Hyperintensionality

First published Mon Feb 8, 2021

Ahyperintensional concept draws a distinction betweennecessarily equivalent contents. If the concept is expressed by anoperator, \(H\), then \(H\) is hyperintensional insofar as \(HA\) and\(HB\) can differ in truth value in spite of \(A\) and \(B\)’sbeing necessarily equivalent. Necessary equivalence of claims isstandardly understood in terms of possible worlds (ways things couldhave been): \(A\) and \(B\) are necessarily equivalent when they aretrue at the same worlds. This is sometimes put in terms of sentencessharing an intension. Anextensional operator (e.g.,Boolean negation) allows substitutionsalva veritate ofsentences with the same extension, that is, truth value: if \(A\) hasthe same truth value as \(B\), then also \({\sim}A\) has the sametruth value as \({\sim}B\). Anintensional operator (e.g.,the box of necessity) allows substitutionsalva veritate ofsentences that express necessary equivalents: if \(A\) is necessarilyequivalent to \(B\), then \(\Box A\) has the same truth value as\(\Box B\). The expression “hyperintensional” is thus foran \(H\) that defies substitutionsalva veritate even ofexpressions with the same intension.

Cresswell (1975) introduced the expression“hyperintensional” to pick out a position in a sentencewhere substitution fails for logical equivalents. Nowadays the term isused more broadly, with unrestrictedly necessary equivalence replacinglogical equivalence. Candidates for unrestricted necessity ofteninclude, besides the logical, mathematical and metaphysical necessity.We don’t discuss whether one of these is reducible to theothers—e.g., the mathematical to the logical, as claimed bylogicists.

Hyperintensionality is pervasive. Take two claims on your prospects ofgetting a job you may apply for:

(1): You have 40% chances of succeeding.
(2): You have 60% chances of failing.

These are necessarily equivalent. However, psychological studies onframing effects (Tversky & Kahneman 1981; Kahneman 2011) show thatpeople are more prone to believe in a positive outcome when presentedwith (1) than with (2), thus more likely to apply.Beliefseems sensitive to hyperintensional distinctions: we can come tobelieve different things depending on how necessarily equivalentoptions are presented to us (Berto 2019). Framing has vast societalimpacts (Thaler & Sunstein 2008).

On to your neighbor Mike:

(3): Mike is Mike.
(4): Mike is Jack the Ripper.

(3) is necessarily true. Suppose (4) is true. If “Jack theRipper” works as a proper name, (4) is necessarily true, too(Barcan Marcus 1947; Kripke 1980). Thus, (3) and (4) are necessarilyequivalent. But (3) is uninformative (you already knew things areself-identical), whereas learning (4) could save your life.Information, this fundamental concept of our age, also seemsto display hyperintensional features. The difference between (3) and(4) is marked by saying that you can know (3), not (4),apriori. The concept ofa priori, then, hashyperintensional features too.

Quine and Davidson’s idea that serious philosophy should useonly extensional notions faded in the latter half of the twentiethcentury, which witnessed an intensional revolution: a collectiveeffort to analyze notions which are fundamental for our understandingof the world and of ourselves, likebelief, information,knowledge, meaning, content, essence, explanation, via a singletheoretical system: employing possible worlds and constructions out ofthem. Standard possible worlds semantics (SPWS) found applications inlogic, linguistics, game theory, artificial intelligence: a successstory of philosophy. However, many of these phenomena (and, more: asurvey inSection 1) are, arguably, hyperintensional, requiring hyperintensional languageto do them justice. And SPWS tends to collapse any distinctions morefine-grained than necessary equivalence. The issue of how best toaddress hyperintensionality has emerged in piecemeal fashion, via aseries of difficulties of SPWS (inSection 1, we list a few). Some have denied that hyperintensionality marks a setof real phenomena and have tried to explain away hyperintensionalintuitions. Some of these general attempts are discussed inSection 2.

Even if genuinely hyperintensional language is accepted into ourphilosophical theorizing, this leaves many further issues open. Onepressing one is the question of “just ‘how hyper’hyperintensions are” (Jespersen & Duží 2015:527): what new distinctions should our theorizing recognize beyondnecessary equivalence? The mathematical structure of the realm ofhyperintensionality is not well understood yet. We come to this inSection 3. However, the early twenty-first century is seeing a hyperintensionalrevolution (Nolan 2014), with different general approaches currentlybeing developed, which qualify as hyperintensional in some sense. Weprovide a critical survey inSection 4, where we also assess the approaches in terms of how they fare withrespect to hyperintensional phenomena singled out in previousSections.

1. The Variety of Hyperintensional Phenomena

We provide a taxonomy of candidate hyperintensional notions bydividing them into two camps: (1)representational conceptspertain to ways of representing the world (1a) in thought and (1b) inlanguage. In the former sub-camp (1a), we find mental states oractivities likebelieving, knowing, imagining, conceiving,being orbecoming informed. In the latter sub-camp(1b), notions such asmeaning,content, andconditionality. (2)Non-representationalhyperintensional concepts include allegedly mind-independent,“worldly” notions such asessence, grounding,metaphysical dependence, explanation.

(The division is not uncontroversial, e.g., some claim thatexplanation and indicative conditionals essentially involve cognition(Lycan 2001), so they would place them in (1a). Some claim that thereexists “environmental” information (Barwise & Seligman1997) that should go into (2). We stick to what we take to bemainstream views on where such concepts should go.)

For each of these notions, we provide (a) a brief explanation of howthey have been dealt with in SPWS; and (b) examples illustrating theirseemingly hyperintensional nature. These, if found persuasive, maywork as data against which the theories described inSection 4 can be evaluated with respect to their hyperintensional aims.

1.1 Hyperintensionality in Representation

1.1.1 Intentional Concepts

Perhaps the most plausible hyperintensional environment is the realmofintentional concepts: notions that involve states of themind which are directed towards contents, situations, etc. The twomost studied such notions are knowledge and belief. Since Hintikka(1962), these have been understood in epistemic logic as modaloperators: restricted quantifiers over possible worlds. Knowledge andbelief are characterized in terms of what is true throughout a set ofworlds which represent ways things could be, for all the cognitiveagent knows, or given the agent’s evidence, etc. If \(K\)expresses the agent’s knowledge or belief state, \(w\) is apossible world, and \(R\) an epistemic accessibility relation on thespace of worlds \(W\), the Hintikkan characterization goes:

(H): \(KA\) is true at \(w\) iff \(A\) is true at all \(w_1\) such thatwRw\(_1\).

Some views in mainstream epistemology follow the same path, e.g., inrelevant alternatives approaches knowledge is “anevidential state in which all relevant alternatives (to what is known)are eliminated” (Dretske 1981: 367). Uneliminated alternativeswork similarly to accessible worlds.

Such treatments of knowledge and belief witnessed an earlymanifestation of the hyperintensionality issue: theproblem oflogical omniscience (Fagin et al. 1995: 335–6)—a setof closure conditions for the relevant \(K\)s, of which perhaps themost important are:

(C1): If \(KA\) and \(A\) entails \(B\), then \(KB\)
(C2): If \(A\) is logically valid, then \(KA\)
(C3): It is not the case that: \(KA\) and \(K{\sim}A\)

(C1), often dubbedClosure under entailment orFullomniscience, has it that one knows or believes all the logicalconsequences of what one knows or believes. (C2),Knowledge of allvalidities orBelief in all validities, has it that oneknows or believes all logical truths. (C3) ensures the consistency ofknowledge or belief. All follow from understanding \(K\) as aquantifier over worlds, as per (H). (C1): if \(A\)’s entailing\(B\) is understood in terms of all \(A\)-worlds’ being\(B\)-worlds (in all the relevant models), and given that \(KA\) meansthat \(A\) is true at all the epistemically accessible worlds, \(B\)will be true there, too, and so \(KB\) will hold. (C2): if\(A\)’s being logically valid is understood as its holding atall worlds (of all the relevant models), then it will hold at all theepistemically accessible ones, and so \(KA\) will hold. (C3): nopossible world, thus no accessible one, will make both \(A\) and\({\sim}A\) true.

These don’t sound right for real humans, as opposed toidealized, logically infallible agents. (C1): we know basic arithmetictruths like Peano’s postulates, and these entail (suppose)Goldbach’s Conjecture (Every even whole number larger than 2 isthe sum of two primes); but we don’t know whetherGoldbach’s Conjecture is true. (C2): “If\({\sim}{\sim}A\), then \(A\)” is (suppose) valid, butintuitionist logicians do not believe it. (C3): we can have, ifimplicitly, inconsistent beliefs.

One may treat “\(KA\)” in (H) as not really standing forknowledge or belief. We should read it as “\(A\) followslogically from the agent’s knowledge”, or “The agentis committed to \(A\), given what it knows/believes”. Then westill lack some logical account ofknowledge andbelief per se, as opposed to some derivative attitude orconditional commitment. The commitment itself may be questioned: arewe committed to believing everything, given that we have inconsistentbeliefs? In what sense must a finite agent believe infinitely manylogical consequences? Such a “must” lacks a corresponding“can”.

Information received a modal treatment in the classicBar-Hillel and Carnap (1952) account. The informative task of asentence \(A\) is to split the totality of worlds into those where\(A\) is true and those where it isn’t. An agent’sbecoming informed of \(A\) is its ruling out the not\(-A\) worlds ascontenders for actuality: before being informed that it’sraining outside, for all you knew, it might have been raining or not.After you are informed, you can rule out all the non-rainy worlds.

But now, “If Kennedy was shot, then Kennedy was shot” and“\(x^n + y^n = z^n\) has no integer solutions for \(n \gt2\)” are necessarily equivalent, given the unrestrictednecessity of logical and mathematical truths, and SPWS. They are trueat the same worlds: all of them. On the Bar-Hillel-Carnap view, onecan rule out no world after learning either. But while one easilydeems the former true, the latter, which expresses Fermat’s LastTheorem, took centuries to be proved and its proof brought significantnews.

Generalizing: a merely intensional analysis of information entailsthatno logical, mathematical, or metaphysical necessarytruth can ever be informative and denies the informativeness oflogical deduction. But logical deductions are informative (Jago 2009,2014). Besides, we can become informed of exactly one of twonecessarily equivalent claims. Of necessity, “All woodchucks arewoodchucks” is true if and only if “All woodchucks arewhistle-pigs” is (Ripley 2012). But the former is trivial,whereas one may learn the latter from a zoologist.

Other attitudes likehoping, fearing, supposing, imaginingdisplay hyperintensional features (Wansing 2017). One may be led tothe general conclusion thatthought is hyperintensional: ourintentional states can treat necessarily equivalent contentsdifferently. Lois Lane can wish that Superman is in love with herwithout wishing that Clark Kent is in love with her, although (ifBarcan Marcus and Kripke are right) it is impossible for Superman tobe other than Clark Kent. We can suppose that \(75 \times 12 = 900\)without supposing Fermat’s Last Theorem. Even intentional statesinvolving perceptual experiences, likeseeing, seemhyperintensional: you can see that Mary is eating an ice cream withoutseeing that Mary is eating an ice cream and John is either eatingchips or not (you may not be seeing John at all). But \(A\) islogically equivalent to \(A \land (B \lor {\sim}B)\) in classicallogic and even in many weaker logics (Barwise & Perry 1983).

1.1.2 Semantic Concepts

Propositions, taken as the contents or meanings of sentences, haveoften been interpreted starting from Wittgenstein’s insight thatunderstanding a sentence is knowing what is the case if it is true(Wittgenstein 1921: 4.024). Truth conditions have been given via setsof possible worlds in the SPWS approaches of Montague (1970),Stalnaker (1976) and many others: the proposition expressed by“St Andrews is rainy” is the set of worlds where StAndrews is rainy.

This makes all necessarily equivalent propositions identical, forpossible worlds can never disagree on them. “If Kennedy wasshot, then Kennedy was shot” and “\(x^n + y^n = z^n\) hasno integer solutions for \(n \gt 2\)” mean the same thing: thetotal set of worlds. This seems wrong, given that they don’t saythe same thing: only one speaks about Diophantine equations. Similarlyfor necessary falsehoods: “Kennedy was both shot and not”and “\(2 + 3 = 6\)” are about different things, but theyalso end up expressing the same proposition: the empty set.

We can locate in the area of semantic concepts hyperintensionalphenomena which have to do with conditionality. We start withindicative conditionals. The extensional material conditional,“\(\supset\)”, validates notoriously paradoxicalinferences:

(1): \({\sim}A\), therefore \(A \supset B\)
(2): \(B\), therefore \(A \supset B\)

If the horseshoe captures the meaning of the indicative“if”, lots of bad-sounding conditionals from ordinarylanguage come out true just because the antecedent is false or theconsequent true: “If Clinton is Australian, then Mars is made ofmarbles”; “If Strasbourg is in Germany, then Clinton isAmerican”.

One may think the issue is that the connection between antecedent andconsequent is too contingent an affair here. One may require, for“If \(A\), then \(B\)” to be true, that it cannot be thecase that the antecedent is true while the consequent is false. Thisgives the so-called strict conditional: a modal operator,“\(\prec\)”, such that “\(A \prec B\)” isunderstood as “At any possible world where \(A\) is true, \(B\)is true”.

This has its paradoxes, too. Where “\(\Box A\)” means that\(A\) is true at all possible worlds, and “\(\Diamond A\)”means that \(A\) is true at some, we get:

(3): \({\sim}\Diamond A\), therefore \(A \prec B\)
(4): \(\Box B\), therefore \(A \prec B\)

When \(A\) is true at no possible world, or \(B\) is true at all,there is no way for \(A\) to be true while \(B\) is false and so“\(A \prec B\)” is true. If the strict conditionalcaptures the meaning of the indicative “if”, lots of badconditionals come out true: “If \(6 + 3 = 11\), then Clinton isAustralian”; “If Clinton is American, then \(6 + 3 =9\)”. These are not obviously true, and there are some reasonsto suppose they are false: there is no connection between antecedentand consequent, for example. Other conditional forms that the strictconditional validates but which are disputable include:

(5): \(A \prec (B \prec B)\)
(6): \(A \prec (B \lor {\sim}B)\)
(7): \((A \land {\sim}A) \prec B\)

As an instance of (6), take “If Mars is made of marbles, theneither St Andrews is in Scotland, or it isn’t”: thematerial constitution of Mars has little to do with St Andrews’location. Some theorists argue that treatments of the conditional thatdo not render conditionals like these automatically true would bepreferable.

On to counterfactuals or subjunctive conditionals—conditionalsof the form, “If it were (had been) the case that \(A\), then itwould be (have been) the case that \(B\)”. A common treatment,due to Stalnaker (1968) and Lewis (1973), analyses them in terms ofpossible worlds. In a simple version, they are true just in case atthe closest worlds where \(A\) is true, \(B\) is true, where closenessis understood as similarity in relevant respects. “If Kangarooshad no tails, they would topple over” is true when, at allworlds where kangaroos have no tails and which are as relevantlysimilar as it gets to the actual world, kangaroos topple over.

This again makes all counterfactuals whose consequent is necessary, orwhose antecedent is impossible, vacuously true: there are no worlds tofalsify the consequent, or to verify the antecedent. Counterfactualswith impossible antecedents are calledcounterpossibles. Theydon’t seem to be all vacuously true. An often-cited example fromNolan (1997):

(8): If Hobbes had (secretly) squared the circle, sick children in themountains of South America at the time would have cared.
(9): If Hobbes had (secretly) squared the circle, sick children in themountains of South America at the time would not have cared.

Squaring the circle is a mathematical impossibility, in spite ofHobbes’ hopes. His achieving it, though, would have made nodifference for those sick children. (9) should come out true for thisreason, not because no possible world verifies the antecedent. Forsimilar reasons, (8) should come out false. Against vacuism, somecounterpossible pairs with opposite consequents should have differenttruth values (for a defense of vacuism, see Williamson 2007; for adiscussion, see Berto et al. 2018).

1.2 Hyperintensionality in the Non-Representational World

Hyperintensionality was initially tied to representation: Cresswell(1975) discussed a semantics for a formal language where substitutionof logical equivalents fails in intentional contexts (he mentioned“says that”, “knows that”, “believesthat”, “deduces that”: 1975: 37). Other casesdiscussed above might contribute to the impression that our languagecontains hyperintensional expressions just because of phenomenainvolving representation.

However, various philosophers have employed hyperintensional resourcesto theorize about aspects of the world beyond representation.Famously, Fine (1994) argues that work on theessences ofthings must play a central role in metaphysics and essence talk mustbe hyperintensional. In the construction “\(x\) is essentially\(F\)”, we cannot be guaranteed to preserve truth bysubstituting necessarily equivalent predicates. Socrates is (suppose)essentially human, but not essentially (human and a member of a set),although necessarily someone is human if and only if they are humanand a member of a set. Fine extends this to talk of real definitiongenerally: it is the real definition of 2 that it is the successor of1, but not the real definition of 2 that it is \(10-8\), even though“the successor of 1” and “\(10-8\)”necessarily have the same extension. (These examples are not in Fine1994, but we take it he would endorse them given his discussion on pp.4–5 and p 14.) Dunn (1990b) also argues that one needshyperintensional resources (in his case, from relevant logic) tocapture essentialist claims.

Nolan (2014) argues for the need to provide hyperintensional accountsof a range of non-representational aspects of reality. As well as atheory of essences, theories of counterfactuals, explanation,metaphysical grounding, of the distinction between intrinsic andextrinsic properties, of dispositions, are all plausiblyhyperintensional. Nolan also mentions the analysis of causation, lawsof nature, objective confirmation, objective chance, and objectiveethics. Other recent proposals in metaphysics that have employedhyperintensional resources include work on truthmakers, omissions,metaphysical dependence, and the distinction between qualitative andnon-qualitative properties. Almost any area of metaphysics wherenecessity or counterfactuals have played a role is a candidate forhyperintensional approaches. We now give a flavor of hyperintensionaltheories of some of these topics.

The hyperintensionality of counterfactuals has been discussed above.Does their truth depend in the right kind of way on facts beyond ourrepresentational abilities? Nolan (2014: 156) argues that it sometimesdoes. Regular platonic solids come in only a handful of sizes (themost sides one can have, in three-dimensional space, is 20.) Still,this seems true:

If there were a piece of steel in the shape of a 36 sided platonicsolid, it would have more sides than any piece of steel in the shapeof a dodecahedron.

On the other hand,

If there were a piece of steel in the shape of a 36 sided platonicsolid, it would have fewer sides than any piece of steel in the shapeof a dodecahedron

seems false. The truth value of these is not just settled by whetherthe antecedent is necessary. They seem to be about pieces of steel andhow they relate to each other. Which blocks of steel have which shapesis not just about us and our representations. It would not be even ifsteel could take shapes that it in fact cannot.

Explanation is plausibly hyperintensional: “explains” canbe flanked by expressions that cannot be substituted with necessaryequivalentssalva veritate. One pure mathematical truth canexplain another, but not every mathematical truth explains everyother, even if every pure mathematical truth is a necessary truth(Baron, Colyvan, & Ripley 2020). Schneider (2011) argues thatsometimes logical equivalents can explain each other, making a casethat “because” is hyperintensional. \({\sim}{\sim}A\) istrue because \(A\) is true, but not vice versa. If the correctness ofexplanations turns out to depend also on how the world is, rather thanonly on what suits our explanatory practices, these cases may be dueto hyperintensional explanatory connections in the world. (How best tounderstand explanations, of course, remains a contentiousphilosophical question.)

Metaphysical grounding is widely regarded as hyperintensional.Sometimes a more fundamental fact necessarily leads to a lessfundamental fact that it grounds, but the two are merely modallyinseparable. A widely discussed example is the grounding of theexistence of a set in the existence of its members. On one view,{Socrates, Plato} automatically exists if Socrates and Plato bothexist, but it exists only in possible worlds where Socrates and Platoboth exist. So the fact that Socrates and Plato both exist grounds thefact that {Socrates, Plato} exists (see Dunn 1990b; Fine 1994). Butthat {Socrates, Plato} exists is not supposed to ground the fact thatSocrates and Plato both exist, even though necessarily one obtainsonly if the other does. On an attractive account, many complex logicaltruths are grounded in simpler ones. \({\sim}{\sim}A\) is grounded in\(A, (A \land A)\), is grounded in \(A\), and many other logicalequivalents of \(A\) are grounded in \(A\), though they are not thesame proposition as \(A\). And as before, while \((A \land A)\) isgrounded in \(A, A\) is not grounded in \((A \land A)\) (seePoggiolesi 2018, 2020 for a formal logic of grounding).

Whether grounding is indeed hyperintensional may depend on howgrounding claims are best understood. Where “…grounds…” is a sentential operator, the cases discussedhad best be captured hyperintensionally. “(Socrates and Platoboth exist) grounds (The set {Socrates, Plato} exists)”, forexample. Things may work differently if we take “…grounds…” as a two-place predicate applying to facts oroccurrences: “The fact that Socrates and Plato both existgrounds the fact that {Socrates, Plato} exists” expresses arelation between two objects, and we might be able to preserve thetruth of this statement by substitution of any other ways of referringto each of those objects. One complication is that one might takethese expressions picking out facts to themselves have constituentsdesignating propositions: “the fact that Socrates and Plato bothexist” may have the logical formthe fact that A, wherethat \(A\) is the proposition that Socrates and Plato exist.Interpreted that way, the expression is functioning hyperintensionallyafter all, since the necessarily equivalent proposition that{Socrates, Plato} exists cannot be substituted into that expressionwithout changing the truth-value of the relevant grounding claims.Finally, one could take the primary way of expressing groundingrelations to be by talking about objects in general, e.g., aboutSocrates and Plato grounding the set {Socrates, Plato} (Schaffer2009). On this third approach it is natural to think that any otherways of designating those objects could be substitutedsalvaveritate so that the grounding-between-objects predicate isextensional; though see Jenkins (2011) for a hyperintensionalobject-theoretic grounding approach that does not allow thissubstitutionsalva veritate.

Many grounding theorists have taken the sentential operator approachas primary: see Fine (2001: 15–16), Schneider (2011:446–7). Those that do not fall on different sides of thequestion of whether expressing the full range of grounding claimsrequires hyperintensional resources. See Duncan, Miller, and Norton(2017) for a discussion of which styles of grounding claims are besttreated as hyperintensional.

The distinction between intrinsic and extrinsic properties features inmany metaphysical theories, and it appears to berepresentation-independent: presumably, if mass is intrinsic it wouldremain such whatever we said or did. One idea behind the distinctionis that a property had intrinsically is had only in virtue of anobject’s own nature, while extrinsic properties are partly amatter of how the object stands to other objects. Some have arguedthat necessarily coextensive properties can differ in whether they areintrinsic or extrinsic (Dunn 1990a; Eddon 2011; Bader 2013; Marshall2015; Hoffmann-Kloss 2015).Being an instance of Barak’sactual favorite property is plausibly extrinsic, buthaving amass may well be intrinsic, even if mass is Barak’sfavorite property, and even though, given it is actually his favoriteproperty, anything in any possible world has the first property iff ithas the second. The above-named philosophers disagree on how tocharacterize the intrinsic/extrinsic distinction, but they share theview that it has to be hyperintensional.

Jenkins and Nolan (2012) argue that disposition ascriptions arehyperintensional. Heidi the gifted mathematician may be disposed tofind the proof of a given conjecture, on condition that it has one.This disposition of Heidi’s can be known in advance of knowingwhether the conjecture has a proof. (“Give this to Heidi: ifthere’s a proof to be found, she’ll find it”.)Suppose the conjecture is in fact false, though the mathematicalcommunity does not yet know that. Then the condition of thedisposition ascription is impossible. Still, we cannot substitute anyimpossibility in the disposition ascription. It is plausibly falsethat Heidi is disposed to find the proof of that conjecture, on thecondition that she was always both identical to the number 7 andinnumerate, for example.

We use necessarily false conditions in disposition ascriptions outsidetalk about agents, too. Working with a model where the number of preykilled is a function of the number of predators, the model might sayadding half a fox to a closed ecosystem, or adding \(1/\pi\) foxes toan ecosystem, produces a certain fraction of additional rabbitskilled. But it is impossible to have, e.g., \(7+1/\pi\) foxes. Still,what the ecosystem is disposed to do on the condition \(7+1/\pi\)foxes are addedand it is idealized in various ways is verydifferent from what it is disposed to do with \(105+1/\pi\) foxes andthe same idealization.

One way to specify probabilities is with propositional complements:the probability that it will rain tomorrow, or that this coin willcome up heads 10 times in a row. Take claims such as “\(\rP(B) =0.6\)”, where the proposition that \(B\) is, e.g., that this(biased) coin will come up heads 10 times in a row. \(\rP\) will behyperintensional if we cannot guarantee that we preserve truth valueby substitution of necessarily equivalent propositions.

If we use \(\rP\) to represent the (degrees of) beliefs of anagent—even a logically idealized one—we might want toallow different probability assignments to necessarily equivalentpropositions: with all her exhaustive knowledge of pure arithmetic andlogic, the agent might still be uncertain on whether water =H₂O before investigating, in spite of certainty on theself-identity of water. If the hyperintensional probability functionrepresented objective chance or frequency, however,hyperintensionality may also be tied to how the world is. Nolan(2016a) and Salmon (2019) have both recently argued that objectivechance ascriptions are hyperintensional.

Plausibly, “ought” is hyperintensional: perhaps you oughtto call your sister, but it is not the case that you ought to callyour sister and be self-identical even though it is impossible to callyour sister without calling your sister and being self-identical, andvice versa. (Perhaps you promised to call: but unless you are aphilosopher you probably did not promise to be self-identical.) Someof these obligations might be dependent on our representationalpractices. But perhaps there are obligations of objective morality aswell. Perhaps your moral obligation to not callously cause sufferingis like this. If there are objective obligations, they might be bestconstrued as hyperintensional: the Moral Law may enjoin you to avoidcallously inflicting suffering, but it may be silent about whether youought to avoid inflicting callous suffering while beingself-identical. See Faroldi (2019) for a recent hyperintensionaltreatment of “ought”.

Truthmakers theorists have been looking for a relationship betweentrue propositions and the aspects of the world that make thosepropositions true. Theories of this relation often make truthmakinghyperintensional: \(A\) and \(B\) may be necessarily equivalent, yetthe worldly bit that makes \(A\) true is distinct from the one thatmakes \(B\) true. As Restall (1996a) points out, this seems to followif one also endorses the Disjunction Thesis: something is a truthmakerfor (\(A\) or \(B\)) iff it is a truthmaker for \(A\) or atruthmaker for \(B\). Independently of the Disjunction Thesis, atruthmaker for one necessary truth may plausibly not be a truthmakerfor all others: “There are infinitely many prime numbers”should have as truthmaker something to do with numbers. “Anyonewho eats and drinks at least eats” seems to not have anything todo with numbers. See MacBride (2013 [2020], the SEP entry ontruthmakers, section 1) for some accounts of truthmaking where necessarily equivalentpropositions need not have the same truthmaker.

Some occurrences are impossible, but may still have explanatory andeven causal consequences. It may be impossible to prove a givenmathematical conjecture: the conjecture might have a counterexamplethat nobody has yet noticed. Still, the fact that the conjecture isnot proved may have various effects: if you have bet that you willprove it by a certain time you will lose the bet; if your tenure casein the mathematics department turns on whether you managed to provethe conjecture, you will miss out on tenure. Bernstein (2016) gives atheory of omissions of impossible outcomes, and argues that they playan explanatory and causal role.

Hoffmann-Kolss (2015) argues that a number of classifications ofproperties needed for metaphysics are hyperintensional. Theintrinsic/extrinsic distinction is one example, discussed above.Hoffman-Kolss also argues the qualitative/non-qualitative distinctionis hyperintensional (the distinction between properties that do notrequire a relationship to a particular and those that do—e.g.,being square vs.being John’s neighbor: seealso Hoffman-Kolss 2019). The dispositional/categorical distinction isarguably hyperintensional (Hoffmann-Kolss 2015: 341–342, 346).(One philosophically controversial example may illustrate: on sometheories, there is a necessarily-existing God that is necessarily theonly omnipotent being. Still, we might thinkbeing omnipotentis dispositional, concerning the powers of that being, whilebeingGod is categorical (2015: 346).) The distinction betweenperfectly natural properties that best carve at the joints, and theothers, is, too (2015: 345). Hoffman-Kolss’s example here is tocontrast the (supposedly) perfectly natural property of being anelectron with the infinitely disjunctive identity property ofbeing \(e_1\) orbeing \(e_2\) or… and so onfor all possible electrons \(e\). If electronhood is essential to theobjects that have it, something is an electron if and only if it isone of \(e_1\), \(e_2\), etc., but Hoffman-Kolss argues this lattercondition is not a perfectly natural one.

Any item on this list might be controversial. You might think thatcounterfactual conditionals, for example, communicate more about howwe explain or predict the world than anything about the world as itis. Or that what explains what is more a matter of the representationswe make of the world, than of the world itself. One could try toexplain away grounding’s putative hyperintensionality in termsof our cognitive structures, as Thompson (2016) suggests. Thompson andByrne (2019) offer a general “conceptualist” alternativeto taking any of the cases we discussed as requiringnon-representational phenomena to be characterized in hyperintensionalterms: an explanation of our use of hyperintensional language tocapture these phenomena comes from an account of what it is todescribe and explain in these areas (2019: 157). Still, if a need torecognize hyperintensional distinctions flows from more than factsabout how we represent the world, hyperintensionality will be ofinterest well beyond philosophy of mind and language.

2. Is Hyperintensionality Real?

WhileSection 4 considers some difficulties for specific hyperintensional theories,here we discuss general objections to the very idea that we need atheory of language or mind that marks hyperintensional distinctions.One kind of argument in this ballpark has to do with a general“Boolean” challenge to formulating a semantics forhyperintensional expressions in the dominant truth-conditionalframework. This style of argument will be discussed in§2.1.

Another kind of argument advanced by defenders of SPWS such asStalnaker (1984) and Lewis (1982, 1986), has it that theprimafacie hyperintensionality of knowledge, belief, or content ismisleading, and that SPWS can still do the job once misleadingintuitions have been explained away. This style of argument will bediscussed in§2.2.

2.1 Hyperintensional Truth Conditions

One way to modify SPWS to give truth conditions for sentences is toresort to sets including both possible and impossible worlds (waysthings couldnot have been). Sentences that agree in truthvalue in every possible world can still be assigned distincttruth-conditions, since those sentences might be true in differentimpossible worlds. (See§4.3). Another way is to usestructured meanings, where sentenceswith the same possible-worlds truth conditions can have differentmeanings because of the way they are built up. (See§4.5). An objection to these approaches can be generalized to any approachthat tries to make room for hyperintensionality in a truth-conditionalframework.

The objection, presented in Cresswell (2002) against structuredpropositions and suggested by Stalnaker (1996) against impossibleworlds, can be illustrated by focusing on the meaning of negation.Given a statement \(A\), what truth-conditions can we offer for\({\sim}A\)? Plausibly, the truth conditions can be exhausted byspecifying that \({\sim}A\) is true when \(A\) is false, and nototherwise. Since this exhausts the meaning of“\({\sim}\)”, it is tempting to identify the proposition\({\sim}A\) as being the proposition with this truth condition.Plausibly, also, the truth conditions for \({\sim}{\sim}A\) will justbe the truth conditions for \(A\). (We may even want to derive thissecond claim from plausible principles about truth, falsehood, andnecessity.) Now any theory which allows for distinct necessarilyequivalent propositions will fall to the same objection: suppose \(C\)and \(D\) are distinct, necessarily equivalent propositions. Then\({\sim}C\) will be the proposition that is true just in case \(C\) isfalse, and \({\sim}D\) will be the proposition that is true just incase \(D\) is false: so \({\sim}C\) and \({\sim}D\) will be the sameproposition. But then there will be a unique proposition with thetruth conditions for \({\sim}{\sim}C (/{\sim}{\sim}D)\), since\({\sim}{\sim}C\) is the negation of \({\sim}C\). So \(C\) and \(D\)must be the same proposition after all. This is areductio ofthe hypothesis that there can be a hyperintensional distinctionbetween \(C\) and \(D\): if they are necessarily equivalent, they areidentical.

We might call this an objection from Booleanism, both becauseuniqueness of truth-conditional complementation is characteristic ofBoolean approaches to negation, and because similar objections can bemotivated by Boolean thoughts about other operations. We might hope todefine conjunction so that necessarily, \((A \land B)\) is true iff\(A\) is true and \(B\) is true, and indeed the truth condition forthe proposition that \((A \land B)\) is just that both \(A\) and \(B\)are true. But then, with other innocuous assumptions, any twonecessarily equivalent sentences will express the same proposition, bya similar chain of reasoning.

Cresswell (1975) was one of the pioneers of structured approaches tosentence meaning to handle hyperintensionality. So it is significantthat he thinks this objection refutes a contention at the heart oforthodox structured meaning projects, though Cresswell thinks that wemay still be interested in semantic structures for reasons other thanhow they might make a difference to the truth-conditions of complexembeddings.

Defenders of hyperintensional meanings need to resist the idea thatfor each proposition \(A\) there is only one proposition true whenever\(A\) is false. This leaves them with a challenge: how are we tounderstand logical operations such as negation and conjunction oncethese Boolean characterizations are rejected?

2.2 The Indispensability of Hyperintensions

Talk of belief, desire, and meaning isprima faciehyperintensional, as discussed above. But perhaps this appearancecould be resisted. One strategy is to employ use/mention distinctionsin novel ways. Instead of hyperintensional operations on propositionsor other meanings, we can treat apparently hyperintensional languageas expressing properties of or relations between sentences or otherpieces of language. If we do this systematically, a theory can do awaywith merely intensional distinctions.

Quine (1953) suggests that modal operators, when we need them at all,are best interpreted as predicates of sentences. Quine (1956) andDavidson (1968) suggest an approach to propositional attitudeascriptions as implicitly talking about connections between people andsentences: “John believes that it is raining” isregimented as a two-place predicate holding between John and thesentence “It is raining”. For Davidson, this in effectasserts that John stands in a certain relation to a sentence thatstands in the same-saying relation to the sentence “It israining”. This supposedly accommodates how non-English speakerscan believe that it is raining. Reconstruing mental attitude reportsas reporting relations to sentences has fallen out of favor incontemporary philosophy, for reasons largely orthogonal tointensionality or hyperintensionality: see Propositional AttitudeAscriptions, section 7. Still, with enough ingenuity techniques likeQuine’s and Davidson’s could be extended to apparentlyhyperintensional pieces of language.

Some philosophers who reject extensionalism nevertheless offertechniques for avoiding treating beliefs and desires ashyperintensional. Stalnaker (1984) tries to rescue the SPWS analysisof propositional content by explaining away hyperintensionalintuitions via a sort of metalinguistic approach: when we fail to seethat a proposition is necessarily true, or that two propositions arenecessarily equivalent, we are actually failing to see whatpropositions are expressed by certain sentences. Given sentence \(A\),one should distinguish (1) the proposition \(P\) expressed by \(A\)(in a context), and (2) the proposition \(P\)*that Aexpresses \(P\), relating the sentence to what it expresses.Ignorance of necessary truths or equivalences is ignorance aboutP*.

Stalnaker maintains that whenever one fails to know or believe anecessary truth, \(A\), one actually fails to fully grasp the sentence\(A\)’s meaning: one doesn’t realize that \(A\) expressesthe total set of worlds. Stalnaker points at the occasional distancebetween the syntactic structure of the relevant \(A\), which can bequite complex, and the expressed proposition—in SPWS, anunstructured entity (a set):

Whenever the structure of sentences is complicated, there will be anontrivial question about the relation between sentences and thepropositions they express, and so there will be room for reasonabledoubt about what proposition is expressed by a given sentence.(Stalnaker 1984: 84)

For Stalnaker’s story to work in general, all cases of ignoranceof necessary truths or equivalences must be instances of meaningignorance. This is difficult to believe. A mathematician who wonderswhether Goldbach’s conjecture is true can hardly be described asmistaken about what “Every even integer larger than two is thesum of two primes” means. They know the meanings of allconstituents of the sentence—they have mastered theelementary-school-math conceptseven integer,largerthan,two,sum,prime number as muchas any of us. As competent speakers of English, they understand howthe syntax of the sentence makes its constituents conjure for it toexpress the content it expresses. What they have doubts about iswhether that content is true. We sometimes seem to fully grasp thecontents of sentences whose status of necessary truths, or falsities,we are ignorant of.

Lewis (1982) proposes a fragmentation strategy to explain awayapparent failures of logical omniscience, in particular, apparentbelief in inconsistencies which is not, however, belief in everything.One’s epistemic state may be split into different “framesof mind”. One may believe \(A\) in one mind frame, \({\sim}A\)in another one, but fail to put them together:

I used to think that Nassau Street ran roughly east-west; that therailroad nearby ran roughly north-south; and that the two were roughlyparallel. […] So each sentence in an inconsistent triple wastrue according to my beliefs, but not everything was true according tomy beliefs. […] My system of beliefs was broken into(overlapping) fragments. Different fragments came into action indifferent situations, and the whole system of beliefs never manifesteditself all at once. […] The inconsistent conjunction of allthree did not belong to, was in no way implied by, and was not trueaccording to, any one fragment. That is why it was not true accordingto my system of beliefs taken as a whole. (Lewis 1982: 436)

Each fragment can be given the SPWS treatment, while trans-fragmentalinconsistencies do not trivialize the belief system.

The account, just like Stalnaker’s, doesn’t seem togeneralize. It seems strange to say that when agents don’tfollow through the logical consequences of what they believe,it’s always because they haven’t conjoined the premiseswhereas, when they do, they suddenly come to believe all theinfinitely many consequences (Jago 2014). It’s sometimes an easylogical step from \(A_1, A_2, A_3\), to \(A_1 \land A_2 \land A_3\).What is often difficult is to see whether a remote \(B\) follows from\(A_1 \land A_2 \land A_3\).

Even if Lewis and Stalnaker can explain away the apparenthyperintensionality of belief, these attempts to not extend in anystraightforward way to allprima facie cases ofhyperintensionality. Stalnaker, at least, seems committed to a projectof explaining away many apparent cases of hyperintensionality,committed as he is to a SPWS account of linguistic meaning and mentalcontent (Stalnaker 1984). So far as we can tell, Stalnaker is willingto take this project piece-meal.

Lewis seems more hospitable to hyperintensional approaches, providedthey can be ultimately explained in terms of possible worlds andset-theoretic constructions out of them. In Lewis (1986: 49–50),he discusses a hyperintensional “trivially” operator thatwould apply differently to different necessary truths. In aposthumously published letter (Lewis 2004) he seems to acknowledgethat there was more than one distinct impossible fiction, andpresumably that “According to fiction \(f, A\)” washyperintensional (since when \(f\) is an impossible fiction, therewill be some impossible claim that is true according to it, but ingeneral not all impossible claims will be).

Philosophers suspicious of hyperintensionality in particular areasshould consider if this stems from features of the specific case or isgeneral. If the latter, they may wish to attempt a general account ofmeaning that ensures we never need to acknowledge hyperintensionality:otherwise, a general suspicion will risk being undermined.

3. Granularity and the Structure of Hyperintensionality

The flip side of hyperintensionality’s pervasiveness is itsencompassing very diverse phenomena. We now investigate what (little)we know about the general structure of hyperintensionality. Theinitial characterization, recall, merely tells us that a concept oroperator is hyperintensional when it is more fine-grained thanintensional or standard modal concepts or operators, marking adistinction invisible to the latter. As Jespersen and Dužíreminded us, little is said on “how hyper” hyperintensionsare. The tag of “granularity problems” (Barwise 1997; Jago2014; Bjerring & Schwarz 2017) labels a set of issues concerningthe right level of fine-grainedness. There are several relatedproblems here, because different hyperintensional concepts seem todisplay different degrees of fine-grainedness. This can be understoodagain in terms of substitutionsalva veritate for therelevant operators: \(X\) is strictly more fine-grained than \(Y\)when all substitutions that go through for \(X\) also do for \(Y,\)but \(X\) fails some, which goes through for \(Y.\)

A requirement often suggested is that a hyperintensional account ofthis or that notion shouldn’t make it as fine-grained as thesyntax of the language one is working with—on pain of givingaway the very point of having a semantics for it. Propositions may bemore fine-grained than sets of possible worlds, but they had betternot be mappable 1:1 to the sentences expressing them. Otherwise, wewould lose one main motivation for having propositions to begin with,namely that syntactically different sentences can say the same thing.It also seems intuitive that, e.g., the content of \(A\) should be thesame as that of \(A \land A\), and that \(A \land B\) should have thesame content as \(B \land A\): “It’s windy and rainytoday” seems to just say the same thing as “It’srainy and windy today” (for further motivations forcoarse-graining, see Bjerring & Schwarz 2017). On the other hand,a limited cognitive agent may fail to make even the simplest inferencestarting from what she knows or believes: she may believe that \(A\land B\), fail to perform Conjunction Elimination, and thus fail tobelieve that \(A\). Psychologically very realistic accounts ofknowledge or belief seem to require extreme fine-grainedness, cuttingmore finely than same-saying. Jespersen (2010) is one example of atheory that has very fine-grained meanings available for someconstructions, while offering more coarse-grained meanings for sometraditional purposes where little or no hyperintensionality isrequired.

One may hope for atotal ordering of hyperintensional notionsby fine-grainedness, i.e., an ordering such that any two given onesare comparable: given any \(X\) and \(Y\), either \(X\) is at least asfine-grained as \(Y\), or vice versa. But one can show that the simplesetting is wrong. One can provide intuitive examples (inspired byHornischer 2017, which also includes a thorough discussion) ofincomparable hyperintensional operators: \(X\) and \(Y\) for whichthere are sentences \(A\), \(B\), \(C\), \(D\), such that \(XA\) and\(XB\) are equivalent but \(YA\) and \(YB\) are not, \(YC\) and \(YD\)are equivalent but \(XC\) and \(XD\) are not.

Take “Lois Lane knows that …” and “…because Clark Kent loves the Guggenheim Museum”. Consider thefour sentences:

\(A =\) Clark Kent loves the Guggenheim Museum
\(B =\) “Clark Kent loves the Guggenheim Museum” istrue
\(C =\) Clark Kent is in New York
\(D =\) Superman is in New York

Lois Lane knows that \(A\) (Clark Kent loves the Guggenheim Museum) ifand only if she knows that \(B\) (“Clark Kent loves theGuggenheim Museum” is true) (surely Lois knows how truthworks!). On the other hand, she knows that \(C\) (Clark Kent is in NewYork), but she doesn’t know that \(D\) (Superman, too, is), forshe is not aware that Superman is Clark Kent.

Also, it is false that \(A\) (Clark Kent loves the Guggenheim Museum)because Clark Kent loves the Guggenheim Museum (facts can’texplain themselves), while it is true that \(B\) (“Clark Kentloves the Guggenheim Museum” is true) because Clark Kent lovesthe Guggenheim Museum (assuming truth supervenes on/is grounded infacts). On the other hand, \(C\) (Clark Kent is in New York) becauseClark Kent loves the Guggenheim Museum if and only if \(D\) (Supermanis in New York) because Clark Kent loves the Guggenheim Museum, forSuperman is Clark Kent.

Soknows andbecause turn out to be incomparable,and, given that they are hyperintensional, there is no total orderingfor hyperintensional notions. This may sustain some skepticism(ventilated, e.g., in Leitgeb 2019) on the possibility of a singletheory for all hyperintensional operators. On the other hand, it makessense to conjecture that these can be at least partially ordered. Itis then an issue worth investigating, whether concepts belonging to asingle sub-family, e.g., the non-representational ones, or therepresentational semantic ones, can be internally totally ordered(notice thatknows is representational, while one may takebecause as belonging to “ontic” explanation). Or,the other way around, the identification of totally ordered subsets ofthe partially ordered set of hyperintensional concepts may be a goodway to group such concepts into families that carve at the joints.

4. General Approaches to Hyperintensionality

We now provide a survey of general hyperintensional approaches. Hardlyany of them has been put forth as a systematic account of allhyperintensional notions. However, we focus on frameworks that aregeneral enough to handle a broad range of phenomena (thus we do notconsider, e.g., hyperintensional epistemic logics that target onlycertain forms of logical omniscience, like justification logic). Webriefly introduce the key ideas of each approach, and we discuss howit handles hyperintensional phenomena such as those introduced inSection 1. We highlight relative merits and open issues.

4.1 Two-Dimensional Semantics

Two-dimensional semantic theories (see entry ontwo-dimensional semantics) build on standard possible worlds semantics to handle a range ofrepresentational hyperintensional phenomena. Originating intwo-dimensional modal logic (Segerberg 1973; van Fraassen 1977; Davies& Humberstone 1980), versions of the view have been presented byKaplan (1989), Jackson (1994), and Chalmers (1996). (Stalnaker [1978]is a key pioneer of the approach, but he applied it to the pragmaticsof communication, rather than to provide a hyperintensional semantics:see Stalnaker 2004.)

Sometimes what a word picks out depends on context. Indexicals(“I”, “you”, “here”,“yesterday”) and demonstratives (“this”,“that”, “over there”) are obvious cases: whichthing someone refers to using “this” depends on what theyare demonstrating at the time. Kaplan (1989) proposed that theseexpressions function as rigid designators on occasions of use: whetherthe proposition expressed by a sentence including one is true at apossible world depends on how things stand with the object, if any,referred to in the actual world. If on 5 March 2020 I say“Yesterday it was sunny”, I express a proposition true ata possible world iff it is sunny on 4 March 2020 in that world. Elvissays “I was born in 1935”: that is true at a possibleworld iffElvis was born in 1935 there.

On this view, “I am Elvis”, said by Elvis, is true just incase Elvis is Elvis, thus necessarily true. Ditto for “Yesterdaywas 4 March 2020”, said on 5 March 2020, which is true just incase 4 March 2020 is 4 March 2020. Expressing those necessary truthscan, however, give contingent information to hearers who already knewthings are self-identical. To capture the difference between “Iam Elvis” and “Elvis is Elvis”, when uttered byElvis as well as by someone else—say, Aretha—we would needa story about the meaning of “I” that did not treat it asa name. Kaplan took such meaning as a function from a context ofutterance to an object, selecting the speaker as the object.Similarly, the meaning of “yesterday” is a function fromthe day of utterance, selecting the day before. This function iscalled thecharacter of the expression. My knowing thecharacter (and facts about context) lets me take new information fromthe utterances. I know “I” picks out the speaker and thatthe speaker said “I am Elvis”. If I trust him, I can workout the speaker is Elvis. I might not have knownthat before,although I knew that Elvis is Elvis.

Meanings, in this view, have two components: acharacter andacontent. The content is normally given via possible worlds.The character is a function from contexts of utterance to contents.The content of “I am Elvis”, uttered by Elvis, is thetotal set of worlds, since Elvis couldn’t but be Elvis. Thecharacter is a function from the context’s speaker to thecontent that that person is Elvis. So it is a function from Elvis tothe content that Elvis is Elvis, from Aretha to the content thatAretha is Elvis, etc.

Traditionally, philosophers have found it useful to represent thisinteraction of character and content with a two-dimensional matrix,hence the expression “two dimensional semantics”. Let usillustrate with a small example. In \(W_1\), Elvis is singing butAretha is not, and in \(W_2\), Aretha is singing but Elvis is not, andin \(W_3\), neither is singing. Consider the sentence “I amsinging”, with two possible speakers, Elvis and Aretha. We getthe following six scenarios, depending on which speaker is speakingand which world the resulting content is being evaluated at:

	\(W_1\)	\(W_2\)	\(W_3\)
Elvis	T	F	F
Aretha	F	T	F

Each horizontal line (row) of the matrix represents a content that canbe associated with “I am singing”, with different rowsrepresenting the different contents produced by different contexts(due to different speakers, in this case). The entire matrixrepresents facts about the character of a sentence. Thesetwo-dimensional matrices become more interesting when more complexconstructions are discussed, but the key ideas for current purposesare more easily presentable without further using matrices.

Now some proponents of two-dimensional semantics extend this settingbeyond indexicals and demonstratives, e.g., toproper namesandnatural kind terms. Take “Robin Hood”. Itscontent, suppose, is just Robin. But its character is a function fromcontexts to such contents. In a world where our use of “RobinHood” derives in the right kind of way from Robin, the characterdelivers our Robin as the content of “Robin Hood”. In aworld where Richard leads a band of outlaws in Sherwood, uses“Robin Hood” as an alias, is famous for robbing for therich to give to the poor, the character of “Robin Hood”picks outRichard. Similarly for expression“water” and H₂O in our world, or“XYZ” in Twin Earth: the character of “water”picks out different substances in different contexts.

Character-sensitive expressions can give rise to hyperintensionality.Take “It isa priori that”: It isapriori that Hesperus is Hesperus; it is nota priorithat Hesperus is Phosphorus. Although “Hesperus” and“Phosphorus” have the same content (the planet Venus),they can differ in character. There may be contingentapriori cases: Elvis says “I am here now” in a contextwhere the assigned content is that Elvis Presley is in Graceland onnoon of 25 December 1975. “I am here now”, some claim, canbe knowna priori by Elvis, but he cannot knowapriori that Elvis is in Graceland on noon of 25 December1975.

Propositional attitude contexts make for a salient case. The Sheriffof Nottingham believes Robin Hood is a dangerous outlaw. He does notbelieve that Robin of Locksley is a dangerous outlaw, although RobinHood is Locksley. We can explain this if we take “believesthat” as character-sensitive. “Robin Hood is a dangerousoutlaw” and “Robin of Locksley is a dangerousoutlaw” coincide in content (they are true in the same worlds).But we can assign different characters to “Robin Hood” and“Robin of Locksley” by looking at differences in how theyare typically acquired, in the causal history leading back to theirfirst uses, or if we tie them to how the Sheriff sees the world. Thiscan be generalized to so-called intensional transitive verbs. TheSheriff can be looking for Robin Hood but not looking for Robin ofLocksley if “looking for” is sensitive to character.

Two-dimensional approaches are somewhat controversial when they movebeyond indexicals and demonstratives. (See the papers inGarcía-Carpintero and Maciá 2006 for arguments for andagainst two-dimensional semantics.) But they can account for theapparent hyperintensionality of “it isa priorithat…” or “… believes that…”without straying too far from a traditional possible-worlds framework.Contents keep being sets of possible worlds. Qua functions fromcontexts to contents, characters are familiar from standard treatmentsof context-dependent expressions such as Kaplan’s.

Two-dimensional semantics also has a nice story on the behavior ofindexicals in belief contexts. “I believe you are Elvis”,said addressing Elvis, appears to mean something different from“I believe Elvis is Elvis”. I would happily utter thelatter but not the former if I believe I am facing an Elvisimpersonator. It is hard to avoid the impression that the character ofexpressions like “you” (or, “yesterday”,“that car”, etc.) makes the difference in such cases.

Two-dimensional semantic theories cannot easily offer a generalaccount of hyperintensionality. Worldly hyperintensionality isespecially recalcitrant. When, e.g., talk of essence is employed, andwe say that Socrates is essentially human but is not essentially(human and \(2 + 2 = 4)\), or not essentially (human and either batshave wings or it is not the case that bats have wings), those twoalternatives to “human” both have the same two-dimensionalintension as “human”. Any context will pick out the samecontent for the predicates “is human”, “is human andsuch that \(2 + 2 = 4\)”, and “is human and either batshave wings or it is not the case that bats have wings”.

The full range of non-trivial counterpossible conditionals are noteasily handled either. Take “If Hobbes had squared the circle,Hobbes would have proved an impressive mathematical result”.This is plausibly true, whereas “If Archimedes had squared thecircle, Hobbes would have proved an impressive mathematicalresult” is false: Archimedes lived well before Hobbes and sowould have beaten him to it, and Hobbes did not otherwise prove anyimpressive results. The necessary falsehoods “Hobbes squared thecircle” and “Archimedes squared the circle” seem todeliver different truth values as antecedents in same-consequentcounterfactuals. Functions from contexts to contents taken as theusual sets of possible worlds don’t deliver the desired result:whichever other person a context picks out for “Hobbes”,it is hard to see how “\(x\) squared the circle” couldpick out a contingent proposition that would be useful. Prospects arenot quite hopeless here: approaches that employ two-dimensionalism asat least part of the picture aboutindicative conditionalshave been suggested by Weatherson (2001) and Nolan (2003). One mighthope for an extension to counterfactuals. Or perhaps some otherapproach might work: recent work on counter-analytic conditionals doneby Locke (forthcoming) and Kocurek, Jerzak, and Rudolph (2020) ispromising.

Two-dimensionalism may be appealing as an account of somehyperintensional phenomena even if it cannot be generalized: it can beseen as one tool among others for constructing semantic theories forhyperintensional expressions, if one is willing to forego aone-size-fits all approach.

4.2 Aboutness

Some theories of propositional content represent a number ofhyperintensional distinctions by combining SPWS with ways of splittingthe modal space. The main work in this area is Yablo’s bookAboutness (2014; and see Osorio-Kupferblum 2016 for acritical discussion). Aboutness is

the relation that meaningful items bear to whatever it is that theyareon orof or that theyaddress orconcern, (Yablo 2014: 1)

namely, theirsubject matter. The subject matter of asentence in context can be seen as a topic, an issue, or a questionthe sentence may be taken, in that context, as answering to. When theissue isthe number of stars, the relevant question may be:“How many stars are there?”. As in inquisitive semantics(Ciardelli, Groenendijk, & Roelofsen 2013), one can associate thequestion with the set of its answers: There are no stars; There is onestar; There are two stars; etc. Answers are propositions in the SPWSsense: sets of possible worlds. We now call thesethinpropositions for the full propositional contents of sentences willturn out to be richer than that. The question, thus, splits thetotality of worlds into sets (the relevant thin propositions). Twoworlds \(w_1\) and \(w_2\) end up in the same set just in case theyagree on the answer—in our example: when the number of stars isthe same at \(w_1\) and \(w_2\). When, as in the example, the questionhas just one correct answer, the thin propositions at issue form apartition of the set of worlds \(W\): a splitting into subsets suchthat their union is all of \(W\), and such that each \(w\) in \(W\) isin exactly one subset or cell. One cell has all the 0-star worlds, oneall the 1-star worlds, one all the 2-star worlds, and so on.

While the idea is already in Lewis (1988) (see also Plebani andSpolaore forthcoming), Yablo proposes a generalization due to somequestions’ having more than one correct answer:“Where’s a good Italian restaurant in Amsterdam?”. Aworld \(w_1\) can be in more than one cell now: it can agree with\(w_2\) by having a good Italian restaurant in Rembrandtplein, with\(w_3\) by having another good Italian restaurant in Keizersgracht.Here the question determines a division of \(W\): a splitting intosubsets whose union is \(W\), but which can overlap.

There’s an intuitive mereology of issues, topics, or subjectmatters: they should be capable of overlapping and of fusing intowholes which inherit the proper features from the parts (Yablo 2014:Section 3.2). The topicphilosophy and the topicmathematics overlap (the overlap including, presumably,logic).How things went in 1882 is included in thelargerhow things went in the nineteenth century. A largersubject matter induces smaller cells: \(w_1\) and \(w_2\) agree onwhat happened in their whole nineteenth century only if they agree onwhat happened in their 1882 to begin with.

Let \(|A|\), a subset of \(W\), be the thin proposition expressed by\(A\). Thethick proposition expressed by \(A\), [\(A\)], isits thin proposition \(|A|\) together with \(A\)’s subjectmatter, \(s(A)\). How does one getthe subject matter of\(A\)? Yablo suggests assigning a positive subject matter, thedivision corresponding to “Why is \(A\) true?”; a negativesubject matter, the division corresponding to “Why is \(A\)false?”; and to identify the overall \(s(A)\) with its positiveand negative subject matters taken together. Worlds agree on theoverall subject matter of \(A\) when either \(A\) is true, or \(A\) isfalse, for the same reasons at them. Yablo calls the reasons for\(A\)’s being true (false), \(A\)’struthmakers(falsemakers). He suggests not to read them in ametaphysically loaded way, as chunks of reality whose obtainingnecessitates truth, and advocates a “semantic” conceptionof truthmaking: truth/falsemakers are cells of divisions of \(W\),thus, just sets of worlds again, i.e., thin propositions.

Thick propositions are hyperintensional. They cut at least as finelyas ordinary sets of worlds: when \(|A|\) differs from \(|B|\), [\(A\)]will differ from [\(B\)]. Additionally, there will be \(A\)’sand \(B\)’s expressing different thick propositions in spite ofbeing true at the same worlds, because they are about differentthings: \(s(A)\) will differ from \(s(B)\) even if \(|A| = |B|\). Forexample, “There is an even number of stars or there isnot” and “There are no non-self-identical Italianrestaurants” are both necessarily true, but they correspond todistinct thick propositions because of their different subjectmatters. The suggestion that adding subject matter to semantics is thekey to handling hyperintensionality is discussed in Gioulatou(2016).

To get a precise answer on which hyperintensional distinctions theframework can make, one needs a full-fledged truthmaker assignment toall sentences of the target language. Yablo gives two “semanticpictures”, areductive and arecursive one,which pull in different directions. We take the recursive route, whichrelies on van Fraassen (1969) and is found more plausible by some(e.g., Hawke 2018; Fine 2020), and leave the details to a consultationof Yablo (2014: 56–9). In a plain sentential language: when\(A\) is an atomic formula \(p\) one can assign it some truthmaker\(\{p^+\}\) and falsemaker \(\{p^-\}\). Negation flips: For \(A ={\sim}B\), what truthmakes \(A\) is what falsemakes \(B\) and viceversa. For \(A = B \land C\), what truthmakes \(A\) is the union ofwhat truthmakes \(B\) and what truthmakes \(C\); what falsemakes \(A\)is what either falsemakes \(B\) or falsemakes \(C\). For \(A = B \lorC\), we flip the truth- and falsemakers of \(B \land C\). Then, e.g.,\(p \land q\) is made true by \(\{p^+, q^+\}\), false by \(\{p^-\}\)and by \(\{q^-\}\); \(p \lor q\) is made true by \(\{p^+\}\) and by\(\{q^+\}\); false by \(\{p^-, q^-\}\).

Now \(p \lor {\sim}p\) and \(q \lor {\sim}q\) (“John is either abachelor or not”, “Either 44 is the sum of two primes ornot”) will express distinct thick propositions insofar as theyare about different things (how things stand John-wise is a differentissue from how things stand 44-wise). When, with distinct \(p\) and\(q\), the truthmakers of \(p \lor {\sim}p\) are \(\{p^+\}\) and\(\{p^-\}\), those of \(q \lor {\sim}q\) are \(\{q^+\}\) and\(\{q^-\}, s(p \lor {\sim}p)\) will differ from \(s(q \lor {\sim}q)\)and so the two thick propositions [\(p \lor {\sim}p\)] and [\(q \lor{\sim}q\)] will differ although \(|p \lor {\sim}p| = |q \lor {\sim}q|= W\). Also, [\(p\)] will differ from [\(p \land (q \lor {\sim}q)\)]as \(p\) is truthmade by \(\{p^+\}, p \land (q \lor {\sim}q)\) by\(\{p^+, q^+\}\) and \(\{p^+, q^-\}\), not by \(\{p^+\}\). So one cancapture the idea that “You see that Mary is eating an icecream” and “You see that Mary is eating an ice cream andJohn is either eating chips or not” don’t say the same, inthe face of the logical equivalence of \(A\) and \(A \land (B \lor{\sim}B\)).

But even lacking a precise treatment for a predicative language, wecan see that distinct, logically atomic necessarily true or falsecontents may be difficult to tell apart. Take “Mike isMike”, \(m = m\), and “Mike is Jack the Ripper”, \(m= j\), which differ in informativeness. Given that Mike is Jack andthe necessity of identity, \(|m = m| = |m = j| = W\); but also thesubject matters will coincide: what makes both claims true is a factabout Mike, Mike’s being Mike. Nothing makes either false.“Hobbes squared the circle”, \(Sh\), and “DanielNolan squared the circle”, \(Sd\), concern different issues: oneis about Hobbes’ mathematical (non)achievements, the other isabout Daniel’s. But \(|Sh| = |Sd| = \emptyset\), andthere’s no way to get truth/falsemakers as sets of possibleworlds that will make \(s(Sh)\) and \(s(Sd)\) differ: no matter howone splits and groups possible worlds into sets, there will be noworlds available where Hobbes squares the circle or where Nolan doesto begin with.

As noticed by Fine (2020), the difficulty may be due to the fact thatthe setting is conservative with respect to SPWS: it starts with theusual possible worlds and just adds ways of splitting and groupingthem (see Hawke 2018 for further criticisms). The next approachpromises to do better by looking at worlds that step beyond thepossible.

4.3 Extended Worlds Semantics

If a theorist starts from a possible worlds framework, and comes underpressure to recognize hyperintensional distinctions, one response isto add impossible worlds (Priest 1992; Zalta 1997; Kiourti 2010; Jago2015; Berto & Jago 2019) taken as ways things could not be, orworlds where some truth of logic, mathematics, or metaphysics fails.We will be brief on this topic here, as one can consult the entry onimpossible worlds for more details.

In impossible worlds semantics, different impossibilities can bedistinguished by being associated with different sets of worlds (onlyimpossible worlds), and different necessary truths can also bedistinguished by different sets of worlds (all possible worlds plussome of the impossible worlds). Each distinct necessary truth willcorrespond to a distinct set containing all possible worlds and someimpossible worlds; and each distinct impossible proposition willcorrespond to a set of worlds that contains no possible worlds, butwhich may contain impossible worlds (Rantala 1982).

Kripke (1965) already introduced “non-normal worlds” forthe semantics of modal systems such as C.I. Lewis’ S2 and S3:such worlds were points where all formulas of the form \(\Diamond A\)are true, and all those of the form \(\Box A\) are false (intuitively:everything is possible, nothing is necessary). Cresswell (1970)offered a logic of belief in which beliefs were modelled with sets ofboth possible and impossible worlds, so as to not treat everyimpossible belief the same as every other, and to allow someone tobelieve some necessary truths without believing them all. (See alsoHintikka 1975). Since then, impossible worlds have been used totheorize about many of the hyperintensional phenomena discussed inthis entry.

A closely related way to extend possible worlds semantics is tointroduce incomplete and inconsistent partial specifications of howthings could be, and could not be. So-calledsituationsemantics (Barwise & Perry 1983) employs these more generalobjects to handle hyperintensional constructions such asspecifications of beliefs or information contents, as well asapplications to avoid paradoxes that can arise from employing possibleworlds. A version of situation semantics known as “truthmakersemantics” has recently been defended as being superior tostandard world semantics (see Fine 2012, 2016, 2017; Hornischer 2020;Moltmann 2020, forthcoming).

With impossible worlds added to a theory, necessarily equivalentexpressions no longer have to have the same semantic values. Apartfrom their widespread use in the semantics for logical systems, workhas continued on using impossible worlds for giving theories ofpsychological attitude ascriptions and of claims about meaning: seeJago (2014) for one recent treatment of a number of issues involvingmental content and reasoning. Impossible worlds also help in theoriesof other kinds of representations: for use in theories of impossiblefictions, see Priest (1997) and Badura and Berto (2019).Epistemologists are starting to find applications for impossibleworlds in theories of knowledge: see, for example, Melchior(forthcoming). And logicians are using them to model non-omniscienthyperintensional agents by combining impossible worlds semantics withdynamic epistemic logic (Bjerring and Skipper (2019); Solaki, Berto,& Smets forthcoming).

Given the limitations of Lewis-Stalnaker theories of counterfactualconditionals when dealing with impossible antecedents (see§1.1.2), it is natural to extend a theory of counterfactual conditionals withimpossible worlds: see Routley (1989), Mares (1997), Nolan (1997),Brogaard and Salerno (2013). Similar theories can be offered ofindicative conditionals: see Nolan (2016b).

Impossible worlds have also been employed to give theories of a rangeof hyperintensional phenomena in metaphysics. One early applicationwas in the theory ofde re necessity, in offering a theory ofhow it could be necessarily true that an artifact could be made ofslightly different material, but not too different, without succumbingto Chisolm’s paradox (Salmon 1984). Impossible worlds can beused to give a hyperintensional theory of properties as sets ofpossible and impossible instances (Nolan 2013: 8). They can be used tooffer a theory of metaphysical explanation (Kment 2014), or in atheory of essence (Brogaard & Salerno 2013: 646–8). They canplay a central role in a theory of omissions of impossibilities(Bernstein 2016).

The literature to date has only scratched the surface of potentialapplications of impossible words in metaphysical theorizing. A rangeof concerns about postulating impossible worlds are discussed in theentry onimpossible worlds, section 6, Nolan (2013), and Berto and Jago (2019), as well as the concernsdiscussed in§2 and§3 above.

4.4 Relevance/Relevant Logic

One should look at the SEP entry onrelevance logic (also called “relevant logic”) for an introduction tothis research program (see the seminal Anderson & Belnap 1975;Anderson, Belnap, & Dunn 1992; and, for a comprehensiveintroduction, Dunn & Restall 2002). We briefly discuss itsconnections to hyperintensionality here. Its origins lay in the aim todevelop logical systems free from the “paradoxes of the materialand strict conditional” discussed above. Another earlymotivation was to construct a conditional that could represent logicaldeducibility within the logical language itself.

In this program, an agreed upon necessary (though, in general, notsufficient) condition for a conditional to be logically valid is thatthere be some connection between antecedent and consequent.Historically, this has often been phrased via a requirement called theVariable Sharing Property (VSP): \(A \rightarrow B\) is validonly when \(A\) and \(B\) share some sentential variable or parameter(cf. Anderson & Belnap 1975: 32–3). VSP delivershyperintensional distinctions: conditionals with (classically)impossible antecedents and necessary consequents are not all triviallyvalid or true. The infamousex falso quodlibet, \((A \land\neg A) \rightarrow B\), for instance, fails (it’s not the casethat anything is implied by a contradiction); whereas \((A \land \negA) \rightarrow A\) holds in most mainstream relevant systems(plausibly, a conjunction relevantly implies its conjuncts).

A key feature of relevance logics, made apparent by their mostdeveloped semantics, the so-called Routley-Meyer frames (Routley &Meyer 1973), is that hyperintensional distinctions are not achieved bysuperimposing on an essentially classical or normal modal logic afilter that screens out irrelevancies. Relevance logics are weakerthan normal modal logics with SPWS or Kripke frames because theirsemantics embeds points of evaluation for formulas different fromclassical (maximally consistent) possible worlds, which are accessible(via aternary accessibility relation, different from thebinary accessibility of Kripke frames) in the evaluation ofconditionals.

Working out the intuitive interpretation of such points, of theternary relation, and indeed of the Routley-Mayer semantics as awhole, has been a difficult task in the history of this researchprogramme (see Copeland 1980; Read 1988; Restall 1996b; Mares 2004).The consensus on the points nowadays seems to be that they beunderstood as non-normal or logically impossible worlds (see Priest2008: Ch, 10; Berto & Jago 2019: Ch. 6). See the SEP entry onimpossible worlds for details. The impossible worlds of (the interpreted) Routley-Meyerframes can be seen as situations which can, e.g., be locallyinconsistent (making true both \(A\) and \(\neg A\), for some \(A)\)without thereby being trivial (making true everything), and partial orlocally incomplete (“taking no stance” on some \(A\) bymaking true neither it nor its negation).

The conditional of most mainstream relevant systems is taken asrepresenting implication or (necessary) entailment, not aceterisparibus or counterfactual conditional. But some of the firsttheories of counterpossible conditionals, addressing the limitationsof the Lewis/Stalnaker account of counterfactuals, were developedagainst a relevance logic background: see Routley (1989); Mares andFuhrmann (1995), and Mares (1997).

Other applications of relevance logic to areas where hyperintensionaldistinctions seem needed include applications to the theory ofconfirmation in science (Goddard 1977; Sylvan & Nola 1991); thelogic of fiction (Routley 1979); and the theory of information (Mares2004; Dunn 2015). Finally, a number of theories in deontic logic havebeen offered using a relevance logic base, including Anderson (1967),Goble (1999) and Tagawa and Cheng (2002).

Relevance logic has also been deployed to give theories ofworldly hyperintensionality. In a series of papers, MichaelDunn developed a theory ofrelevant predication intended toilluminate a range of traditional metaphysical distinctions, includingthe differences between intrinsic and extrinsic properties, essentialand accidental properties, and between essential properties andproperties necessarily had when an object exists (Dunn 1987, 1990a,1990b). And as mentioned above, Restall (1996a) employed relevantlogic to offer a theory of truthmakers.

Overall, the tools provided by the relevance logic program offertheoretical resources for constructing theories of hyperintensionalphenomena, whether representational or non-representational. To date,many of these proposals have been piece-meal, with less of a generalstory about how these techniques could be applied to handlehyperintensional language across the board. Some general strategieshave been outlined for how a theory could be constructed that handlesall hyperintensional phenomena: one such program was RichardRoutley’s “Ultralogic as Universal”, first set outin Routley (1977). See Nolan (2018) and the essays in Routley (2019)for recent evaluations.

4.5 Structured Propositions

After Soames’ (1987) attack against unstructured accounts ofpropositions as sets of truth-supporting circumstances, structuredpropositions accounts (King 1995) have become the main rivals to SPWSviews. Roughly: propositions are taken as structures, composed of theentities which are the semantic values of the correspondingsentences’ syntactic constituents. Take “Robin lovesBatman”. It’s composed of a noun, “Robin”, anda verb phrase, “loves Batman”, itself made of a verb and anoun. Let the semantic values of the lexical items be: [Robin],[Batman], [loving]. Then the corresponding proposition can betaken as an ordered triple \(\langle\)[Robin], [loving],[Batman]\(\rangle\). Ordering is important: Batman may not return thefeeling, and “Batman loves Robin” expresses a differentproposition: \(\langle\)[Batman], [loving],[Robin]\(\rangle\). (A proposition is notjust an orderedlist, but we leave aside the difficult metaphysical issue of what tiesits constituents together: see Gaskin 2008.)

What makes for the constituents? BroadlyRussellianstructured accounts, developed by authors like Salmon (1986), Soames(2008), are typically paired with direct reference or Millian accountsof names: thus, [Batman] is Batman and [Robin] is Robin. The semanticvalues of predicates and verb phrases can be taken as properties orrelations. The logical vocabulary is typically interpreted as denotinglogical operations, e.g., [\({\sim}\)] can be the unary function thatflips truth and falsity, [\(\land\)] the binary function that outputstruth only if both inputs are truths, etc.

This delivers hyperintensional distinctions, e.g., “John iseither a bachelor or not” and “Either 44 is the sum of twoprimes or not” will express different propositions: only onewill include [John], i.e., John, as a constituent. Necessarilytrue/false atomic sentences will also be differentiated, even whentheir syntactic structure is the same (“Hobbes squared thecircle” and “Daniel Nolan squared the circle”:[Daniel Nolan] isn’t [Hobbes]; “\(3 + 3 = 6\)” and“\(2 + 2 = 4\)”: [3] isn’t [2]). More challengingcases for the view come with the necessarya posteriori.“Mike is Mike” and “Mike is Jack the Ripper”express the same proposition, because [Mike] = [Jack the Ripper]. Samefor “All woodchucks are woodchucks”, “All woodchucksare whistlepigs”, as [woodchuck] = [whistlepig]. This willcreate issues in intentional contexts:

Suppose Tama is familiar with both woodchucks and whistlepigs, butisn’t sure that they are the same kind of critter. He’snoticed the similarities, though, and so he has his suspicions.Suppose further that Tama knows he is allergic to whistlepigs, andknows that he has just been bitten by a woodchuck. In this scenario,[“Tama fears that all woodchucks are whistlepigs”] islikely true, while [“Tama fears that all woodchucks arewoodchucks”] is almost certainly false. (Ripley 2012: 9)

Soames (1987) proposes a pragmatic-metalinguistic strategy similar tothe one tried by Stalnaker (1984), discussed in§2.2: one should distinguish attitudes towards sentences and towardspropositions. Soames comes up with examples showing that the formercan be an unreliable guide to the latter, e.g., one can sometimesbelieve a proposition, but only assent to one of two distinctsentences expressing it. Youdo know that Mike is Jack theRipper by knowing that Mike is Mike, for it’s the sameproposition. But you will assent to, or assert, only the sentence thatreports your knowledge in the trivial clothing. Then, the ancientsbelieved that Hesperus is Phosphorus, Tama fears that all woodchucksare woodchucks, and so on. If you think otherwise, it’s becauseyou confuse belief reports with reports of linguistic practices.Ripley (2012) argues that it’s difficult to account forphenomena concerning such attitude reports by throwing them in the boxof pragmatics and conversational implicatures: failures ofsubstitutivitysalva veritate behave in systematics ways incontexts of iterated embeddings, and this calls for a systematic,compositional treatment.

Salmon also resorts to a distinction between “semanticallyencoded” and “pragmatically imparted information”(Salmon 1986: 78) for the “Mike is Mike” vs. “Mikeis Jack the Ripper” cases. He introduces differentrepresentational guises under which one can have attitudes to theproposition expressed by different sentences. For direct reference towork, we need [Mike] to be [Jack the Ripper], but we can allowMike-guises to differ from Jack-the-Ripper-guises, and claim that(de dicto) intentional attitudes are relations topropositions mediated by guises: Lois Lane loves Kal-El under theguise of Superman, not under the guise of Clark Kent. Guisesdon’t belong in semantics but in pragmatics, and don’tdetermine denotations.

Salmon’s view can in principle account for compositionalphenomena involving embeddings. It has been criticized because ofthis: Forbes (1987) argues that guises look too much like Fregeansenses in disguise. Branquinho (1990) argues that the correspondencemay not reduce Salmon’s account to a relabelled Fregeanview.

Guises are supposed to be mobilized within attitude reports. Beyondattitude reports, as argued by Ripley (2012), Jago (2014), there arehyperintensional phenomena involving indicative and counterfactualconditionals structuralism doesn’t seem to handle. As Jago(2014: 76–7) has it, we can be at the 1972 Ziggy Stardust Tourand wonder if Ziggy is David Bowie. We come up with the following:

(1): If Ziggy isn’t Bowie, then Bowie isn’t Bowie.
(2): If Bowie isn’t Bowie, then Bowie isn’t Bowie.
(3): If Ziggy weren’t Bowie, then Bowie wouldn’t beBowie.
(4): If Bowie weren’t Bowie, then Bowie wouldn’t beBowie.

All of (1)–(4) have metaphysically impossible antecedents. (1)and (3) are plausibly false (unless one is a vacuist: see Williamson2007), but (2) and (4) are trivial truths. (A nonvacuist conditionalsemantics using impossible worlds delivers just these results: seeBerto et al. 2018.) But the structuralist has (1)–(2) and(3)–(4), respectively, express the same propositions, given that[Ziggy] = [Bowie]. If conditionals in general, or even onlycounterfactuals, don’t involve intentionality, guises will offerlittle help if they are not supposed to come into play here. However,there are treatments of conditionality along the lines of epistemicmodals (Kratzer 1986; Lycan 2001), and the Russellian structuralistmay appeal to one of them, or come up with a new one.

Could a broadlyFregean account of structured propositions dobetter? In this view, we don’t take the constituents ofstructured propositions as denotations (think objects and properties),rather as Fregean senses. Much now depends on how these areunderstood. If they are taken à la Carnap, as functions fromthe possible worlds of SPWS to extensions, there is little hope toaccount for “Mike is Mike” vs. “Mike is Jack theRipper” cases. Now [Mike] and [Jack the Ripper] are functionsfrom possible worlds to individuals; but unless one clashes with theKripkean criticisms of descriptivism on names (Kripke 1980), they hadbetter be constant functions: given that they both actually pick outMike, they should output Mike at all possible worlds (at which Mikeexists). But then [Mike] and [Jack the Ripper] will be defined on thesame set of things and will give the same output for any input.

One may enlarge or change the set of circumstances that provides theinputs, admitting metaphysically impossible situations where Mike isdistinct from Jack the Ripper, and have the two as the outputs of[Mike] and [Jack the Ripper] there, respectively. But then, asremarked by Ripley (2012) and Jago (2014) again, it is not so muchstructuralism that does the job of giving the requiredhyperintensional distinctions, but rather the fact that we are notworking (only) with standard (metaphysically) possible worldsanymore.

A recent interesting proposal to improve on the Fregean situationcomes from Hawke (2018), combining a theory of subject matter with astructuralist account of propositions resorting toFregean-senses-lookalikes. Subject matter sensitivity allows Hawke tomake hyperintensional distinction unavailable to standard Fregeanstructuralism.

One logically sophisticated, neo-Fregean structured account, offeringa systematic analysis of a range of hyperintensional phenomena, is theTransparent Intensional Logic approach. The view, pioneeredby Tichý (1968, 1971, 1988), treats the meanings of expressionsas given by structural procedures, calledconstructions,built out of entities that are somewhat like Fregean senses. Inparticular, different names and different predicates, even if theynecessarily co-designate, may be associated with different senses, sothe meanings of “Robin Hood” and “Robin ofLocksley”, or of “furze” and “gorse”,may be distinct even if those meanings are not built up out of others.This gives the system resources to handle many hyperintensionalcontexts straightforwardly. In particular, it manages to give apowerful compositional account where other approaches have to resortto pragmatics. The approach is less popular than it should be incontemporary semantics, possibly due its resorting to a technicalapparatus of typed lambda-calculus. For hyperintensional applicationsand a comprehensive discussion of Transparent Intensional Logic, seeDuží, Jespersen, and Materna (2010) and Dužíand Jespersen (2015).

We mention one final proposal in the Fregean camp by Schellenberg2012, and which was further defended by Skipper and Bjerring (2020).This pivots on another Fregean notion, namely that ofcognitiveequipollence: \(P\) and \(Q\) are cognitively equipollent whenone cannot rationally regard either as true and the other as false .This gives a cognitive or epistemic notion of sentential meaningwhich, the authors claim, is more fine-grained than SPWS intensions,but less than extra-fine-grained extended (e.g., impossible) worldssemantics.

4.6 Algebraic and Object Theory Approaches

A final family of approaches to hyperintensionality to be discussed isthe algebraic. Algebraic approaches to semantics typically do not tryto give an account of meaning in terms of anything else at all.Instead, they treat meanings as primitive and focus on therelationships between them to give a semantic theory.

By defining in a direct way how the meanings of different parts oflanguage go together to yield sentence meanings, these approaches arenot under pressure to identify the meanings of sentences true in allthe same possible circumstances, or even sentences which are logicallyequivalent. So it is easy to see how a general algebraic approach canmake room for hyperintensional differences between parts of language.Even if sentence meanings are necessarily equivalent, if they relatedifferently to a “believes that” operator they do not haveto be substitutable in “John believes that \(A\)”contexts. Bealer (1979) touts this flexibility as an advantage indealing with puzzles like the semantics of belief attributions, asdoes Menzel (1993).

Some algebraic approaches do not take as much advantage of thisflexibility as one might expect with respect to hyperintensionalitypuzzles. Keenan and Faltz (1984)’s algebraic approach identifiesnecessary equivalents. Bacon (2018) adopts a Boolean account in whichthe propositions associated with logical equivalents in predicatelogic are identified with each other (e.g., the proposition that\({\sim}({\sim}A \lor {\sim}B)\) is identical to the proposition that\((A \land B))\). This leaves room for distinguishing merelynecessarily equivalent propositions, e.g., the proposition that\(2+2=4\) and the proposition that it rains if and only if it rains.But it does little to help with cases where we want to discriminatebetween logical equivalents: if we were to think that \(A\) explainswhy \({\sim}{\sim}A\), or capture the belief content of the confusedman who believes something of the form \((A \land B)\) but does notbelieve \({\sim}({\sim}A \lor {\sim}B)\). So it is not the algebraicapproachper se that yields solutions to particular puzzlesabout hyperintensionality, but the algebraic approach combined withtheoretical choices about the structure of the algebra.

Algebraic approaches are largely silent on the question of whatmeanings are, and how meanings come to stand in the relations they do.(Or, if every structure of relations between meanings can be foundsomewhere in Platonic heaven, how we come to be associated with ourparticular structure rather than others.) Of course, theories whichsay more about the nature of meanings (e.g., that they areset-theoretic constructions from worlds, or they are ordered sets ofobjects and properties)also face the challenge of explaininghow our concrete practices explain which structures of meanings areassociated with our speech and writing. So it may be that theseconcerns are general, and algebraic semantics can employ the samekinds of answers to these questions as their rivals.

A close relative of algebraic theories is the object theory of Zalta(1988). It also supplies distinct fine-grained meanings to, e.g., bethe suitably distinct inputs so that “John believes alltriangles have internal angles that add up to 180°” and“John believes all trilaterals have internal angles that add upto 180°” express different propositions. Zalta says a littlemore about the identity conditions of hishyperintensionally-distinguished properties: according to Zalta,properties are identical if and only if they areencoded bythe same objects.

Algebraic accounts of meaning may have little to say about worldlyhyperintensionality. There is nothing in the general motivations ofalgebraic theories that would rule out the hypothesis that we needhyperintensional language to best capture the non-representationalaspects of reality, but nor does there appear to be anything in thegeneral motivations that would indicate why we would, or how thatmight manifest in our language. This is not really an objection toalgebraic approaches, however. One might think that a theory ofmeaningshould be silent on substantive questions about whatthe world is like. We do not want our theory of meaning to tell uswhich physics or chemical theory to adopt. Likewise, it might be goodfor a theory of meaning to be neutral about how to understandessences, or intrinsicness of properties, or causation. Of course, atheory of meaning may not be able to be completely neutral aboutmetaphysical questions—it should have something to say aboutwhat meanings are, whether there are abstract representations, and soon—but perhaps algebraic theories not taking a stand on worldlyhyperintensionality is an attractive feature rather than adrawback.

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Acknowledgments

Versions of this entry have been presented at the ArchéMetaphysics and Logic research seminar at the University of StAndrews, at the Logic of Conceivability seminar at the University ofAmsterdam, and at the 2020 Hamburg Summer School, at the Institute ofPhilosophy of the University of Hamburg. We are grateful to theparticipants, as well as Sara Bernstein, Max Cresswell, Greg Restalland two referees of the Stanford Encyclopedia of Philosophy, for theirvaluable feedback. Franz Berto’s research for this entry hasbeen funded by the European Research Council (ERC CoG), Grant Number681404.

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