ,S . I leave open what kind of reading the posited higher necessity modal is to be given.

is correctly predicts that () is true atAM, and thus consistentwith (): roughly,() is true at AM iff ‘you have tomeet Alice’ is true at AM iff⋂{¬(h∩l),¬h, l} ⊆ l.

. Outweighing and undercuttingW was a case where one consideration was outweighed by an-other con icting consideration. Now consider a case where the applicability of onepremise undercuts, or excludes, the applicability of another premise. Suppose Bettyis a cadet, and her Captain orders her to clean the barracks. Ordinarily, this wouldimply that Betty has to clean the barracks. But the Major, who outranks them both,orders Betty to ignore the Captain’s command. Intuitively, () is true.

() Betty doesn’t have to clean the barracks.

e Major’s command, it is oen claimed, isn’t an ordinary, weightier rst-orderreason; rather, it undercuts the consideration about the Captain’s command frombearing on Betty’s deliberation.

Let b be the proposition that Betty cleans the barracks, c be the proposition thatthe Captain ordered Betty to clean the barracks, and m be the proposition that theMajor ordered Betty to ignore the Captain’s command. Let CM be the world as itis described by the case, and CM be an otherwise similar world in which the Majordoesn’t order Betty to ignore theCaptain. We can capture the contents of the relevantnorms at play with an unsaturated premise set with the following properties:

() GCM = ∅GCM = {b}

enormative import of theCaptain’s command is re ected inG’s assigning a premiseset that includes b to (c∧¬m)-worlds in which the Major doesn’t interfere. e un-dercutting role of the Major’s command is re ected in G’s assigning a premise setthat fails to include b to (c ∧m)-worlds. is correctly predicts that () is true inthe given context (i.e., at CM): ⋂{c,m} ⊈ b.

at the Captain’s command is undercut, and not outweighed, is re ected in thefact that the Major needn’t forbid Betty from cleaning the barracks or order her toperform some alternative action. Hemight just want to undermine the Captain’s au-

Compare the distinction between “rebutting” defeat and “undercutting” defeat in the litera-ture on epistemic reasons (P , ), and the notions of “exclusionary” reasons (R/, G , P ) and “overridden” requirements (C , ) inethics.

thority. e undercutting role of the Major’s command can be further reinforced byconsidering aminor extension of the case. Adapting an example fromH (:–), suppose that the situation is as before, but Betty also received an orderfrom her Lieutenant to do drills. Since the Captain outranks the Lieutenant, ordi-narily Betty would have to obey the Captain’s command and not do drills (assumingshe can’t both clean the barracks and do drills). But given the Major’s command toignore the Captain, there is now nothing excluding the Lieutenant’s command fromapplying. e contents of the relevant norms can be re ected as follows, where d isthe proposition that Betty does drills, l is the proposition that the Lieutenant orderedBetty to do drills, and Lxx is an l-world.

() GLCM = {d}GLCM = {b}GLCM = ∅

is correctly predicts that () and () are true in the revised context (i.e., atLCM): {c,m, l,¬(b ∩ d),d} entails d but doesn’t entail b.

() Betty must do drills.

More generally, the contrasting ways in which considerations can be undercutand outweighed is represented in terms of what premise sets are assigned at certainminimally different worlds. Let a and b be relevant conditions, AB be a relevant(a ∧ b)-world, and AB be a relevant (a ∧ ¬b)-world. (For simplicity, suppose thatGAB andGAB are each consistent, and bracket the role of F.) Suppose that⋂GAB ⊆ ϕ,re ecting that given a and absent some defeating condition b, ϕ is necessary. epremise that ϕ, with applicability condition a, is outweighed if there is a premise ψwith applicability condition b, where ϕ∩ψ = ∅, such that⋂GAB ⊆ ψ and⋂GAB ⊈ ϕ.By contrast, the premise that ϕ is undercut by a background condition b simply if⋂GAB ⊈ ϕ.

. Epistemic readingsSo far we have been focusing on root modals. e proposals for capturing prioritiesamong premises, and for distinguishing outweighing from undercutting, apply toepistemic readings of modals as well. First, consider a familiar epistemic analog of . Suppose you are looking at a ball. It seems red to you, but apeer tells you that the ball isn’t actually red. Your sense perception is general reli-able; typically, the fact that an object seems red is good evidence that the object is

red. But your peer is eminently trustworthy and might have access to informationthat you don’t— for example, perhaps there are unusual lighting conditions. e re-liability of your peer, let’s suppose, is even greater than that of your sense perception.Intuitively, this outweighs your reason for thinking that the ball is red— indeed, itgives you reason for thinking the ball is not red. Suppose that there is no other rele-vant possible evidence that may bear on the color of the ball. () seems true in thisscenario.

() e ball must not be red.

We can capture this as follows. Let l be the proposition that the ball looks red; r bethe proposition that the ball is red; and t be the proposition that your peer told youthat the ball isn’t red. Consider the worlds LT and LT, characterized in the expectedway as above. We can capture the priorities among the relevant evidential normswith an unsaturated premise set with the following properties.

() GLT = {¬r}GLT = {r}

eevidential import of the object’s looking red is re ected inG’s assigning a premiseset that includes r to ¬t-worlds where the ball looks red. e priority of your peer’stestimony is re ected inG’s assigning a premise set that includes¬r to t-worlds. ispredicts that () is true in the given context (i.e., at LT): ⋂{l, t,¬r} ⊆ ¬r.

Now consider a case of epistemic undercutting. Suppose that rather thanhearingfrom your peer that the object is not red, you realize that you have taken a drug thatmakes everything look red. Intuitively, this undercuts your reason for concludingthat the ball is red. () is false in this scenario.

() e ball must be red.

Let d be the proposition that you took the drug, and consider the worlds LD andLD, characterized in the expectedway. We can capture the probabilistic informationencoded in the relevant epistemic norms with an unsaturated premise set with thefollowing properties:

() GLD = ∅GLD = {r}

eevidential import of the object’s looking red is re ected inG’s assigning a premiseset that includes r to ¬d-worlds in which the ball looks red. e undercutting role of

taking the drug is re ected in G’s assigning the empty set to d-worlds. is predictsthat () is false in the given context (i.e., at LD): ⋂{l,d} ⊈ r.

Modals and comparativesAn adequate general treatment of prioritiesmust extend to the case of comparatives,like ().

() It is better for me to keep my promise to my mother than for me to keepmy promise to Alice.

Indeed, in the passages fromL cited in §, Lassiter’s central criticism ofthe implementation in K a, , is that its proliferation of incon-sistencies in premise sets leaves it unable to capture the truth of such comparatives.e primary aim of this paper has been to capture how intuitions about prioritiesamong considerations can affect the truth conditions of modal sentences. e pro-posed strategy is compatible with various views on the semantics of comparativeslike () and the semantic relation between modals and comparatives. Neverthe-less I would like to brie y mention one way of extending the account from §§– tocapture certain data involving comparatives.

I suggest that comparatives like () have a kind of counterfactual element totheir meaning: in considering whether an option ϕ is better than another option ψ,one looks at relevant possibilities in whichϕ orψ is necessary (in the relevant sense),and assesses whether ϕ is necessary in those possibilities. Informally, () seems tomean something like “If I had to keep my promise to my mother or to Alice— andconditions were otherwise normal, expected, or as desired— I would have to keepmy promise to my mother.” Roughly, ‘ϕ is better than ψ’ is true at w iff for all closest(maximally similar) relevant worlds w′ to w at which ‘Must ϕ ∨ ψ’ is true, ‘Must ϕ’is true. More formally:

De nition . ‘ϕ is a better possibility than ψ’ is true at w (written: ϕ ≺w ψ) iff for allclosest relevant worlds w′ to w such that ∀P ∈ (Fw′ ,Gw′) ∶⋂P ⊆ ϕ ∨ ψ, it’s thecase that ∀P ∈ (Fw′ ,Gw′) ∶⋂P ⊆ ϕ

For simplicity, I will treat ‘ϕ is at least as good asψ’ as true when it’s not the case thatψ ≺ ϕ.

ere may be reasons for strengthening De nition to require not simply that ψ ⊀w ϕ, butrather than for all closest relevant w′ to w at which ‘Must ϕ ∨ ψ’ is true, ‘Must ψ’ is false. is

De nition . ‘ϕ is at least as good a possibility as ψ’ is true at w (written: ϕ ⪯w ψ) iffψ ⊀w ϕesede nitions correctly predict that () is true. Reconsider the proposed premisesets for at world AM in which you both promise to meet Aliceand promise to help your mother:

() (FAM,GAM) = {{¬(h ∩ l),h}}

e closest world to the evaluation world AM at which ‘Must h ∨ l’ is true is AMitself, givenWeak Centering for the similarity relation among worlds (i.e., given thatif u is a ϕ-world, u is the closest ϕ-world to u). Assuming AM is among the closest“relevant” suchworlds, the truth of the comparative () follows straightaway: ‘Musth’ is true at AM.

Seth Yalcin () and Daniel Lassiter (, ) have argued that the follow-ing problematic inference is validated on Kratzer’s (a, , ) semanticsfor comparative modals.

() Disjunctive Inference:a. p is at least as likely as qb. p is at least as likely as rc. ∴ p is at least as likely as q ∨ r

e semantics in De nition – avoids validating this inference. Here is a model:Suppose (i) the closest relevant worlds i at which ‘Must p ∨ q’ is true are such that (Fi,Gi) = {{s ∧ ¬t ∧ ¬u, p}}; (ii) the closest worlds j at which ‘Must p ∨ r’ istrue are such that (Fj,Gj) = {{¬s ∧ t ∧ ¬u, p}}; and (iii) the closest worlds k atwhich ‘Must p ∨ q ∨ r’ is true are such that (Fk,Gk) = {{s ∨ t ∨ u, q ∨ r}}. en(a) and (b) are true, but (c) is false. For instance, let p be the proposition thatx wins the tournament; q be the proposition that y wins the tournament; r be theproposition that zwins the tournament; s be the proposition that teams x and y playin the nals; t be the proposition that teams x and z play in the nals; and u be theproposition that teams y and z play in the nals.

Lassiter has also objected to Kratzer’s semantics on the grounds that it fails tocapture certain apparent entailment relations among modals and comparatives. For

would follow on the assumption that all the worlds w′ being considered agree in what is possible andnecessary (permitted and required)—perhaps understood as a sort of “comparability” assumptionbetween ϕ and ψ. e two de nitions would collapse given conditional excluded middle (L: –). I assume that the relevant sense of “closeness” is the same (context-dependent) senseused in interpreting subjunctive conditionals, however it is ultimately to be cashed out.

instance, it fails to validate inferences like the following (see, e.g., his : –;there he focuses speci cally on epistemic readings, but the points extend to deonticreadings as well).

() a. Must ϕb. ψ ⪯ ϕc. ∴Must ψ

e semantics proposed here also doesn’t validate this inference (though slight vari-ants would do so). However, contrary to Lassiter, I take this to be a feature, not a bug.I am doubtful about whether we should treat the inference in () as semanticallyvalid. First, it is intuitively invalid for deontic readings. It excludes the possibility ofsupererogatory acts— acts that go “beyond the call of duty,” acts that are permittedbut not required and better than what is minimally required—as re ected in ().

() a. I must give of my income to charity.b. It’s better for me to give of my income to charity than for me to

give of my income to charity.c. ∴ I must give of my income to charity.

Even if there are no supererogatory acts, this should be determined on the basis ofsubstantive normative theory, not logic or semantics.

Second, the inference pattern in () is arguably invalid even in the case of epis-temic readings. Epistemic ‘must’ sentences make claims about what follows from arelevant body of evidence. Considered in this light, there may seem something puz-zling about the constraint on bodies of evidence semantically required by (): whyshould we expect the fact that ψ is at least as likely as ϕ to imply that, necessarily,any body of evidence that entails ϕ must also entail ψ? is is abstract; consider aconcrete example. Suppose you are inside and see a bunch of people coming in withwet umbrellas. You infer, plausibly enough, that it must be raining; you accept ().

() It must be raining.

You also accept ().

() It’s at least as likely to be cloudy as it is to be raining.

Aer all, it’s nearly always cloudy when it rains. Nevertheless, you are inside withno access to windows. Your evidence doesn’t itself say anything about whether it iscloudy. For all your evidence suggests, there is a sun shower. It is hard to see why

it would be incoherent, or a re ection of semantic incompetence, for you to thinkthat it might not be cloudy and fail to accept ().

() It must be cloudy.

My aim in this section has been modest. My claim is simply that we can extendthe premise semantic treatment of priorities from §§– to give a preliminary anal-ysis of certain expressions of graded modality. is analysis avoids problems facingprevious attempts. However, there may ultimately be reasons for giving expressionsof comparative possibility an overtly probabilistic semantics. More thorough inves-tigation of (e.g.) inferences involving modals and comparatives, similarities and dif-ferences between epistemic readings and deontic readings, and the relation betweenexpressions of comparative possibility and overt probabilistic language is required.Such additional investigations must be le for future work.

Concluding re ectionsOn a standard premise semantics for modals, modals are interpreted with respect toa contextually supplied function from worlds to premise sets. is function—whatI have called an “unsaturated premise set”—determines the intended reading forthemodal. In this paper I have offered an interpretation of these unsaturated premisesets as encoding the content of a body of conditional norms, values, preferences,expectations, etc. (“considerations”). e value of an unsaturated premise set ata world of evaluation w represents what follows from such a body of considera-tions given the circumstances that obtain in w. is way of thinking about premisesets makes empirically correct predictions about various ways in which weights andpriorities can affect the truth conditions of modal sentences. e project has notbeen to argue that no other theory can get the data concerning weights and priori-ties right. It has been to motivate one way of doing so that is empirically adequate,methodologically conservative, and theoretically attractive.

In this spirit I would like to close by brie y returning to an alternative strategyfor capturing the role of weights and priorities raised in §. Consider a prioritizeddefault theory in the style of, say, H (, ), consisting of a set of back-ground facts F, a set of default rules representing defeasible norms or generaliza-tions, and an ordering relation < on the set of default rules re ecting their relativepriority. Following Horty’s terminology, say that a default rule A → B whose an-

See, e.g., P , L , , K .

tecedent is entailed by F is triggered, and that a triggered default is defeated if there issome other triggered default that is given higher priority according to <. To a veryrough rst approximation, we could then say that ‘Must ϕ’ is true iff every maxi-mally consistent set of conclusions of non-defeated triggered defaults entails ϕ (cf.nn. , ). For example, in , let pm → h be the default rule dthat you help your mother given that you promised to do so; let pa → l be the defaultrule d that you meet Alice for lunch given that you promised to do so; and supposed < d, re ecting that your promise to your mother has priority over your promiseto Alice. Since both defaults are triggered, but d is defeated by d, the set of conclu-sions of non-defeated triggered defaults is {h}. us, ‘You must help your mother’is true, and ‘You must meet Alice for lunch’ is false.

e premise semantics in § simply utilizes a function from worlds to premisesets, where the premises in a premise set represent what is ultimately enjoined giventhe circumstances that obtain in the evaluation world. Information about the ap-plicability conditions of particular norms and about the priority relations amongnorms is encoded in, but not recoverable from, a deontic unsaturated premise set.e above kind of default-theoretic analysis, by contrast, explicitly represents a fac-tual condition for each conclusion and a priority relation among defaults. And itmakes explicit how the set of conclusions of non-defeated triggered defaults is gen-erated. Given the independent motivation for introducing this sort of extra struc-ture into a logic of practical and theoretical reasoning, why not introduce it into oursemantics for modals?

ere may ultimately be empirical reasons for incorporating the above sort ofapparatus into our semantics for modals. e strategy taken up in § may not gen-eralize to more complex cases. But if it does, we should be cautious about buildingthe extra structure into the conventional meanings of modals. At least for the exam-ples considered in this paper, it has been sufficient to utilize unsaturated premisesets that encode the contents of considerations (conditional norms, etc.) in terms ofwhat premise sets are returned at certain worlds, given the relevant circumstancesin those worlds. For the purposes of capturing the intuitively correct truth condi-tions we have been able to bracket the substantive normative and logical questionsof how to generate unsaturated premise sets from intuitive bodies of considerationsand weights, and how to derive premise sets from sets of background conditionsand considerations. is is not simply a point about theoretical economy. By beingneutral on these issues, we can avoid building controversial logical and normativeviews into the conventional meanings of modals— e.g., concerning the proper logi-cal representations of certain cases, the proper order in which to apply defaults, and

consequence (non)monotonicity. If this is right, our best logical theories can beseen as providing an explicit account of something that is taken as given by the com-positional semantics: how unsaturated premise sets are generated from the intuitiveconsiderations and priorities relevant in concrete discourse contexts. is suggestsan attractive way of situating the respective work in deontic logic and the semanticsof modals in an overall theory of modality and modal language.

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