## 1. The Variety of Hyperintensional Phenomena



We provide a taxonomy of candidate hyperintensional notions by
dividing them into two camps: (1) *representational* concepts
pertain to ways of representing the world (1a) in thought and (1b) in
language. In the former sub-camp (1a), we find mental states or
activities like *believing, knowing, imagining, conceiving*,
*being* or *becoming informed*. In the latter sub-camp
(1b), notions such as *meaning*, *content,* and
*conditionality*. (2) *Non-representational*
hyperintensional concepts include allegedly mind-independent,
"worldly" notions such as *essence, grounding,
metaphysical dependence, explanation*.



(The division is not uncontroversial, e.g., some claim that
explanation and indicative conditionals essentially involve cognition
(Lycan 2001), so they would place them in (1a). Some claim that there
exists "environmental" information (Barwise & Seligman
1997) that should go into (2). We stick to what we take to be
mainstream views on where such concepts should go.)



For each of these notions, we provide (a) a brief explanation of how
they have been dealt with in SPWS; and (b) examples illustrating their
seemingly hyperintensional nature. These, if found persuasive, may
work as data against which the theories described in
 Section 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 realm
of *intentional* concepts: notions that involve states of the
mind which are directed towards contents, situations, etc. The two
most studied such notions are knowledge and belief. Since Hintikka
(1962), these have been understood in epistemic logic as modal
operators: restricted quantifiers over possible worlds. Knowledge and
belief are characterized in terms of what is true throughout a set of
worlds which represent ways things could be, for all the cognitive
agent knows, or given the agent's evidence, etc. If \(K\)
expresses the agent's knowledge or belief state, \(w\) is a
possible world, and \(R\) an epistemic accessibility relation on the
space of worlds \(W\), the Hintikkan characterization goes:



(H)
\(KA\) is true at \(w\) iff \(A\) is true at all \(w\_1\) such that
*wRw*\(\_1\).


Some views in mainstream epistemology follow the same path, e.g., in
*relevant alternatives* approaches knowledge is "an
evidential state in which all relevant alternatives (to what is known)
are eliminated" (Dretske 1981: 367). Uneliminated alternatives
work similarly to accessible worlds.



Such treatments of knowledge and belief witnessed an early
manifestation of the hyperintensionality issue: the *problem of
logical omniscience* (Fagin et al. 1995: 335-6)--a set
of closure conditions for the relevant \(K\)s, of which perhaps the
most 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 dubbed *Closure under entailment* or *Full
omniscience*, has it that one knows or believes all the logical
consequences of what one knows or believes. (C2), *Knowledge of all
validities* or *Belief in all validities*, has it that one
knows or believes all logical truths. (C3) ensures the consistency of
knowledge or belief. All follow from understanding \(K\) as a
quantifier 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 \(KA\) means
that \(A\) 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 at
all worlds (of all the relevant models), then it will hold at all the
epistemically accessible ones, and so \(KA\) will hold. (C3): no
possible world, thus no accessible one, will make both \(A\) and
\({\sim}A\) true.



These don't sound right for real humans, as opposed to
idealized, logically infallible agents. (C1): we know basic arithmetic
truths like Peano's postulates, and these entail (suppose)
Goldbach's Conjecture (Every even whole number larger than 2 is
the sum of two primes); but we don't know whether
Goldbach's Conjecture is true. (C2): "If
\({\sim}{\sim}A\), then \(A\)" is (suppose) valid, but
intuitionist logicians do not believe it. (C3): we can have, if
implicitly, inconsistent beliefs.



One may treat "\(KA\)" in (H) as not really standing for
knowledge or belief. We should read it as "\(A\) follows
logically from the agent's knowledge", or "The agent
is committed to \(A\), given what it knows/believes". Then we
still lack some logical account of *knowledge* and
*belief* per se, as opposed to some derivative attitude or
conditional commitment. The commitment itself may be questioned: are
we committed to believing everything, given that we have inconsistent
beliefs? In what sense must a finite agent believe infinitely many
logical consequences? Such a "must" lacks a corresponding
"can".



*Information* received a modal treatment in the classic
Bar-Hillel and Carnap (1952) account. The informative task of a
sentence \(A\) is to split the totality of worlds into those where
\(A\) is true and those where it isn't. An agent's
becoming informed of \(A\) is its ruling out the not\(-A\) worlds as
contenders for actuality: before being informed that it's
raining 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 \(n \gt
2\)" are necessarily equivalent, given the unrestricted
necessity of logical and mathematical truths, and SPWS. They are true
at the same worlds: all of them. On the Bar-Hillel-Carnap view, one
can rule out no world after learning either. But while one easily
deems the former true, the latter, which expresses Fermat's Last
Theorem, took centuries to be proved and its proof brought significant
news.



Generalizing: a merely intensional analysis of information entails
that *no* logical, mathematical, or metaphysical necessary
truth can ever be informative and denies the informativeness of
logical deduction. But logical deductions are informative (Jago 2009,
2014). Besides, we can become informed of exactly one of two
necessarily equivalent claims. Of necessity, "All woodchucks are
woodchucks" is true if and only if "All woodchucks are
whistle-pigs" is (Ripley 2012). But the former is trivial,
whereas one may learn the latter from a zoologist.



Other attitudes like *hoping, fearing, supposing, imagining*
display hyperintensional features (Wansing 2017). One may be led to
the general conclusion that *thought* is hyperintensional: our
intentional states can treat necessarily equivalent contents
differently. Lois Lane can wish that Superman is in love with her
without wishing that Clark Kent is in love with her, although (if
Barcan Marcus and Kripke are right) it is impossible for Superman to
be other than Clark Kent. We can suppose that \(75 \times 12 = 900\)
without supposing Fermat's Last Theorem. Even intentional states
involving perceptual experiences, like *seeing*, seem
hyperintensional: you can see that Mary is eating an ice cream without seeing
that Mary is eating an ice cream and John is either eating chips or
not (you may not be seeing John at all). But \(A\) is logically
equivalent to \(A \land (B \lor {\sim}B)\) in classical logic and even
in many weaker logics (Barwise & Perry 1983).


#### 1.1.2 Semantic Concepts



Propositions, taken as the contents or meanings of sentences, have
often been interpreted starting from Wittgenstein's insight that
understanding a sentence is knowing what is the case if it is true
(Wittgenstein 1921: 4.024). Truth conditions have been given via sets
of 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 St
Andrews is rainy.



This makes all necessarily equivalent propositions identical, for
possible worlds can never disagree on them. "If Kennedy was
shot, then Kennedy was shot" and "\(x^n + y^n = z^n\) has
no integer solutions for \(n \gt 2\)" mean the same thing: the
total set of worlds. This seems wrong, given that they don't say
the same thing: only one speaks about Diophantine equations. Similarly
for necessary falsehoods: "Kennedy was both shot and not"
and "\(2 + 3 = 6\)" are about different things, but they
also end up expressing the same proposition: the empty set.



We can locate in the area of semantic concepts hyperintensional
phenomena which have to do with conditionality. We start with
indicative conditionals. The extensional material conditional,
"\(\supset\)", validates notoriously paradoxical
inferences:



(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 ordinary
language come out true just because the antecedent is false or the
consequent true: "If Clinton is Australian, then Mars is made of
marbles"; "If Strasbourg is in Germany, then Clinton is
American".



One may think the issue is that the connection between antecedent and
consequent is too contingent an affair here. One may require, for
"If \(A\), then \(B\)" to be true, that it cannot be the
case that the antecedent is true while the consequent is false. This
gives the so-called strict conditional: a modal operator,
"\(\prec\)", such that "\(A \prec B\)" is
understood 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 conditional
captures the meaning of the indicative "if", lots of bad
conditionals come out true: "If \(6 + 3 = 11\), then Clinton is
Australian"; "If Clinton is American, then \(6 + 3 =
9\)". These are not obviously true, and there are some reasons
to suppose they are false: there is no connection between antecedent
and consequent, for example. Other conditional forms that the strict
conditional 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, then
either St Andrews is in Scotland, or it isn't": the
material constitution of Mars has little to do with St Andrews'
location. Some theorists argue that treatments of the conditional that
do not render conditionals like these automatically true would be
preferable.



On to counterfactuals or subjunctive conditionals--conditionals
of the form, "If it were (had been) the case that \(A\), then it
would be (have been) the case that \(B\)". A common treatment,
due to Stalnaker (1968) and Lewis (1973), analyses them in terms of
possible worlds. In a simple version, they are true just in case at
the closest worlds where \(A\) is true, \(B\) is true, where closeness
is understood as similarity in relevant respects. "If Kangaroos
had no tails, they would topple over" is true when, at all
worlds where kangaroos have no tails and which are as relevantly
similar as it gets to the actual world, kangaroos topple over.



This again makes all counterfactuals whose consequent is necessary, or
whose antecedent is impossible, vacuously true: there are no worlds to
falsify the consequent, or to verify the antecedent. Counterfactuals
with impossible antecedents are called *counterpossibles.* They
don't seem to be all vacuously true. An often-cited example from
Nolan (1997):



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


Squaring the circle is a mathematical impossibility, in spite of
Hobbes' hopes. His achieving it, though, would have made no
difference for those sick children. (9) should come out true for this
reason, not because no possible world verifies the antecedent. For
similar reasons, (8) should come out false. Against vacuism, some
counterpossible pairs with opposite consequents should have different
truth values (for a defense of vacuism, see Williamson 2007; for a
discussion, 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 substitution
of logical equivalents fails in intentional contexts (he mentioned
"says that", "knows that", "believes
that", "deduces that": 1975: 37). Other cases
discussed above might contribute to the impression that our language
contains hyperintensional expressions just because of phenomena
involving representation.



However, various philosophers have employed hyperintensional resources
to theorize about aspects of the world beyond representation.
Famously, Fine (1994) argues that work on the *essences* of
things must play a central role in metaphysics and essence talk must
be hyperintensional. In the construction "\(x\) is essentially
\(F\)", we cannot be guaranteed to preserve truth by
substituting 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 human
and a member of a set. Fine extends this to talk of real definition
generally: it is the real definition of 2 that it is the successor of
1, 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 Fine
1994, but we take it he would endorse them given his discussion on pp.
4-5 and p 14.) Dunn (1990b) also argues that one needs
hyperintensional resources (in his case, from relevant logic) to
capture essentialist claims.



Nolan (2014) argues for the need to provide hyperintensional accounts
of a range of non-representational aspects of reality. As well as a
theory of essences, theories of counterfactuals, explanation,
metaphysical grounding, of the distinction between intrinsic and
extrinsic properties, of dispositions, are all plausibly
hyperintensional. Nolan also mentions the analysis of causation, laws
of nature, objective confirmation, objective chance, and objective
ethics. Other recent proposals in metaphysics that have employed
hyperintensional resources include work on truthmakers, omissions,
metaphysical dependence, and the distinction between qualitative and
non-qualitative properties. Almost any area of metaphysics where
necessity or counterfactuals have played a role is a candidate for
hyperintensional approaches. We now give a flavor of hyperintensional
theories 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 our
representational abilities? Nolan (2014: 156) argues that it sometimes
does. Regular platonic solids come in only a handful of sizes (the
most 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 platonic
> solid, it would have more sides than any piece of steel in the shape
> of a dodecahedron.
> 
> 
> 



On the other hand,



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



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



Explanation is plausibly hyperintensional: "explains" can
be flanked by expressions that cannot be substituted with necessary
equivalents *salva veritate*. One pure mathematical truth can
explain another, but not every mathematical truth explains every
other, even if every pure mathematical truth is a necessary truth
(Baron, Colyvan, & Ripley 2020). Schneider (2011) argues that
sometimes logical equivalents can explain each other, making a case
that "because" is hyperintensional. \({\sim}{\sim}A\) is
true because \(A\) is true, but not vice versa. If the correctness of
explanations turns out to depend also on how the world is, rather than
only on what suits our explanatory practices, these cases may be due
to hyperintensional explanatory connections in the world. (How best to
understand explanations, of course, remains a contentious
philosophical question.)



Metaphysical grounding is widely regarded as hyperintensional.
Sometimes a more fundamental fact necessarily leads to a less
fundamental fact that it grounds, but the two are merely modally
inseparable. A widely discussed example is the grounding of the
existence of a set in the existence of its members. On one view,
{Socrates, Plato} automatically exists if Socrates and Plato both
exist, but it exists only in possible worlds where Socrates and Plato
both exist. So the fact that Socrates and Plato both exist grounds the
fact that {Socrates, Plato} exists (see Dunn 1990b; Fine 1994). But
that {Socrates, Plato} exists is not supposed to ground the fact that
Socrates and Plato both exist, even though necessarily one obtains
only if the other does. On an attractive account, many complex logical
truths are grounded in simpler ones. \({\sim}{\sim}A\) is grounded in
\(A, (A \land A)\), is grounded in \(A\), and many other logical
equivalents of \(A\) are grounded in \(A\), though they are not the
same proposition as \(A\). And as before, while \((A \land A)\) is
grounded in \(A, A\) is not grounded in \((A \land A)\) (see
Poggiolesi 2018, 2020 for a formal logic of grounding).



Whether grounding is indeed hyperintensional may depend on how
grounding claims are best understood. Where "...
grounds..." is a sentential operator, the cases discussed
had best be captured hyperintensionally. "(Socrates and Plato
both exist) grounds (The set {Socrates, Plato} exists)", for
example. Things may work differently if we take "...
grounds..." as a two-place predicate applying to facts or
occurrences: "The fact that Socrates and Plato both exist
grounds the fact that {Socrates, Plato} exists" expresses a
relation between two objects, and we might be able to preserve the
truth of this statement by substitution of any other ways of referring
to each of those objects. One complication is that one might take
these expressions picking out facts to themselves have constituents
designating propositions: "the fact that Socrates and Plato both
exist" may have the logical form *the fact that A*, where
*that* \(A\) is the proposition that Socrates and Plato exist.
Interpreted that way, the expression is functioning hyperintensionally
after all, since the necessarily equivalent proposition that
{Socrates, Plato} exists cannot be substituted into that expression
without changing the truth-value of the relevant grounding claims.
Finally, one could take the primary way of expressing grounding
relations to be by talking about objects in general, e.g., about
Socrates and Plato grounding the set {Socrates, Plato} (Schaffer
2009). On this third approach it is natural to think that any other
ways of designating those objects could be substituted *salva
veritate* so that the grounding-between-objects predicate is
extensional; though see Jenkins (2011) for a hyperintensional
object-theoretic grounding approach that does not allow this
substitution *salva veritate*.



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



The distinction between intrinsic and extrinsic properties features in
many metaphysical theories, and it appears to be
representation-independent: presumably, if mass is intrinsic it would
remain such whatever we said or did. One idea behind the distinction
is that a property had intrinsically is had only in virtue of an
object's own nature, while extrinsic properties are partly a
matter of how the object stands to other objects. Some have argued
that necessarily coextensive properties can differ in whether they are
intrinsic or extrinsic (Dunn 1990a; Eddon 2011; Bader 2013; Marshall
2015; Hoffmann-Kloss 2015). *Being an instance of Barak's
actual favorite property* is plausibly extrinsic, but *having a
mass* may well be intrinsic, even if mass is Barak's
favorite property, and even though, given it is actually his favorite
property, anything in any possible world has the first property iff it
has the second. The above-named philosophers disagree on how to
characterize the intrinsic/extrinsic distinction, but they share the
view that it has to be hyperintensional.



Jenkins and Nolan (2012) argue that disposition ascriptions are
hyperintensional. Heidi the gifted mathematician may be disposed to
find the proof of a given conjecture, on condition that it has one.
This disposition of Heidi's can be known in advance of knowing
whether the conjecture has a proof. ("Give this to Heidi: if
there's a proof to be found, she'll find it".)
Suppose the conjecture is in fact false, though the mathematical
community does not yet know that. Then the condition of the
disposition ascription is impossible. Still, we cannot substitute any
impossibility in the disposition ascription. It is plausibly false
that Heidi is disposed to find the proof of that conjecture, on the
condition that she was always both identical to the number 7 and
innumerate, for example.



We use necessarily false conditions in disposition ascriptions outside
talk about agents, too. Working with a model where the number of prey
killed is a function of the number of predators, the model might say
adding half a fox to a closed ecosystem, or adding \(1/\pi\) foxes to
an ecosystem, produces a certain fraction of additional rabbits
killed. 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 added *and* it is idealized in various ways is very
different from what it is disposed to do with \(105+1/\pi\) foxes and
the same idealization.



One way to specify probabilities is with propositional complements:
the probability that it will rain tomorrow, or that this coin will
come 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 be
hyperintensional if we cannot guarantee that we preserve truth value
by substitution of necessarily equivalent propositions.



If we use \(\rP\) to represent the (degrees of) beliefs of an
agent--even a logically idealized one--we might want to
allow different probability assignments to necessarily equivalent
propositions: with all her exhaustive knowledge of pure arithmetic and
logic, the agent might still be uncertain on whether water =
H2O before investigating, in spite of certainty on the
self-identity of water. If the hyperintensional probability function
represented 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 objective
chance ascriptions are hyperintensional.



Plausibly, "ought" is hyperintensional: perhaps you ought
to call your sister, but it is not the case that you ought to call
your sister and be self-identical even though it is impossible to call
your sister without calling your sister and being self-identical, and
vice versa. (Perhaps you promised to call: but unless you are a
philosopher you probably did not promise to be self-identical.) Some
of these obligations might be dependent on our representational
practices. But perhaps there are obligations of objective morality as
well. Perhaps your moral obligation to not callously cause suffering
is like this. If there are objective obligations, they might be best
construed as hyperintensional: the Moral Law may enjoin you to avoid
callously inflicting suffering, but it may be silent about whether you
ought to avoid inflicting callous suffering while being
self-identical. See Faroldi (2019) for a recent hyperintensional
treatment of "ought".



Truthmakers theorists have been looking for a relationship between
true propositions and the aspects of the world that make those
propositions true. Theories of this relation often make truthmaking
hyperintensional: \(A\) and \(B\) may be necessarily equivalent, yet
the worldly bit that makes \(A\) true is distinct from the one that
makes \(B\) true. As Restall (1996a) points out, this seems to follow
if one also endorses the Disjunction Thesis: something is a truthmaker
for (*\(A\) or \(B\)*) iff it is a truthmaker for \(A\) or a
truthmaker for \(B\). Independently of the Disjunction Thesis, a
truthmaker for one necessary truth may plausibly not be a truthmaker
for all others: "There are infinitely many prime numbers"
should have as truthmaker something to do with numbers. "Anyone
who eats and drinks at least eats" seems to not have anything to
do with numbers. See MacBride (2013 [2020], the SEP entry on
 truthmakers, section 1)
 for some accounts of truthmaking where necessarily equivalent
propositions need not have the same truthmaker.



Some occurrences are impossible, but may still have explanatory and
even causal consequences. It may be impossible to prove a given
mathematical conjecture: the conjecture might have a counterexample
that nobody has yet noticed. Still, the fact that the conjecture is
not proved may have various effects: if you have bet that you will
prove it by a certain time you will lose the bet; if your tenure case
in the mathematics department turns on whether you managed to prove
the conjecture, you will miss out on tenure. Bernstein (2016) gives a
theory of omissions of impossible outcomes, and argues that they play
an explanatory and causal role.



Hoffmann-Kolss (2015) argues that a number of classifications of
properties needed for metaphysics are hyperintensional. The
intrinsic/extrinsic distinction is one example, discussed above.
Hoffman-Kolss also argues the qualitative/non-qualitative distinction
is hyperintensional (the distinction between properties that do not
require a relationship to a particular and those that do--e.g.,
*being square* vs. *being John's neighbor*: see
also Hoffman-Kolss 2019). The dispositional/categorical distinction is
arguably hyperintensional (Hoffmann-Kolss 2015: 341-342, 346).
(One philosophically controversial example may illustrate: on some
theories, there is a necessarily-existing God that is necessarily the
only omnipotent being. Still, we might think *being omnipotent*
is dispositional, concerning the powers of that being, while *being
God* is categorical (2015: 346).) The distinction between
perfectly natural properties that best carve at the joints, and the
others, is, too (2015: 345). Hoffman-Kolss's example here is to
contrast the (supposedly) perfectly natural property of being an
electron with the infinitely disjunctive identity property of
*being \(e\_1\)* or *being \(e\_2\)* or... and so on
for all possible electrons \(e\). If electronhood is essential to the
objects that have it, something is an electron if and only if it is
one of \(e\_1\), \(e\_2\), etc., but Hoffman-Kolss argues this latter
condition is not a perfectly natural one.



Any item on this list might be controversial. You might think that
counterfactual conditionals, for example, communicate more about how
we explain or predict the world than anything about the world as it
is. Or that what explains what is more a matter of the representations
we make of the world, than of the world itself. One could try to
explain away grounding's putative hyperintensionality in terms
of our cognitive structures, as Thompson (2016) suggests. Thompson and
Byrne (2019) offer a general "conceptualist" alternative
to taking any of the cases we discussed as requiring
non-representational phenomena to be characterized in hyperintensional
terms: an explanation of our use of hyperintensional language to
capture these phenomena comes from an account of what it is to
describe and explain in these areas (2019: 157). Still, if a need to
recognize hyperintensional distinctions flows from more than facts
about how we represent the world, hyperintensionality will be of
interest well beyond philosophy of mind and language.


## 2. Is Hyperintensionality Real?



While
 Section 4
 considers some difficulties for specific hyperintensional theories,
here we discuss general objections to the very idea that we need a
theory 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 for
hyperintensional expressions in the dominant truth-conditional
framework. This style of argument will be discussed in
 SS2.1.



Another kind of argument advanced by defenders of SPWS such as
Stalnaker (1984) and Lewis (1982, 1986), has it that the *prima
facie* hyperintensionality of knowledge, belief, or content is
misleading, and that SPWS can still do the job once misleading
intuitions have been explained away. This style of argument will be
discussed in
 SS2.2.


### 2.1 Hyperintensional Truth Conditions



One way to modify SPWS to give truth conditions for sentences is to
resort to sets including both possible and impossible worlds (ways
things could *not* have been). Sentences that agree in truth
value in very possible world can still be assigned distinct
truth-conditions, since those sentences might be true in different
impossible worlds. (See
 SS4.3).
 Another way is to use *structured meanings*, where sentences
with the same possible-worlds truth conditions can have different
meanings because of the way they are built up. (See
 SS4.5).
 An objection to these approaches can be generalized to any approach
that tries to make room for hyperintensionality in a truth-conditional
framework.



The objection, presented in Cresswell (2002) against structured
propositions and suggested by Stalnaker (1996) against impossible
worlds, 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 by
specifying that \({\sim}A\) is true when \(A\) is false, and not
otherwise. 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 just
be the truth conditions for \(A\). (We may even want to derive this
second claim from plausible principles about truth, falsehood, and
necessity.) Now any theory which allows for distinct necessarily
equivalent 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 \(C\) is
false, and \({\sim}D\) will be the proposition that is true just in
case \(D\) is false: so \({\sim}C\) and \({\sim}D\) will be the same
proposition. But then there will be a unique proposition with the
truth 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 a *reductio* of
the hypothesis that there can be a hyperintensional distinction
between \(C\) and \(D\): if they are necessarily equivalent, they are
identical.



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



Cresswell (1975) was one of the pioneers of structured approaches to
sentence meaning to handle hyperintensionality. So it is significant
that he thinks this objection refutes a contention at the heart of
orthodox structured meaning projects, though Cresswell thinks that we
may still be interested in semantic structures for reasons other than
how they might make a difference to the truth-conditions of complex
embeddings.



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


### 2.2 The Indispensability of Hyperintensions



Talk of belief, desire, and meaning is *prima facie*
hyperintensional, as discussed above. But perhaps this appearance
could be resisted. One strategy is to employ use/mention distinctions
in novel ways. Instead of hyperintensional operations on propositions
or other meanings, we can treat apparently hyperintensional language
as expressing properties of or relations between sentences or other
pieces of language. If we do this systematically, a theory can do away
with merely intensional distinctions.



Quine (1953) suggests that modal operators, when we need them at all,
are best interpreted as predicates of sentences. Quine (1956) and
Davidson (1968) suggest an approach to propositional attitude
ascriptions as implicitly talking about connections between people and
sentences: "John believes that it is raining" is
regimented as a two-place predicate holding between John and the
sentence "It is raining". For Davidson, this in effect
asserts that John stands in a certain relation to a sentence that
stands in the same-saying relation to the sentence "It is
raining". This supposedly accommodates how non-English speakers
can believe that it is raining. Reconstruing mental attitude reports
as reporting relations to sentences has fallen out of favor in
contemporary philosophy, for reasons largely orthogonal to
intensionality or hyperintensionality: see Propositional Attitude
Ascriptions, section 7. Still, with enough ingenuity techniques like
Quine's and Davidson's could be extended to apparently
hyperintensional pieces of language.



Some philosophers who reject extensionalism nevertheless offer
techniques for avoiding treating beliefs and desires as
hyperintensional. Stalnaker (1984) tries to rescue the SPWS analysis
of propositional content by explaining away hyperintensional
intuitions via a sort of metalinguistic approach: when we fail to see
that a proposition is necessarily true, or that two propositions are
necessarily equivalent, we are actually failing to see what
propositions 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 A
expresses* \(P\), relating the sentence to what it expresses.
Ignorance of necessary truths or equivalences is ignorance about
*P\*.*



Stalnaker maintains that whenever one fails to know or believe a necessary truth, \(A\),
one actually fails to fully grasp the sentence \(A\)'s meaning:
one doesn't realize that \(A\) expresses the total set of
worlds. Stalnaker points at the occasional distance between the
syntactic structure of the relevant \(A\), which can be quite complex,
and the expressed proposition--in SPWS, an unstructured entity (a
set):



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



For Stalnaker's story to work in general, all cases of ignorance
of necessary truths or equivalences must be instances of meaning
ignorance. This is difficult to believe. A mathematician who wonders
whether Goldbach's conjecture is true can hardly be described as
mistaken about what "Every even integer larger than two is the
sum of two primes" means. They know the meanings of all
constituents of the sentence--they have mastered the
elementary-school-math concepts *even integer*, *larger
than*, *two*, *sum*, *prime number* as much
as any of us. As competent speakers of English, they understand how
the syntax of the sentence makes its constituents conjure for it to
express the content it expresses. What they have doubts about is
whether that content is true. We sometimes seem to fully grasp the
contents of sentences whose status of necessary truths, or falsities,
we are ignorant of.



Lewis (1982) proposes a fragmentation strategy to explain away
apparent failures of logical omniscience, in particular, apparent
belief in inconsistencies which is not, however, belief in everything.
One's epistemic state may be split into different "frames
of 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 the
> railroad nearby ran roughly north-south; and that the two were roughly
> parallel. [...] So each sentence in an inconsistent triple was
> true according to my beliefs, but not everything was true according to
> my beliefs. [...] My system of beliefs was broken into
> (overlapping) fragments. Different fragments came into action in
> different situations, and the whole system of beliefs never manifested
> itself all at once. [...] The inconsistent conjunction of all
> three did not belong to, was in no way implied by, and was not true
> according to, any one fragment. That is why it was not true according
> to my system of beliefs taken as a whole. (Lewis 1982: 436)
> 
> 
> 



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



The account, just like Stalnaker's, doesn't seem to
generalize. It seems strange to say that when agents don't
follow through the logical consequences of what they believe,
it's always because they haven't conjoined the premises
whereas, when they do, they suddenly come to believe all the
infinitely many consequences (Jago 2014). It's sometimes an easy
logical 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 apparent
hyperintensionality of belief, these attempts to not extend in any
straightforward way to all *prima facie* cases of
hyperintensionality. Stalnaker, at least, seems committed to a project
of explaining away many apparent cases of hyperintensionality,
committed as he is to a SPWS account of linguistic meaning and mental
content (Stalnaker 1984). So far as we can tell, Stalnaker is willing
to take this project piece-meal.



Lewis seems more hospitable to hyperintensional approaches, provided
they can be ultimately explained in terms of possible worlds and
set-theoretic constructions out of them. In Lewis (1986: 49-50),
he discusses a hyperintensional "trivially" operator that
would apply differently to different necessary truths. In a
posthumously published letter (Lewis 2004) he seems to acknowledge
that there was more than one distinct impossible fiction, and
presumably that "According to fiction \(f, A\)" was
hyperintensional (since when \(f\) is an impossible fiction, there
will be some impossible claim that is true according to it, but in
general not all impossible claims will be).



Philosophers suspicious of hyperintensionality in particular areas
should consider if this stems from features of the specific case or is
general. If the latter, they may wish to attempt a general account of
meaning 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 its
encompassing very diverse phenomena. We now investigate what (little)
we know about the general structure of hyperintensionality. The
initial characterization, recall, merely tells us that a concept or
operator is hyperintensional when it is more fine-grained than
intensional or standard modal concepts or operators, marking a
distinction invisible to the latter. As Jespersen and Duzi
reminded us, little is said on "how hyper" hyperintensions
are. The tag of "granularity problems" (Barwise 1997; Jago
2014; Bjerring & Schwarz 2017) labels a set of issues concerning
the right level of fine-grainedness. There are several related
problems here, because different hyperintensional concepts seem to
display different degrees of fine-grainedness. This can be understood
again in terms of substitution *salva veritate* for the
relevant 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 of
this or that notion shouldn't make it as fine-grained as the
syntax of the language one is working with--on pain of giving
away the very point of having a semantics for it. Propositions may be
more fine-grained than sets of possible worlds, but they had better
not be mappable 1:1 to the sentences expressing them. Otherwise, we
would 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 the
same as that of \(A \land A\), and that \(A \land B\) should have the
same content as \(B \land A\): "It's windy and rainy
today" seems to just say the same thing as "It's
rainy and windy today" (for further motivations for
coarse-graining, see Bjerring & Schwarz 2017). On the other hand,
a limited cognitive agent may fail to make even the simplest inference
starting from what she knows or believes: she may believe that \(A
\land B\), fail to perform Conjunction Elimination, and thus fail to
believe that \(A\). Psychologically very realistic accounts of
knowledge or belief seem to require extreme fine-grainedness, cutting
more finely than same-saying. Jespersen (2010) is one example of a
theory that has very fine-grained meanings available for some
constructions, while offering more coarse-grained meanings for some
traditional purposes where little or no hyperintensionality is
required.



One may hope for a *total* ordering of hyperintensional notions
by fine-grainedness, i.e., an ordering such that any two given ones
are comparable: given any \(X\) and \(Y\), either \(X\) is at least as
fine-grained as \(Y\), or vice versa. But one can show that the simple
setting is wrong. One can provide intuitive examples (inspired by
Hornischer 2017, which also includes a thorough discussion) of
incomparable hyperintensional operators: \(X\) and \(Y\) for which
there 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 the
four sentences:


* \(A =\) Clark Kent loves the Guggenheim Museum
* \(B =\) "Clark Kent loves the Guggenheim Museum" is
true
* \(C =\) Clark Kent is in New York
* \(D =\) Superman is in New York



Lois Lane knows that \(A\) (Clark Kent loves the Guggenheim Museum) if
and only if she knows that \(B\) ("Clark Kent loves the
Guggenheim Museum" is true) (surely Lois knows how truth
works!). On the other hand, she knows that \(C\) (Clark Kent is in New
York), but she doesn't know that \(D\) (Superman, too, is), for
she 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't
explain themselves), while it is true that \(B\) ("Clark Kent
loves the Guggenheim Museum" is true) because Clark Kent loves
the Guggenheim Museum (assuming truth supervenes on/is grounded in
facts). On the other hand, \(C\) (Clark Kent is in New York) because
Clark Kent loves the Guggenheim Museum if and only if \(D\) (Superman
is in New York) because Clark Kent loves the Guggenheim Museum, for
Superman is Clark Kent.



So *knows* and *because* turn out to be incomparable,
and, given that they are hyperintensional, there is no total ordering
for hyperintensional notions. This may sustain some skepticism
(ventilated, e.g., in Leitgeb 2019) on the possibility of a single
theory for all hyperintensional operators. On the other hand, it makes
sense to conjecture that these can be at least partially ordered. It
is then an issue worth investigating, whether concepts belonging to a
single sub-family, e.g., the non-representational ones, or the
representational semantic ones, can be internally totally ordered
(notice that *knows* is representational, while one may take
*because* as belonging to "ontic" explanation). Or,
the other way around, the identification of totally ordered subsets of
the partially ordered set of hyperintensional concepts may be a good
way 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. Hardly
any of them has been put forth as a systematic account of all
hyperintensional notions. However, we focus on frameworks that are
general enough to handle a broad range of phenomena (thus we do not
consider, e.g., hyperintensional epistemic logics that target only
certain forms of logical omniscience, like justification logic). We
briefly introduce the key ideas of each approach, and we discuss how
it handles hyperintensional phenomena such as those introduced in
 Section 1.
 We highlight relative merits and open issues.


### 4.1 Two-Dimensional Semantics



Two-dimensional semantic theories (see entry on
 two-dimensional semantics)
 build on standard possible worlds semantics to handle a range of
representational hyperintensional phenomena. Originating in
two-dimensional modal logic (Segerberg 1973; van Fraassen 1977; Davies
& Humberstone 1980), versions of the view have been presented by
Kaplan (1989), Jackson (1994), and Chalmers (1996). (Stalnaker [1978]
is a key pioneer of the approach, but he applied it to the pragmatics
of 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: which
thing someone refers to using "this" depends on what they
are demonstrating at the time. Kaplan (1989) proposed that these
expressions function as rigid designators on occasions of use: whether
the proposition expressed by a sentence including one is true at a
possible 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 at
a possible world iff it is sunny on 4 March 2020 in that world. Elvis
says "I was born in 1935": that is true at a possible
world iff *Elvis* was born in 1935 there.



On this view, "I am Elvis", said by Elvis, is true just in
case Elvis is Elvis, thus necessarily true. Ditto for "Yesterday
was 4 March 2020", said on 5 March 2020, which is true just in
case 4 March 2020 is 4 March 2020. Expressing those necessary truths
can, however, give contingent information to hearers who already knew
things are self-identical. To capture the difference between "I
am Elvis" and "Elvis is Elvis", when uttered by
Elvis as well as by someone else--say, Aretha--we would need
a story about the meaning of "I" that did not treat it as
a name. Kaplan took such meaning as a function from a context of
utterance to an object, selecting the speaker as the object.
Similarly, the meaning of "yesterday" is a function from
the day of utterance, selecting the day before. This function is
called the *character* of the expression. My knowing the
character (and facts about context) lets me take new information from
the utterances. I know "I" picks out the speaker and that
the speaker said "I am Elvis". If I trust him, I can work
out the speaker is Elvis. I might not have known *that* before,
although I knew that Elvis is Elvis.



Meanings, in this view, have two components: a *character* and
a *content*. 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 the
total set of worlds, since Elvis couldn't but be Elvis. The
character is a function from the context's speaker to the
content that that person is Elvis. So it is a function from Elvis to
the content that Elvis is Elvis, from Aretha to the content that
Aretha is Elvis, etc.



Traditionally, philosophers have found it useful to represent this
interaction of character and content with a two-dimensional matrix,
hence the expression "two dimensional semantics". Let us
illustrate with a small example. In \(W\_1\), Elvis is singing but
Aretha is not, and in \(W\_2\), Aretha is singing but Elvis is not, and
in \(W\_3\), neither is singing. Consider the sentence "I am
singing", with two possible speakers, Elvis and Aretha. We get
the following six scenarios, depending on which speaker is speaking
and 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 can
be associated with "I am singing", with different rows
representing the different contents produced by different contexts
(due to different speakers, in this case). The entire matrix
represents facts about the character of a sentence. These
two-dimensional matrices become more interesting when more complex
constructions are discussed, but the key ideas for current purposes
are more easily presentable without further using matrices.



Now some proponents of two-dimensional semantics extend this setting
beyond indexicals and demonstratives, e.g., to *proper names*
and *natural kind terms*. Take "Robin Hood". Its
content, suppose, is just Robin. But its character is a function from
contexts to such contents. In a world where our use of "Robin
Hood" derives in the right kind of way from Robin, the character
delivers our Robin as the content of "Robin Hood". In a
world where Richard leads a band of outlaws in Sherwood, uses
"Robin Hood" as an alias, is famous for robbing for the
rich to give to the poor, the character of "Robin Hood"
picks out *Richard*. 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 is *a priori* that": It is *a
priori* that Hesperus is Hesperus; it is not *a priori*
that Hesperus is Phosphorus. Although "Hesperus" and
"Phosphorus" have the same content (the planet Venus),
they can differ in character. There may be contingent *a
priori* cases: Elvis says "I am here now" in a context
where the assigned content is that Elvis Presley is in Graceland on
noon of 25 December 1975. "I am here now", some claim, can
be known *a priori* by Elvis, but he cannot know *a
priori* that Elvis is in Graceland on noon of 25 December
1975.



Propositional attitude contexts make for a salient case. The Sheriff
of Nottingham believes Robin Hood is a dangerous outlaw. He does not
believe that Robin of Locksley is a dangerous outlaw, although Robin
Hood is Locksley. We can explain this if we take "believes
that" as character-sensitive. "Robin Hood is a dangerous
outlaw" and "Robin of Locksley is a dangerous
outlaw" 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 they
are typically acquired, in the causal history leading back to their
first uses, or if we tie them to how the Sheriff sees the world. This
can be generalized to so-called intensional transitive verbs. The
Sheriff can be looking for Robin Hood but not looking for Robin of
Locksley if "looking for" is sensitive to character.



Two-dimensional approaches are somewhat controversial when they move
beyond indexicals and demonstratives. (See the papers in
Garcia-Carpintero and Macia 2006 for arguments for and
against two-dimensional semantics.) But they can account for the
apparent hyperintensionality of "it is *a priori*
that..." or "... believes that..."
without straying too far from a traditional possible-worlds framework.
Contents keep being sets of possible worlds. Qua functions from
contexts to contents, characters are familiar from standard treatments
of context-dependent expressions such as Kaplan's.



Two-dimensional semantics also has a nice story on the behavior of
indexicals 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 the
latter but not the former if I believe I am facing an Elvis
impersonator. It is hard to avoid the impression that the character of
expressions like "you" (or, "yesterday",
"that car", etc.) makes the difference in such cases.



Two-dimensional semantic theories cannot easily offer a general
account of hyperintensionality. Worldly hyperintensionality is
especially recalcitrant. When, e.g., talk of essence is employed, and
we say that Socrates is essentially human but is not essentially
(human and \(2 + 2 = 4)\), or not essentially (human and either bats
have wings or it is not the case that bats have wings), those two
alternatives to "human" both have the same two-dimensional
intension as "human". Any context will pick out the same
content for the predicates "is human", "is human and
such that \(2 + 2 = 4\)", and "is human and either bats
have wings or it is not the case that bats have wings".



The full range of non-trivial counterpossible conditionals are not
easily 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 the
circle, Hobbes would have proved an impressive mathematical
result" is false: Archimedes lived well before Hobbes and so
would have beaten him to it, and Hobbes did not otherwise prove any
impressive results. The necessary falsehoods "Hobbes squared the
circle" and "Archimedes squared the circle" seem to
deliver different truth values as antecedents in same-consequent
counterfactuals. Functions from contexts to contents taken as the
usual 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" could
pick out a contingent proposition that would be useful. Prospects are
not quite hopeless here: approaches that employ two-dimensionalism as
at least part of the picture about *indicative* conditionals
have been suggested by Weatherson (2001) and Nolan (2003). One might
hope for an extension to counterfactuals. Or perhaps some other
approach might work: recent work on counter-analytic conditionals done
by Locke (forthcoming) and Kocurek, Jerzak, and Rudolph (2020) is
promising.



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


### 4.2 Aboutness



Some theories of propositional content represent a number of
hyperintensional distinctions by combining SPWS with ways of splitting
the modal space. The main work in this area is Yablo's book
*Aboutness* (2014; and see Osorio-Kupferblum 2016 for a
critical discussion). Aboutness is



> 
> 
> the relation that meaningful items bear to whatever it is that they
> are *on* or *of* or that they *address* or
> *concern*, (Yablo 2014: 1)
> 
> 
> 



namely, their *subject matter*. The subject matter of a
sentence in context can be seen as a topic, an issue, or a question
the sentence may be taken, in that context, as answering to. When the
issue is *the number of stars*, the relevant question may be:
"How many stars are there?". As in inquisitive semantics
(Ciardelli, Groenendijk, & Roelofsen 2013), one can associate the
question with the set of its answers: There are no stars; There is one
star; There are two stars; etc. Answers are propositions in the SPWS
sense: sets of possible worlds. We now call these *thin*
propositions for the full propositional contents of sentences will
turn out to be richer than that. The question, thus, splits the
totality of worlds into sets (the relevant thin propositions). Two
worlds \(w\_1\) and \(w\_2\) end up in the same set just in case they
agree on the answer--in our example: when the number of stars is
the same at \(w\_1\) and \(w\_2\). When, as in the example, the question
has just one correct answer, the thin propositions at issue form a
partition of the set of worlds \(W\): a splitting into subsets such
that their union is all of \(W\), and such that each \(w\) in
\(W\) is in exactly one subset or cell. One cell has all the 0-star
worlds, one all the 1-star worlds, one all the 2-star worlds, and so
on.



While the idea is already in Lewis (1988) (see also Plebani and
Spolaore forthcoming), Yablo proposes a generalization due to some
questions' having more than one correct answer:
"Where's a good Italian restaurant in Amsterdam?". A
world \(w\_1\) 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 into
subsets whose union is \(W\), but which can overlap.



There's an intuitive mereology of issues, topics, or subject
matters: they should be capable of overlapping and of fusing into
wholes which inherit the proper features from the parts (Yablo 2014:
Section 3.2). The topic *philosophy* and the topic
*mathematics* overlap (the overlap including, presumably,
*logic*). *How things went in 1882* is included in the
larger *how things went in the nineteenth century*. A larger
subject matter induces smaller cells: \(w\_1\) and \(w\_2\) agree on
what happened in their whole nineteenth century only if they agree on
what happened in their 1882 to begin with.



Let \(|A|\), a subset of \(W\), be the thin proposition expressed by
\(A\). The *thick* proposition expressed by \(A\), [\(A\)], is
its thin proposition \(|A|\) together with \(A\)'s subject
matter, \(s(A)\). How does one get *the* subject matter of
\(A\)? Yablo suggests assigning a positive subject matter, the
division corresponding to "Why is \(A\) true?"; a negative
subject matter, the division corresponding to "Why is \(A\)
false?"; and to identify the overall \(s(A)\) with its positive
and negative subject matters taken together. Worlds agree on the
overall subject matter of \(A\) when either \(A\) is true, or \(A\) is
false, for the same reasons at them. Yablo calls the reasons for
\(A\)'s being true (false), \(A\)'s *truthmakers*
(*falsemakers*). He suggests not to read them in a
metaphysically loaded way, as chunks of reality whose obtaining
necessitates truth, and advocates a "semantic" conception
of 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 finely
as ordinary sets of worlds: when \(|A|\) differs from \(|B|\), [\(A\)]
will differ from [\(B\)]. Additionally, there will be \(A\)'s
and \(B\)'s expressing different thick propositions in spite of
being true at the same worlds, because they are about different
things: \(s(A)\) will differ from \(s(B)\) even if \(|A| = |B|\). For
example, "There is an even number of stars or there is
not" and "There are no non-self-identical Italian
restaurants" are both necessarily true, but they correspond to
distinct thick propositions because of their different subject
matters. The suggestion that adding subject matter to semantics is the
key to handling hyperintensionality is discussed in Gioulatou
(2016).



To get a precise answer on which hyperintensional distinctions the
framework can make, one needs a full-fledged truthmaker assignment to
all sentences of the target language. Yablo gives two "semantic
pictures", a *reductive* and a *recursive* one,
which pull in different directions. We take the recursive route, which
relies on van Fraassen (1969) and is found more plausible by some
(e.g., Hawke 2018; Fine 2020), and leave the details to a consultation
of 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 vice
versa. For \(A = B \land C\), what truthmakes \(A\) is the union of
what truthmakes \(B\) and what truthmakes \(C\); what falsemakes \(A\)
is what either falsemakes \(B\) or falsemakes \(C\). For \(A = B \lor
C\), 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 a
bachelor or not", "Either 44 is the sum of two primes or
not") will express distinct thick propositions insofar as they
are about different things (how things stand John-wise is a different
issue 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 can
capture the idea that "You see that Mary is eating an ice
cream" and "You see that Mary is eating an ice cream and
John is either eating chips or not" don't say the same, in
the face of the logical equivalence of \(A\) and \(A \land (B \lor
{\sim}B\)).



But even lacking a precise treatment for a predicative language, we
can see that distinct, logically atomic necessarily true or false
contents may be difficult to tell apart. Take "Mike is
Mike", \(m = m\), and "Mike is Jack the Ripper", \(m
= j\), which differ in informativeness. Given that Mike is Jack and
the necessity of identity, \(|m = m| = |m = j| = W\); but also the
subject matters will coincide: what makes both claims true is a fact
about Mike, Mike's being Mike. Nothing makes either false.
"Hobbes squared the circle", \(Sh\), and "Daniel
Nolan squared the circle", \(Sd\), concern different issues: one
is about Hobbes' mathematical (non)achievements, the other is
about Daniel's. But \(|Sh| = |Sd| = \emptyset\), and
there's no way to get truth/falsemakers as sets of possible
worlds that will make \(s(Sh)\) and \(s(Sd)\) differ: no matter how
one splits and groups possible worlds into sets, there will be no
worlds available where Hobbes squares the circle or where Nolan does
to begin with.



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


### 4.3 Extended Worlds Semantics



If a theorist starts from a possible worlds framework, and comes under
pressure to recognize hyperintensional distinctions, one response is
to add impossible worlds (Priest 1992; Zalta 1997; Kiourti 2010; Jago
2015; Berto & Jago 2019) taken as ways things could not be, or
worlds where some truth of logic, mathematics, or metaphysics fails.
We will be brief on this topic here, as one can consult the entry on
 impossible worlds
 for more details.



In impossible worlds semantics, different impossibilities can be
distinguished by being associated with different sets of worlds (only
impossible worlds), and different necessary truths can also be
distinguished by different sets of worlds (all possible worlds plus
some of the impossible worlds). Each distinct necessary truth will
correspond to a distinct set containing all possible worlds and some
impossible worlds; and each distinct impossible proposition will
correspond to a set of worlds that contains no possible worlds, but
which may contain impossible worlds (Rantala 1982).



Kripke (1965) already introduced "non-normal worlds" for
the 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 of
both possible and impossible worlds, so as to not treat every
impossible belief the same as every other, and to allow someone to
believe some necessary truths without believing them all. (See also
Hintikka 1975). Since then, impossible worlds have been used to
theorize about many of the hyperintensional phenomena discussed in
this entry.



A closely related way to extend possible worlds semantics is to
introduce incomplete and inconsistent partial specifications of how
things could be, and could not be. So-called *situation
semantics* (Barwise & Perry 1983) employs these more general
objects to handle hyperintensional constructions such as
specifications of beliefs or information contents, as well as
applications to avoid paradoxes that can arise from employing possible
worlds. A version of situation semantics known as "truthmaker
semantics" has recently been defended as being superior to
standard world semantics (see Fine 2012, 2016, 2017; Hornischer 2020;
Moltmann 2020, forthcoming).



With impossible worlds added to a theory, necessarily equivalent
expressions no longer have to have the same semantic values. Apart
from their widespread use in the semantics for logical systems, work
has continued on using impossible worlds for giving theories of
psychological attitude ascriptions and of claims about meaning: see
Jago (2014) for one recent treatment of a number of issues involving
mental content and reasoning. Impossible worlds also help in theories
of other kinds of representations: for use in theories of impossible
fictions, see Priest (1997) and Badura and Berto (2019).
Epistemologists are starting to find applications for impossible
worlds in theories of knowledge: see, for example, Melchior
(forthcoming). And logicians are using them to model non-omniscient
hyperintensional agents by combining impossible worlds semantics with
dynamic epistemic logic (Bjerring and Skipper (2019); Solaki, Berto,
& Smets forthcoming).



Given the limitations of Lewis-Stalnaker theories of counterfactual
conditionals when dealing with impossible antecedents (see
 SS1.1.2),
 it is natural to extend a theory of counterfactual conditionals with
impossible worlds: see Routley (1989), Mares (1997), Nolan (1997),
Brogaard and Salerno (2013). Similar theories can be offered of
indicative conditionals: see Nolan (2016b).



Impossible worlds have also been employed to give theories of a range
of hyperintensional phenomena in metaphysics. One early application
was in the theory of *de re* necessity, in offering a theory of
how it could be necessarily true that an artifact could be made of
slightly different material, but not too different, without succumbing
to Chisolm's paradox (Salmon 1984). Impossible worlds can be
used to give a hyperintensional theory of properties as sets of
possible and impossible instances (Nolan 2013: 8). They can be used to
offer a theory of metaphysical explanation (Kment 2014), or in a
theory of essence (Brogaard & Salerno 2013: 646-8). They can
play a central role in a theory of omissions of impossibilities
(Bernstein 2016).



The literature to date has only scratched the surface of potential
applications of impossible words in metaphysical theorizing. A range
of concerns about postulating impossible worlds are discussed in the
entry on
 impossible worlds, section 6,
 Nolan (2013), and Berto and Jago (2019), as well as the concerns
discussed in
 SS2
 and
 SS3
 above.


### 4.4 Relevance/Relevant Logic



One should look at the SEP entry on
 relevance logic
 (also called "relevant logic") for an introduction to
this research program (see the seminal Anderson & Belnap 1975;
Anderson, Belnap, & Dunn 1992; and, for a comprehensive
introduction, Dunn & Restall 2002). We briefly discuss its
connections to hyperintensionality here. Its origins lay in the aim to
develop logical systems free from the "paradoxes of the material
and strict conditional" discussed above. Another early
motivation was to construct a conditional that could represent logical
deducibility within the logical language itself.



In this program, an agreed upon necessary (though, in general, not
sufficient) condition for a conditional to be logically valid is that
there be some connection between antecedent and consequent.
Historically, this has often been phrased via a requirement called the
*Variable Sharing Property* (VSP): \(A \rightarrow B\) is valid
only when \(A\) and \(B\) share some sentential variable or parameter
(cf. Anderson & Belnap 1975: 32-3). VSP delivers
hyperintensional distinctions: conditionals with (classically)
impossible antecedents and necessary consequents are not all trivially
valid or true. The infamous *ex falso quodlibet*, \((A \land
\neg A) \rightarrow B\), for instance, fails (it's not the case
that anything is implied by a contradiction); whereas \((A \land \neg
A) \rightarrow A\) holds in most mainstream relevant systems
(plausibly, a conjunction relevantly implies its conjuncts).



A key feature of relevance logics, made apparent by their most
developed semantics, the so-called Routley-Meyer frames (Routley &
Meyer 1973), is that hyperintensional distinctions are not achieved by
superimposing on an essentially classical or normal modal logic a
filter that screens out irrelevancies. Relevance logics are weaker
than normal modal logics with SPWS or Kripke frames because their
semantics embeds points of evaluation for formulas different from
classical (maximally consistent) possible worlds, which are accessible
(via a *ternary* accessibility relation, different from the
binary accessibility of Kripke frames) in the evaluation of
conditionals.



Working out the intuitive interpretation of such points, of the
ternary relation, and indeed of the Routley-Mayer semantics as a
whole, has been a difficult task in the history of this research
programme (see Copeland 1980; Read 1988; Restall 1996b; Mares 2004).
The consensus on the points nowadays seems to be that they be
understood as non-normal or logically impossible worlds (see Priest
2008: Ch, 10; Berto & Jago 2019: Ch. 6). See the SEP entry on
 impossible worlds
 for details. The impossible worlds of (the interpreted) Routley-Meyer
frames can be seen as situations which can, e.g., be locally
inconsistent (making true both \(A\) and \(\neg A\), for some \(A)\)
without thereby being trivial (making true everything), and partial or
locally incomplete ("taking no stance" on some \(A\) by
making true neither it nor its negation).



The conditional of most mainstream relevant systems is taken as
representing implication or (necessary) entailment, not a *ceteris
paribus* or counterfactual conditional. But some of the first
theories of counterpossible conditionals, addressing the limitations
of the Lewis/Stalnaker account of counterfactuals, were developed
against a relevance logic background: see Routley (1989); Mares and
Fuhrmann (1995), and Mares (1997).



Other applications of relevance logic to areas where hyperintensional
distinctions seem needed include applications to the theory of
confirmation in science (Goddard 1977; Sylvan & Nola 1991); the
logic of fiction (Routley 1979); and the theory of information (Mares
2004; Dunn 2015). Finally, a number of theories in deontic logic have
been 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 of
*worldly* hyperintensionality. In a series of papers, Michael
Dunn developed a theory of *relevant predication* intended to
illuminate a range of traditional metaphysical distinctions, including
the differences between intrinsic and extrinsic properties, essential
and accidental properties, and between essential properties and
properties necessarily had when an object exists (Dunn 1987, 1990a,
1990b). And as mentioned above, Restall (1996a) employed relevant
logic to offer a theory of truthmakers.



Overall, the tools provided by the relevance logic program offer
theoretical resources for constructing theories of hyperintensional
phenomena, whether representational or non-representational. To date,
many of these proposals have been piece-meal, with less of a general
story about how these techniques could be applied to handle
hyperintensional language across the board. Some general strategies
have been outlined for how a theory could be constructed that handles
all hyperintensional phenomena: one such program was Richard
Routley's "Ultralogic as Universal", first set out
in 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 of
propositions as sets of truth-supporting circumstances, structured
propositions accounts (King 1995) have become the main rivals to SPWS
views. Roughly: propositions are taken as structures, composed of the
entities which are the semantic values of the corresponding
sentences' syntactic constituents. Take "Robin loves
Batman". It's composed of a noun, "Robin", and
a verb phrase, "loves Batman", itself made of a verb and a
noun. Let the semantic values of the lexical items be: [Robin],
[Batman], [*loving*]. Then the corresponding proposition can be
taken as an ordered triple \(\langle\)[Robin], [*loving*],
[Batman]\(\rangle\). Ordering is important: Batman may not return the
feeling, and "Batman loves Robin" expresses a different
proposition: \(\langle\)[Batman], [*loving*],
[Robin]\(\rangle\). (A proposition is not *just* an ordered
list, but we leave aside the difficult metaphysical issue of what ties
its constituents together: see Gaskin 2008.)



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



This delivers hyperintensional distinctions, e.g., "John is
either a bachelor or not" and "Either 44 is the sum of two
primes or not" will express different propositions: only one
will include [John], i.e., John, as a constituent. Necessarily
true/false atomic sentences will also be differentiated, even when
their syntactic structure is the same ("Hobbes squared the
circle" and "Daniel Nolan squared the circle":
[Daniel Nolan] isn't [Hobbes]; "\(3 + 3 = 6\)" and
"\(2 + 2 = 4\)": [3] isn't [2]). More challenging
cases for the view come with the necessary *a posteriori*.
"Mike is Mike" and "Mike is Jack the Ripper"
express the same proposition, because [Mike] = [Jack the Ripper]. Same
for "All woodchucks are woodchucks", "All woodchucks
are whistlepigs", as [woodchuck] = [whistlepig]. This will
create issues in intentional contexts:



> 
> 
> Suppose Tama is familiar with both woodchucks and whistlepigs, but
> isn't sure that they are the same kind of critter. He's
> noticed the similarities, though, and so he has his suspicions.
> Suppose further that Tama knows he is allergic to whistlepigs, and
> knows that he has just been bitten by a woodchuck. In this scenario,
> ["Tama fears that all woodchucks are whistlepigs"] is
> likely true, while ["Tama fears that all woodchucks are
> woodchucks"] is almost certainly false. (Ripley 2012: 9)
> 
> 
> 



Soames (1987) proposes a pragmatic-metalinguistic strategy similar to
the one tried by Stalnaker (1984), discussed in
 SS2.2:
 one should distinguish attitudes towards sentences and towards
propositions. Soames comes up with examples showing that the former
can be an unreliable guide to the latter, e.g., one can sometimes
believe a proposition, but only assent to one of two distinct
sentences expressing it. You *do* know that Mike is Jack the
Ripper by knowing that Mike is Mike, for it's the same
proposition. But you will assent to, or assert, only the sentence that
reports your knowledge in the trivial clothing. Then, the ancients
believed that Hesperus is Phosphorus, Tama fears that all woodchucks
are woodchucks, and so on. If you think otherwise, it's because
you confuse belief reports with reports of linguistic practices.
Ripley (2012) argues that it's difficult to account for
phenomena concerning such attitude reports by throwing them in the box
of pragmatics and conversational implicatures: failures of
substitutivity *salva veritate* behave in systematics ways in
contexts of iterated embeddings, and this calls for a systematic,
compositional treatment.



Salmon also resorts to a distinction between "semantically
encoded" and "pragmatically imparted information"
(Salmon 1986: 78) for the "Mike is Mike" vs. "Mike
is Jack the Ripper" cases. He introduces different
representational guises under which one can have attitudes to the
proposition expressed by different sentences. For direct reference to
work, we need [Mike] to be [Jack the Ripper], but we can allow
Mike-guises to differ from Jack-the-Ripper-guises, and claim that
(*de dicto*) intentional attitudes are relations to
propositions mediated by guises: Lois Lane loves Kal-El under the
guise of Superman, not under the guise of Clark Kent. Guises
don't belong in semantics but in pragmatics, and don't
determine denotations.



Salmon's view can in principle account for compositional
phenomena involving embeddings. It has been criticized because of
this: Forbes (1987) argues that guises look too much like Fregean
senses in disguise. Branquinho (1990) argues that the correspondence
may not reduce Salmon's account to a relabelled Fregean
view.



Guises are supposed to be mobilized within attitude reports. Beyond
attitude reports, as argued by Ripley (2012), Jago (2014), there are
hyperintensional phenomena involving indicative and counterfactual
conditionals structuralism doesn't seem to handle. As Jago
(2014: 76-7) has it, we can be at the 1972 Ziggy Stardust Tour
and 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 be
Bowie.
(4)
If Bowie weren't Bowie, then Bowie wouldn't be
Bowie.


All of (1)-(4) have metaphysically impossible antecedents. (1)
and (3) are plausibly false (unless one is a vacuist: see Williamson
2007), but (2) and (4) are trivial truths. (A nonvacuist conditional
semantics using impossible worlds delivers just these results: see
Berto 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 only
counterfactuals, don't involve intentionality, guises will offer
little help if they are not supposed to come into play here. However,
there are treatments of conditionality along the lines of epistemic
modals (Kratzer 1986; Lycan 2001), and the Russellian structuralist
may appeal to one of them, or come up with a new one.



Could a broadly *Fregean* account of structured propositions do
better? In this view, we don't take the constituents of
structured propositions as denotations (think objects and properties),
rather as Fregean senses. Much now depends on how these are
understood. If they are taken a la Carnap, as functions from
the possible worlds of SPWS to extensions, there is little hope to
account for "Mike is Mike" vs. "Mike is Jack the
Ripper" cases. Now [Mike] and [Jack the Ripper] are functions
from possible worlds to individuals; but unless one clashes with the
Kripkean criticisms of descriptivism on names (Kripke 1980), they had
better be constant functions: given that they both actually pick out
Mike, they should output Mike at all possible worlds (at which Mike
exists). But then [Mike] and [Jack the Ripper] will be defined on the
same set of things and will give the same output for any input.



One may enlarge or change the set of circumstances that provides the
inputs, admitting metaphysically impossible situations where Mike is
distinct from Jack the Ripper, and have the two as the outputs of
[Mike] and [Jack the Ripper] there, respectively. But then, as
remarked by Ripley (2012) and Jago (2014) again, it is not so much
structuralism that does the job of giving the required
hyperintensional distinctions, but rather the fact that we are not
working (only) with standard (metaphysically) possible worlds
anymore.



A recent interesting proposal to improve on the Fregean situation
comes from Hawke (2018), combining a theory of subject matter with a
structuralist account of propositions resorting to
Fregean-senses-lookalikes. Subject matter sensitivity allows Hawke to
make hyperintensional distinction unavailable to standard Fregean
structuralism.



One logically sophisticated, neo-Fregean structured account, offering
a systematic analysis of a range of hyperintensional phenomena, is the
*Transparent Intensional Logic* approach. The view, pioneered
by Tichy (1968, 1971, 1988), treats the meanings of expressions
as given by structural procedures, called *constructions*,
built out of entities that are somewhat like Fregean senses. In
particular, different names and different predicates, even if they
necessarily co-designate, may be associated with different senses, so
the meanings of "Robin Hood" and "Robin of
Locksley", 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 hyperintensional
contexts straightforwardly. In particular, it manages to give a
powerful compositional account where other approaches have to resort
to pragmatics. The approach is less popular than it should be in
contemporary semantics, possibly due its resorting to a technical
apparatus of typed lambda-calculus. For hyperintensional applications
and a comprehensive discussion of Transparent Intensional Logic, see
Duzi, Jespersen, and Materna (2010) and Duzi
and Jespersen (2015).



We mention one final, recent proposal in the Fregean camp, by Skipper
and Bjerring (2020). This pivots on another Fregean notion, namely
that of *cognitive equipollence*: \(P\) and \(Q\) are
cognitively equipollent when one cannot rationally regard either as
true and the other as false (see Schellenberg 2012). This gives a
cognitive or epistemic notion of sentential meaning which, the authors
claim, is more fine-grained than SPWS intensions, but less than
extra-fine-grained extended (e.g., impossible) worlds semantics.


### 4.6 Algebraic and Object Theory Approaches



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



By defining in a direct way how the meanings of different parts of
language go together to yield sentence meanings, these approaches are
not under pressure to identify the meanings of sentences true in all
the same possible circumstances, or even sentences which are logically
equivalent. So it is easy to see how a general algebraic approach can
make room for hyperintensional differences between parts of language.
Even if sentence meanings are necessarily equivalent, if they relate
differently to a "believes that" operator they do not have
to be substitutable in "John believes that \(A\)"
contexts. Bealer (1979) touts this flexibility as an advantage in
dealing with puzzles like the semantics of belief attributions, as
does Menzel (1993).



Some algebraic approaches do not take as much advantage of this
flexibility as one might expect with respect to hyperintensionality
puzzles. Keenan and Faltz (1984)'s algebraic approach identifies
necessary equivalents. Bacon (2018) adopts a Boolean account in which
the propositions associated with logical equivalents in predicate
logic 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 merely
necessarily 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 discriminate
between logical equivalents: if we were to think that \(A\) explains
why \({\sim}{\sim}A\), or capture the belief content of the confused
man who believes something of the form \((A \land B)\) but does not
believe \({\sim}({\sim}A \lor {\sim}B)\). So it is not the algebraic
approach *per se* that yields solutions to particular puzzles
about hyperintensionality, but the algebraic approach combined with
theoretical choices about the structure of the algebra.



Algebraic approaches are largely silent on the question of what
meanings are, and how meanings come to stand in the relations they do.
(Or, if every structure of relations between meanings can be found
somewhere in Platonic heaven, how we come to be associated with our
particular structure rather than others.) Of course, theories which
say more about the nature of meanings (e.g., that they are
set-theoretic constructions from worlds, or they are ordered sets of
objects and properties) *also* face the challenge of explaining
how our concrete practices explain which structures of meanings are
associated with our speech and writing. So it may be that these
concerns are general, and algebraic semantics can employ the same
kinds 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., be
the suitably distinct inputs so that "John believes all
triangles have internal angles that add up to 180deg" and
"John believes all trilaterals have internal angles that add up
to 180deg" express different propositions. Zalta says a little
more about the identity conditions of his
hyperintensionally-distinguished properties: according to Zalta,
properties are identical if and only if they are *encoded* by
the same objects.



Algebraic accounts of meaning may have little to say about worldly
hyperintensionality. There is nothing in the general motivations of
algebraic theories that would rule out the hypothesis that we need
hyperintensional language to best capture the non-representational
aspects of reality, but nor does there appear to be anything in the
general motivations that would indicate why we would, or how that
might manifest in our language. This is not really an objection to
algebraic approaches, however. One might think that a theory of
meaning *should* be silent on substantive questions about what
the world is like. We do not want our theory of meaning to tell us
which physics or chemical theory to adopt. Likewise, it might be good
for a theory of meaning to be neutral about how to understand
essences, or intrinsicness of properties, or causation. Of course, a
theory of meaning may not be able to be completely neutral about
metaphysical questions--it should have something to say about
what meanings are, whether there are abstract representations, and so
on--but perhaps algebraic theories not taking a stand on worldly
hyperintensionality is an attractive feature rather than a
drawback.