## 1. The Focus of the Debate



The debate between possibilists and actualists is at root
*ontological*. It is not--fundamentally, at least--a
debate about meaning, or the proper linguistic primitives of our modal
discourse, or the model theory of certain formal languages, or the
permissibility of certain inferences. It is a disagreement over
*what there is*, about the kinds of things that reality
includes. Characterizing the nature of the debate precisely, however,
is challenging. As noted in the preceding introduction, the debate
centers around the question of whether, in addition to such things as
you and me, reality includes *possibilia*, such as
Bergoglio's merely possible children, the children he does not
*actually* have but *could have* had if only things had
gone rather differently. Clearly, if there *are* such things in
some sense, they differ from us rather dramatically: a merely possible
child is not *actually* anyone's child and indeed, we are
inclined to say, does not *actually* exist at all and, hence,
is not actually conscious but only could have been, does not actually
have a body but only could have had one, and so on. As we will see in
more detail in the following section, in many historical and more
contemporary discussions, this alleged difference is characterized in
terms of distinct *ways*, or *modes*, of being, and it
will be useful, for now, to continue to frame the debate in these
terms. Specifically, on this *bi-modal* conception of the
debate, on the one hand, there is the more substantial and robust mode
of *actuality* (or, often, *existence*) that you and I
enjoy. And, on the other hand, there is the more rarefied mode
(sometimes called *subsistence*) exhibited by things that fail
to be
 actual.[4]
 While (as we will see) there has been some disagreement over the
location of abstract objects like the natural numbers in this scheme,
what uniquely distinguishes *possibilia* on the bi-modal
conception is that, for the possibilist, they are
*contingently* non-actual:



**Poss**:
There are *possibilia*, that is, things that are not actual
but could have been.


It is clarifying to represent matters more formally. Accordingly,
where \(\sfA!\) is the *actuality* predicate, we have:



**Poss**\(\_{\sfA!}\):
\(\exists \sfx(\neg \sfA!\sfx \land \Diamond \sfA!\sfx)\)


As noted in the above introduction, in its simplest form, actualism is
just the denial of possibilism: there are no mere *possibilia*.
However, the actualist's denial is meant to be stronger: that
possibilism is false is not a mere historical accident; rather, for
the actualist, not only are there not in fact any *possibilia*,
there *couldn't* be any:



**Act**:
There could not have been any *possibilia*, that is, any
things that are not actual but could have been


or, again, more formally:



**Act**\(\_{\sfA!}\):
 \(\neg\Diamond\exists \sfx(\neg \sfA!\sfx \land \Diamond
\sfA!\sfx)\)


For the actualist, then, the possibilist's purported mode of
contingent non-actuality, or *mere possibility*, is empty:
necessarily, anything that *could have been* actual already
*is* actual. Otherwise put, necessarily, there are no
contingently non-actual
 things.[5]



Actualism is quite clearly the preferred common sense position here:
on a first hearing, the idea that the Pope's unborn children
dwell in some shadowy corner of reality is to most ears ludicrous on
its face--and defending against this common sense intuition
remains arguably the central challenge facing the possibilist. But the
motivations for possibilism are surprisingly strong. We have already
noted what is perhaps the strongest of these: possibilism yields a
straightforward, unified semantics for our modal discourse. By
appealing to *possibilia*, the possibilist can provide
satisfying truth conditions for otherwise semantically problematic
modal statements like
 (2)
 along the lines of those for
 (1)
 that *ground* their intuitive truth in the modal properties of
individuals. As we might also put it: possibilism provides
 *truthmakers*
 for such statements as
 (2)
 that actualism is apparently unable to supply. But a powerful second
motivation is that possibilism falls out as a *consequence* of
the most natural quantified modal logic. In particular, in that logic,
 (4)
 is an immediate logical consequence of
 (2).
 (We will spell out these motivations further in the following two
sections.) These motivations in turn present a ***twofold
challenge for the actualist***:


1. to provide a systematic and philosophically satisfying account of
truth conditions for the likes of
 (2);
 and
2. to develop a robust quantified modal logic that invalidates such
inferences as the one from
 (2)
 to
 (4).



What makes the possibilism-actualism debate so interesting is that,
while actualism is the overwhelming metaphysical choice of most
philosophers, there is no easy or obvious way for the actualist to
meet these challenges. We will explore a number of attempts in this
essay.

 
## 2. The Origins and Nature of *Possibilia*


### 2.1 Possibilism and the Bi-Modal Conception of Being



To fully appreciate what is at issue in the possibilism-actualism
debate, as well as its framing in those terms, it is important to see
its origins--more generally, the origins of the bi-modal
conception of being--in the ancient, and vexing, philosophical
problem of
 non-existent objects.
 Inklings of the bi-modal conception arguably trace back to the dawn
of western philosophy in the goddess's enigmatic warning to
 Parmenides
 not to be deceived by the "unmanageable" idea "that
things that are not
 *are*".[6]
 It finds clearer expression in Seneca's description of the
Stoics, for whom being included both physical objects and
"incorporeals"
(asomata) that "have a
derivative kind of reality" (de Harven 2015: 406):



> 
> 
> The Stoics want to place above this [the existent] yet another, more
> primary genus.... Some Stoics consider "something"
> the first genus, and I shall add the reason why they do. In nature,
> they say, some things exist, some do not exist. But nature includes
> even those which do not exist--things which enter the mind, such
> as centaurs, giants, and whatever else falsely formed by thought takes
> on some image despite lacking substance. (Seneca, *Letters*
> 58:13-15, quoted in Long & Sedley 1987: 162; see also Caston
> 1999.)
> 
> 
> 



The Stoics' motivation for their bifurcation of
"nature" was clearly to explain the
 *intentionality*
 of our thought and discourse, our apparent ability to think and talk
as coherently about the creatures of mythology and fiction as about
ordinary physical existents. Later variations broadened the
Stoics' realm of incorporeal intentional objects to include
*possibilia*. In the early medieval period, the Islamic
Mutazilite theologians distinguished between *thing*
(*shay*') and *existent* (*mawjud*)) to
comport with two passages of the Qu'ran (16:40, 36:82) suggesting
that God *commands* non-existent things into existence
(Wisnovski 2003: 147). Avicenna (1926: 54-56) under the influence of
the Mutazilates, was more explicit still that these non-existent
things are *possibilia*:



> 
> 
> It is necessary with respect to everything that came into existence
> that before it came into existence, it was in itself possibly
> existent. For if it had not been possibly existent in itself, it never
> would exist at all. Moreover, the possibility of its existence does
> not consist in the fact that an agent could produce it or that an
> agent has power over it. Indeed, an agent would scarcely have power
> over it, if the thing itself were not possible in
>  itself.[7]
> 
> 
> 



Similar ideas were espoused by a number of other prominent medieval
philosophers including
 Giles of Rome,
 Henry of Ghent,
 John Duns Scotus,
 William of Ockham, and
 Francisco Suarez.[8]



Although the grounds of possibility and intentionality were prominent
themes in the modern
 period,[9]
 the idea of explaining them in terms of any sort of bifurcation of
being largely receded until the early nineteenth century beginning,
notably, with the remarkable work of
 Bernard Bolzano.
 Specifically, in his work on the ground of possibility, Bolzano
developed a sophisticated bi-modal account of being on which
everything there *is* divides into those things
(*Dinge*) with *Wirklichkeit*, usually rendered
"actuality" by Bolzano's
 translators,[10]
 and those with *Bestand*, typically rendered
"subsistence". Importantly, for Bolzano as well as for
many of his successors, to be actual (*wirklich*) is to be part
of the causal order and, hence, typically at any rate, to occupy a
position in space and time--in a word (as commonly understood),
to be
 *concrete*.[11]
 Subsistence, by contrast, is the mode of being shared by non-concrete
objects, which in particular for Bolzano included both
*possibilia* and *abstracta* like numbers and
propositions. Unlike *abstracta*, however, Bolzano's
*possibilia* are only *contingently* subsistent,
*contingently* non-concrete, and, hence, are capable of
*Wirklichkeit*:



> 
> 
> [I]n addition to those things that have actuality
> (*Wirklichkeit*)...there are others that have mere
> possibility (*blosse Moglichkeit*), as well as those
> that could never make the transition to actuality, e.g., propositions
> and truths as such (*an sich*). (Bolzano 1837: SS483, pp.
> 184-5)
> 
> 
> 



Motivated chiefly to provide a ground for intentionality, several
decades later,
 Alexius Meinong
 (1904b [1960], 1907) famously postulated a rich class of non-existent
objects to explain our apparent ability to conceive and talk about,
not only creatures of myth and fiction, but impossible objects like
the round square. Roughly, in Meinong's theory, for any class of
"ordinary" properties (*konstitutorische
Bestimmungen*) there is an intentional object
(*Gegenstand*) having *exactly* those
 properties.[12]
 Presumably, then, although he did not broach the issue of possible
objects
 directly,[13]
 among these objects would be those having (perhaps among others) the
property *being a possible child of Bergoglio*. However,
importantly, although he broadly accepted Bolzano's bi-modal
division of being into concrete and subsistent
 objects,[14]
 Meinong ascribed no variety of being whatsoever to his intentional
objects, not even the subsistence (*Bestand*) enjoyed by
*abstracta* like mathematical objects and
 propositions.[15]
 Although strongly influenced by Meinong, the early Russell (1903)
spurned the idea that intentional objects are being-less but not the
objects themselves, save for *impossibilia* like the round
square. Rather, he simply moved them all into the subsistent realm
alongside the objects of mathematics, reserving the "world of
existence" for "actual objects" in the
spatio-temporal causal order (*ibid*., pp.
 449-50):[16]



> 
> 
> *Being* is that which belongs to...every possible object
> of thought....Numbers, the Homeric gods, relations, chimeras and
> four-dimensional spaces all have being, for if they were not entities
> of a kind, we could make no propositions about them. Thus being is a
> general attribute of everything, and to mention anything is to show
> that it is....*Existence*, on the contrary, is the
> prerogative of some only amongst beings....[T]his distinction
> [between being and existence] is essential, if we are ever to deny the
> existence of anything. For what does not exist must be something, or
> it would be meaningless to deny its existence; and hence we need the
> concept of being, as that which belongs even to the non-existent.
> 
> 
> 



Russell's thoroughgoing modal skepticism led him to exclude
*possibilia* from his non-existent
 objects.[17]
 Nonetheless, were he to have admitted them into his ontology, the
subsistent realm would be their natural place in his bifurcated
ontology. Quine (1948: 22), in the voice of his fictional possibilist
metaphysician Wyman--and undoubtedly influenced at the time by
recent work of C. I. Lewis (1943) and Rudolf Carnap
 (1947)[18]--made
 exactly this move in a justly famous and widely-cited paper that
played a pivotal role in both framing the philosophical issues and
fixing modern terminology:



> 
> 
> Pegasus...has his being as an unactualized possible. When we say
> of Pegasus that there is no such thing, we are saying, more precisely,
> that Pegasus does not have the special attribute of actuality. Saying
> that Pegasus is not actual is on a par, logically, with saying that
> the Parthenon is not red; in either case we are saying something about
> an entity whose being is unquestioned. (1948: 22)
> 
> 
> 



To be a mere *possibile*, then, according to Wyman, is to be
*unactualized*, i.e., it is, unqualifiedly, to *be* but
to fail to exemplify "the special attribute of actuality";
it is to *subsist* rather than to *exist* (Quine 1948:
 23).[19]
 But perhaps equally important for fixing the nature of the modern
possibilism-actualism debate was Quine's explicit break from his
nineteenth and early twentieth century predecessors on the ontological
status of *abstracta*; speaking now in his own voice, he
says:



> 
> 
> If Pegasus existed he would indeed be in space and time, but only
> because the word "Pegasus" has spatio-temporal
> connotations, and not because "exists" has spatio-temporal
> connotations. If spatio-temporal reference is lacking when we affirm
> the existence of the cube root of 27, this is simply because a cube
> root is not a spatio-temporal kind of thing, and not because we are
> being ambiguous in our use of "exist". (1948: 23)
> 
> 
> 



For Quine, that is, their necessary non-concreteness notwithstanding,
abstract objects exist no less robustly than we and, hence, in the
context of the possibilism-actualism debate, are fully
 *actual*.[20]
 A particular advantage of this view of *abstracta* for the
debate is that, assuming that there could be no *necessarily*
non-actual objects, *possibilia* are the only things lacking
actuality in the possibilist's universe and, hence, actualism
can take a particularly common and familiar form:



**Act**\*:
Necessarily, everything is actual.


That is, more formally put once again:



**Act**\(\astAbang\):
 \(\Box\forall \sfx\,\sfA!\sfx\)


But the most important consequence of this wholesale shift in the
ontological status of *abstracta* is that it opened the door to
perhaps the most prominent form of contemporary actualism--dubbed
(rather tendentiously) *ersatz modal realism* by David Lewis
(1986: SS3.1)--on which modal phenomena are understood in
terms of *abstracta* of various sorts and, hence, in terms of
actually existing things only, as per
 **Act**\*.[21]
 (Ersatz modal realism is discussed in more detail in
 SS4.2
 and
 SS4.4
 below and in SS2.2 of the *Encyclopedia*'s entry on
 possible worlds.)



One final matter requires attention. Quine's choice of Pegasus
as his paradigmatic *possibile* in the above quote highlights
the fact that even relatively modern discussions often conflate the
two motivations for non-existent objects and, consequently, conflate
fictional objects and *possibilia*, and this leads to an
important confusion about the nature of *possibilia* that is
important to avoid. It is particularly evident in another well-known
Quinean passage. Returning to his own voice, in an attempt to show
that possibilism is ultimately incoherent, Quine asks a series of
rhetorical questions, beginning with the following:



> 
> 
> Take, for instance, the [merely] possible fat man in that doorway;
> and, again, the [merely] possible bald man in that doorway. Are they
> the same possible man, or two possible men? How do we decide? How many
> possible men are there in that doorway? (1948: 23)
> 
> 
> 



That is, on this characterization of possibilism, a merely possible
*F* is something that (contingently) fails to be actual but,
nonetheless, like a Meinongian intentional object, is
*actually* an *F*--a merely possible man in that
doorway has the property of being in the given doorway; hence the
purportedly unanswerable questions about whether or not he is
identical with the indefinitely many other merely possible but
somewhat differently described men who can also be said to occupy the
same space. However, as Linsky and Zalta (1994: 445) emphasize, a
merely possible *F* needn't (and indeed typically
won't) be an *F*. Rather, a merely possible *F* is
(typically) something that is *not in fact* an *F* but
rather only *could be* an
 *F*.[22]
 In particular, a merely possible bald man in that doorway is neither
bald, nor a man, nor in that doorway--indeed, it has the
complements of all those properties. Rather, it is only something that
*could* have those properties.



With that problem corrected, this Quinean framing has by and large
become the dominant conception of the possibilism-actualism
distinction in the contemporary
 literature.[23]


### 2.2 Possibilism Without Bi-Modalism



We have characterized actualism as, first and foremost, the denial of
possibilism, defined as the thesis
 (**Poss**)
 that there are things that contingently fail to be actual. Following
the historical precedents just detailed, actuality has been depicted
as the more robust of two purported modes of being. However, work by
Linsky and Zalta (1994, 1996) and Williamson (1998, 2013) casts doubt
upon the viability of the possibilism-actualism distinction under this
bi-modal conception. Williamson questions its coherence: what,
exactly, is the nature of the "robustness" that allegedly
distinguishes actuality from the merely possible; as Williamson (2013:
23ff) puts it: "being actual had better be actually doing
something harder than just being.... But what is that harder
thing...?" Convinced there is no cogent answer to the
question, Williamson proposes scotching the possibilism-actualism
distinction entirely in favor of an allegedly much clearer distinction
between *necessitism* and *contingentism*, that is,
between the thesis that, necessarily, all things exist necessarily
(\(\Box\forall \sfx \Box\exists \sfy\,\sfy=\sfx\)) and its denial.
(This distinction will be discussed in greater detail
 below.)[24]



Linsky and Zalta (1994: SS4) do not so much question the coherence
of the possibilism-actualism distinction (under the bi-modal
conception) as dissolve
 it.[25]
 Specifically, they show that, for all that the thesis
 (**Act**\*)
 that, necessarily, everything is actual tells us, there is nothing to
prevent avowed possibilists like
 themselves[26]
 from simply rejecting the idea that Bergoglio's merely possible
children "have a [mode] of being that is less than the
full-fledged existence" that we enjoy, and insisting instead
that they too are fully actual and, hence, exist as robustly as we
 do.[27]
 It's just that we, by sheer happenstance, are *concrete*
and they are not; we, that is, happen to exist in the spatio-temporal
causal order and they do not. But things might just as well have been
the other way 'round: existence-wise, we are all on an
ontological par. There are thus no *possibilia* in the sense of
 **Poss**
 at all; necessarily, everything is actual, as per
 **Act**\*.
 On this telling, then, Linsky and Zalta and their ilk all turn out to
be fully-fledged actualists, their ontological commitments
notwithstanding.



Clearly, however, even if one is skeptical of the bi-modal conception,
the core of the intended debate remains: whether or not to countenance
the likes of Bergoglio's possible children. In response to those
who do, actualists define themselves simply as those who do not. The
historical trajectory sketched above explains why the debate has often
come to be framed in terms of distinct modes of
being--*actuality* and *mere possibility*. But this
bi-modal framing is inessential: when the intended debate is kept in
the foreground, "actuality", for the actualist, is best
understood simply as a placeholder for whatever it is that allegedly
distinguishes the likes of us (and abstract objects such as the
numbers) from the likes of Bergoglio's merely possible children.
Linsky and Zalta (1994: SS4) and Williamson (2013: SS1.2)
themselves, in fact, just re-introduce what is essentially
Bolzano's characterization of *possibilia*: they are
*contingently non-concrete*. On this characterization,
possibilism can take a form that clarifies the debate without any
mention of modes of being and, hence, avoids the above critiques:



**PossC**:
There are *possibilia*, that is, things that are not
concrete but could have been


or, more formally, where \(\sfC!\) is *concreteness*:



**Poss**\(\_{\sfC!}\):
\(\exists \sfx(\neg \sfC!\sfx \land \Diamond \sfC!\sfx)\)


Accordingly, actualism becomes:



**ActC**:
There could not have been any *possibilia*, that is, any
things that are not concrete but could have been


or, more formally:



**Act**\(\_{\sfC!}\):
\(\neg\Diamond\exists \sfx(\neg \sfC!\sfx \land \Diamond
\sfC!\sfx)\)


So on this framing, for the possibilist, what allegedly distinguishes
the likes of us (and the likes of the natural numbers) from
*possibilia* in the sense of **PossC**
is: *not* being contingently non-concrete. Taking this, then,
to be what "actuality" signifies, with a bit of
propositional logic, we have:



**A!Def**:
\(\sfA!\tau \eqdf \sfC!\tau \lor \Box \neg \sfC!\tau,\) for any
term \(\tau\)


That is, on this framing, to be actual is to be either concrete or
necessarily non-concrete; or, more simply put, it is to be either
concrete or abstract. It is a straightforward exercise to show that,
under this definition of actuality, **PossC**
is equivalent to the original definition
 **Poss**
 and that **ActC** is equivalent to the
principle
 **Act**\*
 that, necessarily, everything is
 actual.[28]



Of course, possibilists who are more inclined toward the bi-modal
conception will still prefer the original framing of the debate in
terms of a primitive notion of actuality as per **Poss**
and **Act**. But even the most committed bi-modalist will
agree that contingent non-concreteness is at least necessarily
coextensive with mere possibility and, hence, its complement with
actuality. So, for those skeptical of the original framing, nothing
essential to the debate is lost if it is simply understood in terms of
concreteness as per **PossC** and
**ActC**.



**Note on David Lewis**. The influential and highly
original late twentieth century philosopher
 David Lewis
 also rejected bi-modalism and famously defended a view that is often
characterized as a variety of possibilism. In fact, Lewis's
possibilism is orthogonal to the classical possibilism-actualism
debate under discussion in this entry. See the supplemental document
 Classical Possibilism and Lewisian Possibilism
 for details.

 
## 3. Possibilism and Possible World Semantics



Possibilism would almost surely not be taken as seriously as it is
were it not for the dramatic development of possible world semantics
for modal logic in the second half of the twentieth century. For it
not only enables the possibilist to formulate truth modal conditions
with particular clarity and cogency, it yields a natural and elegant
quantified modal logic, known as SQML, in which possibilism's
fundamental metaphysical principles fall out as logical truths. In
order to appreciate the cogency of possibilism, therefore, it is
important to understand basic possible world semantics.

 
### 3.1 Basic Possible World Semantics



Possible world semantics is built upon
 Tarski's
 (1936, 1944) epochal theory of truth in the first half of the
twentieth century. Tarski's theory provided a rigorous account
of the fundamental semantic connections between the languages of
classical logic and non-linguistic reality that determine the truth
conditions for the sentences of those languages. By generalizing
Tarski's theory to modal
 languages,[29]
 possible world semantics promised an equally rigorous account of the
semantic connections between those languages and *modal*
reality, and thereby an equally rigorous account of modal truth
conditions. It is illuminating therefore to start with an account of
Tarskian semantics.


#### 3.1.1 Tarskian Interpretations



Given a standard first-order language \(\scrL\) with the truth
functional operators \(\neg,\,\to,\) a distinguished identity
predicate \(=,\) and the universal quantifier \(\forall,\) a
*Tarskian interpretation* \(\calI\) *for* \(\scrL\)
specifies a nonempty set \(D\)--the *universe* of
\(\calI\)--for the quantifiers of \(\scrL\) to range over and
assigns appropriate semantic values to the terms (i.e., the
(individual) constants and variables) and predicates of \(\scrL\).
Specifically, to each term \(\tau\) of \(\scrL,\) \(\calI\) assigns a
referent \(\tau^\calI \in D\) and, to each *n*-place predicate
\(\pi\) of \(\scrL,\) \(\calI\) assigns a set \(\pi^\calI \subseteq
D^n\) of *n*-tuples of members of \(D,\) often referred to as
the *extension* of \(\pi\) in
 \(\calI\).[30]
 The extension \(=^\calI\) assigned to the identity predicate, of
course, is always stipulated to be the "real" identity
relation for \(D,\) i.e., the set \(\lbrace\langle a,a\rangle:a \in
D\rbrace\). Those assignments to the terms and predicates of
\(\scrL,\) in turn, completely determine the truth values of all the
formulas of \(\scrL\) by means of a familiar set of recursive clauses.
To facilitate the quantificational clause, let
\(\calI[\frac{\nu}{a}]\) be the interpretation that assigns the
individual \(a\) to the variable \(\nu\) and is otherwise exactly like
\(\calI\). This apparatus then delivers a
 *compositional*
 theory of meaning for \(\scrL,\) that is, an account on which the
meaning (in this case, the truth value) of a complex formula is
determined by its grammatical structure and the meanings of its
semantically significant parts and hence, ultimately, in the Tarskian
case, by the semantic values assigned to the terms and predicates of
the language:


* An atomic formula \(\pi\tau\_1\ldots \tau\_n\) is *true
in* \(\calI\)--*true*\(^\calI\),
for short--if and
only if \(\langle \tau\_1^\calI,\ldots,\tau\_n^\calI\rangle \in
\pi^\calI.\)
* A negation \(\neg\psi\) is true\(\calI\) if and only if
\(\psi\) is not true\(\_{\calI}\).
* A conditional \(\psi \to \theta\) is true\(\calI\) iff
\(\theta\) is true\(\calI\) if \(\psi\) is.
* A universally quantified formula \(\forall\nu\psi\) is
true\(^\calI\) if and only if, for all individuals \(a \in D,\)
\(\psi\) is true\(^{\calI[\frac{\nu}{a}]}\).



Clauses for the other standard truth-functional operators and the
existential quantifier under their usual definitions follow
straightaway from these clauses. In particular, where



\(\exists\)**Def**:
\(\exists\nu\psi \eqdf \neg\forall\nu\neg\psi\)


it follows that:


* An existentially quantified sentence
\(\exists\nu\psi\) is true\(^\calI\) if and only if, for some
individual \(a \in D,\) \(\psi\) is
true\(^{\calI[\frac{\nu}{a}]}\).



A set \(\Sigma\) of formulas of \(\scrL\) is said to be
*satisfiable* if there is an interpretation for \(\scrL\) in
which every member of \(\Sigma\) is true. A formula \(\varphi\) is
*valid*, or a *logical truth*--written
\(\vDash\varphi\)--if it is true in every interpretation for
\(\scrL\).



The above definitions yield a *logic*, in one standard sense: a
class of formal languages for which we've provided a model
theoretic semantics that determines a rigorous notion of logical
 truth.[31]
 And the logic they define is *classical (first-order) predicate
logic*.


#### 3.1.2 Truth *Simpliciter*



Strictly speaking, truth in an interpretation is a purely mathematical
relation between the formulas of a formal language and rigorously
defined mathematical objects of a certain type. However, in practice,
most formal languages are *applied* languages and this will
enable us to define an objective notion of truth *simpliciter*
for classical predicate logic. More specifically, an applied formal
language \(\scrL\) is designed to clearly and unambiguously formalize
a range of discourse about some real world domain (e.g., the stars and
planets, the US electorate on 6 November 2020, the natural numbers,
etc)--call this the *intended domain* of \(\scrL\). Hence,
each constant of \(\scrL\) symbolizes a name in the given discourse
and each predicate of \(\scrL\) symbolizes a predicate of the
 discourse.[32]
 An interpretation \(\calI\) for \(\scrL\) will be *intended*,
then, just in case its universe \(D\) comprises exactly the
individuals in the intended domain of \(\scrL\) and \(\calI\) assigns
to the constants and predicates of \(\scrL\) the actual semantic
values of the names and natural language predicates they are meant to
symbolize. The compositional truth condition of a sentence \(\varphi\)
in an intended interpretation thus traces \(\varphi\)'s
truth value down to the basic
atomic facts on which it ultimately depends or, as we might put it,
the atomic facts on which its truth value is *grounded*. Thus,
a sentence \(\varphi\) of an applied language \(\scrL\) will be
*true* just in case it is true\(^\calI,\) for some intended
interpretation \(\calI\) for \(\scrL\).


#### 3.1.3 SQML Interpretations



Intuitively, a Tarskian interpretation of an applied non-modal
language represents a possible world, a way in which the properties
and relations expressed by the predicates of the language might be
exemplified by the things in the universe of the interpretation. The
idea underlying possible world semantics is simply to interpret a
modal language \(\scrL\_\Box\) by bringing a collection of Tarskian
interpretations together to represent a modal space of many possible
worlds in a single interpretation of
 \(\scrL\_\Box\).[33]
 So let \(\scrL\_\Box\) be the result of adding the modal operator
\(\Box\) to some standard first-order language \(\scrL\). As with a
Tarskian interpretation of \(\scrL,\) an *SQML interpretation
\(\calM\) for* \(\scrL\_\Box\) specifies a nonempty set \(D\) to
serve as its universe. Also as in Tarskian semantics, \(\calM\)
assigns each term \(\tau\) of \(\scrL\_\Box\) a semantic value
\(\tau^\calM \in D\). Additionally, however, \(\calM\) specifies a
nonempty set \(W\)--these are typically called the set of
"possible worlds" of \(\calM\) but can be any nonempty
set. One member \(w^\ast\) of \(W\) is designated as the "actual
world" of \(\calM\). To give substance and structure to these
"worlds", \(\calM\) then assigns extensions to the
predicates of \(\scrL\_\Box\) *relative to* each
world--that is, for every *n*-place predicate \(\pi\) of
\(\scrL\) and each world \(w \in W,\) \(\calM\) assigns a set
\(\pi\_w^\calM \subseteq D^n\) of *n*-tuples of members of
\(D,\) the *extension of* \(\pi\) *at*
 \(w\);[34]
 in particular, the extension \(=\_w^\calM\) of the identity predicate
at all worlds \(w\) is stipulated to be \(\lbrace\langle a,a\rangle:a
\in D\rbrace\). In this way \(\calM\) represents the different ways
that the properties and relations expressed by those predicates can
change (or not) from world to world.



Given an SQML interpretation \(\calM,\) then, the Tarskian truth
conditions above are generalized by relativizing them to worlds as
follows: for any possible world \(w\) of \(\calM\) (the *world of
evaluation*),


* An atomic formula \(\pi\tau\_1\ldots\tau\_n\) is *true
at \(w\) in* \(\calM\)--*true*\(\_w^\calM\),
for short--if and
only if \(\langle \tau\_1^\calM,\ldots,\tau\_n^\calM\rangle \in
\pi\_w^\calM.\)
* A negation \(\neg\psi\) is
 true\(\_w^\calM\) if and only if \(\psi\) is
not true\(\_w^\calM\).
* A conditional \(\psi \to \theta\) is true\(\_w^\calM\)
iff \(\theta\) is true\(\_w^\calM\)
if \(\psi\) is.
* A universally quantified formula \(\forall\nu\psi\) is true\(\_w^\calM\)
if and only if, for all
individuals \(a \in D,\) \(\psi\) is true\(\_w^{\calM[\frac{\nu}{a}]}\).



Putting the clause for universally quantified formulas together with
the clause for negated formulas and the definition
 **[?]Def**
 of the existential quantifier, we have


* An existentially quantified formula
\(\exists\nu\psi\) is true\(\_w^\calM\) if
and only if, for some individual \(a \in D,\) \(\psi\) is true\(\_w^{\calM[\frac{\nu}{a}]}\).



And to these, of course, is added the critical modal case that
explicitly interprets the operator \(\Box\) to be a quantifier over
possible worlds:


* A necessitation \(\Box\psi\) is true\(\_w^\calM\)
if and only if, for all worlds
\(u \in W,\) \(\psi\) is true\(\_u^\calM\).



The possibility operator \(\Diamond\) is defined as usual in terms of
\(\Box\):



\(\Diamond\)**Def**:
\(\Diamond\psi \eqdf \neg\Box\neg\psi\)


That is, intuitively, to say that a statement is possible is just to
say that its negation isn't necessary. The structural similarity
between \(\Diamond\textbf{Def}\) and [?]**Def**
should be unsurprising given that, semantically, the necessity
operator \(\Box\) is literally a universal quantifier over the set of
worlds. Accordingly, it follows from \(\Diamond\textbf{Def}\) and the
semantic clause for necessitations that


* \(\Diamond\psi\) is true\(\_w^\calM\)
if and only if, for some world
\(u \in W,\) \(\psi\) is true\(\_u^{\calM}\).



We say that a formula \(\varphi\) of \(\scrL\_\Box\) is *true
in* an SQML interpretation \(\calM\) for \(\scrL\_\Box\) if it is
true\(\_{w^\ast}^\calM\), i.e., true in
\(\calM\) at the "actual world" \(w^\ast\) of \(\calM\).
Satisfiability and logical truth are defined exactly as they are
 above
 for classical predicate logic, albeit relative to modal languages
\(\scrL\_\Box\) and the preceding notion of truth in an SQML
interpretation; in particular, a formula \(\varphi\) of \(\scrL\_\Box\)
is logically true if it is true in every SQML interpretation. The
logic so defined is, of course, SQML.


#### 3.1.4 Modal Truth *Simpliciter*



The definition of truth *simpliciter* above for formulas of an
applied non-modal language \(\scrL\) stems from the fact that a
Tarskian interpretation \(\calI\) for \(\scrL\) can take the things in
(some relevant chunk of) the actual world and represent how they
actually exemplify the properties and relations expressed by the
predicates of \(\scrL\). But if we can represent the actual world (or
a relevant chunk of it) by means of an intended Tarskian
interpretation, then there is no reason that we can't represent
a merely *possible* world in which those same things exist but
have different properties and stand in different relations and, hence,
define objective notions of truth at a world and of truth
*simpliciter* for the formulas of a modal language as well,
that is, notions of truth that are not simply relative to formal,
mathematical interpretations but, rather, correspond to objective
modal reality. So let \(\scrL\_\Box\) be an applied modal language
whose individual constants and predicates represent those in some
ordinary range of modal discourse and let \(D\) be its intended
domain. Say that \(\calM\) is an *intended* interpretation of
\(\scrL\_\Box\) if


1. its set \(W\) of "possible worlds" is in fact a
sufficiently comprehensive set of honest-to-goodness possible
 worlds,[35]
2. its designated "actual world" is in fact the actual
world,
3. its universe is the intended domain \(D\) of \(\scrL\_\Box,\)
and
4. the referents assigned to the constants of \(\scrL\_\Box\) are the
ones they actually refer to and the extensions assigned to the
predicates of \(\scrL\_\Box\) at each world \(w \in W\) are the ones
they in fact have at \(w\).



Then, where \(\calM\) is an intended interpretation of \(\scrL\_\Box,\)
we can say that a formula \(\varphi\) of \(\scrL\_\Box\) is *true
at* a world \(w\)--*true*\(\_w\)--just
in case \(w \in W\)
and \(\varphi\) is true\(\_w^\calM\), and
that \(\varphi\) is *true* just in case it is true\(\_{w^\ast}\).



### 3.2 *SQML*: A Deductive System for SQML



Ideally, a logic \(\mathfrak L\) has a sound and complete proof
theory, that is, an accompanying deductive system whose
theorems--the formulas provable in the system--are exactly
the logical truths of \(\mathfrak
 L\).[36]
 There is such a system for SQML--for convenience, we'll
refer to it by setting "SQML" in italics: *SQML*.
(We will follow this convention for logics generally.)



"SQML" is an acronym for "simplest quantified modal
logic", and it is so-called because it is a straightforward
amalgam of the most popular and semantically least complicated
propositional
 modal logic
 S5 and
 classical first-order logic
 (with
 identity)--FOL,
 for
 short.[37]
 The deductive system *SQML*, accordingly, is an amalgam of the
corresponding deductive systems *S5* and
 *FOL*.[38]
 S5 builds on the foundation of classical propositional
logic--PL, for short--whose deductive system *PL*
takes every instance of the following schemas as its axioms and Modus
Ponens as its inference rule:


* *Propositional Axiom Schemas*

**P1**:
\(\varphi \to (\psi \to \varphi)\)
**P2**:
\((\varphi \to (\psi \to \theta)) \to ((\varphi \to \psi) \to
(\varphi \to \theta))\)
**P3**:
\((\varphi \to \psi) \to ((\varphi \to \neg \psi) \to \neg
\varphi)\)
* *Rule of Inference*

**MP**:
\(\psi\) follows from \(\varphi\) and \(\varphi \to \psi\)



On top of this foundation the system *S5* adds the rule of
Necessitation and every instance of the following three schemas:


* *Modal Axiom Schemas*

**K**:
\(\Box( \varphi \to \psi) \to (\Box \varphi \to \Box \psi)\)
**T**:
\(\Box \varphi \to \varphi\)
**5**:
\(\Diamond \varphi \to \Box\Diamond \varphi\)
* *Rule of Inference*

**Nec**:
\(\Box\psi\) follows from \(\psi\)



**K** is a fundamental principle common to all the modal
logics we will survey here: if a conditional is necessary, then its
consequent is necessary if its antecedent is. **T**
expresses that necessity implies truth, and **5**
expresses that what is possible is not a mere matter of happenstance;
no possibility could have turned out to be impossible; or again: what
is possible in the actual world is possible in every world. The basic
*normal* deductive system *K* is the result of adding
(every instance of) **K** and the rule of Necessitation
to *PL*; the system *T* is the result of adding all
instances of **T** to *K*; and the system
*S5* is the result of adding all instances of
**5** to
 *T*.[39]



Two important principles (that is, every instance of them) can be
proved in
 *S5*:[40]



**4**:
\(\Box \varphi \to \Box\Box \varphi \)
**B**:
\(\varphi \to \Box\Diamond\varphi\)


**4** says of necessities what **5** says
about possibilities: necessity is not a matter of happenstance; the
necessary truths of our world are necessary in every world.
**B** says that anything that is in fact the case had to
have been possible; the truths in our world are, at the least,
possibilities in every other world.



Schemas **T**, **5**, **4**,
and **B** all have common equivalent forms it is useful
to note:



**T***:
\(\varphi \to \Diamond\varphi\)
**5***:
\(\Diamond\Box\varphi \to \Box\varphi\)
**4***:
\(\Diamond\Diamond \varphi \to \Diamond\varphi\)
**B***:
\(\Diamond\Box\varphi \to \varphi\)


We obtain the full deductive system *SQML* by adding the
quantificational and identity axioms of *FOL* and the rule of
Generalization to *S5*:


* *Quantificational Axiom Schemas*

**Q1**:
\(\forall \nu ( \varphi \to \psi)\to(\forall \nu \varphi \to
\forall \nu \psi)\)
**Q2**:
\(\forall \nu \varphi \to \varphi^\nu\_\tau,\) where \(\tau\) is a
term that is substitutable for \(\nu\) in \(\varphi\) and
\(\varphi^\nu\_\tau\) is the result of replacing every free occurrence
of \(\nu\) in \(\varphi\) with an occurrence of
 \(\tau\)[41]
**Q3**:
\( \varphi \to \forall \nu \varphi ,\) if there are no free
occurrences of \( \nu \) in \( \varphi \).
* *Identity Axiom Schemas*

**Id1**:
\(\nu = \nu\)
**Id2**:
\(\nu = \nu'\to (\varphi \to \varphi'),\) where \(\nu'\) is
substitutable for \(\nu\) in \(\varphi\) and \(\varphi'\) is the
result of replacing some or all free occurrences of \(\nu\) in
\(\varphi\) with occurrences of \(\nu'\)
* *Rule of Inference*

**Gen**:
\(\forall\nu\psi\) follows from \(\psi\)



*Proofs and Theorems*
  

The notions of proof and theoremhood are defined as usual; we define
them generally as they will apply to each of the deductive systems
discussed in this entry. Specifically, for any deductive system
*S*, a *proof* in *S* is a finite sequence of
formulas of the language \(\scrL\) of *S*--\(\scrL\_\Box,\)
in the case of *SQML*--such that each formula is either an
axiom of *S* or follows from preceding formulas in the sequence
by a rule of inference of *S*. A proof is a proof *of*
the formula occurring last in the sequence. A formula \(\varphi\) of
\(\scrL\) is a *theorem of S*--\(\vdash\_{S}
\varphi\)--if there is a proof of it in *S*. And if
\(\Gamma\) is a set of formulas of \(\scrL,\) then \(\varphi\) is a
*theorem of* \(\Gamma\) (*in S*)--\(\Gamma
\vdash\_{S} \varphi\)--if, for some finite subset
\(\{\psi\_1,\ldots,\psi\_n\}\) of \(\Gamma,\) \(\psi\_1 \to (\ldots \to
(\psi\_n \to \varphi)\ldots)\) is a theorem of *S*.



As the role of an individual constant in a theorem of *SQML* is
essentially indistinguishable from that of a free variable, it is
useful to note the following:



**Metatheorem**: Let \(\psi\) be a formula of
\(\scrL\_\Box\) that contains an individual constant \(\kappa\) of
\(\scrL\_\Box\) and let \(\nu\) be a variable that doesn't occur
in \(\psi\). Then if \(\psi\) is a theorem of *SQML*, so is
 \(\forall\nu\psi^\kappa\_\nu.\)[42]



The metatheorem justifies the following derived rule of inference,
where \(\varphi,\) \(\kappa,\) and \(\nu\) are as indicated there:



**Gen**\*:
\(\forall\nu\psi^\kappa\_\nu\) follows from \(\psi.\)


That is, informally put, if a formula containing an individual
constant is a theorem, we can effectively generalize on it as if it
were a free variable.


### 3.3 Possibilism, Necessitism, and Logical Truth



In addition to the degree of clarity it brings to the issues, a
compelling reason for expressing the possibilism-actualism debate in
formal terms is how starkly it illustrates the inextricable link
between logic and metaphysics, in particular, the dramatic impact our
metaphysical choices have on quantified modal logic. Arguably the best
known illustration of this is seen in the validity of the inference
from
 (2)
 to
 (4)
 in SQML and, more generally, in the validity of the *Barcan
 Formula*:[43]



**BF**:
\(\Diamond\exists \nu \varphi \to \exists \nu
\Diamond\varphi\)


That is, informally, if there *could be* something satisfying
any given description \(\varphi,\) then *there is* something
that *could* satisfy that description, a thing that is
*possibly* \(\varphi.\) The validity of **BF** in
SQML rests on two facts:
 first,
 that in the model theory of SQML (as in all varieties of possible
world semantics), the possibility operator \(\Diamond\) is literally
an existential quantifier ranging over all possible worlds; and
 second,
 that, in evaluating an existentially quantified formula
\(\exists\nu\varphi\) at a possible world \(w,\) the initial
occurrence of \(\exists\nu\) in the formula ranges unrestrictedly over
all individuals. Hence, switching the order of adjacent occurrences of
\(\Diamond\) and \(\exists\nu\) in a formula will not alter its truth
value; to say that some world and some object are thus and so is to
say no more and no less than that some object and some world are thus
and
 so.[44]
 And this is what warrants, in particular, the inference from
 (2)
 to
 (4).
 Letting "\(\sfB\)" represent the predicate "is
Bergoglio's child", the logical form of
 (2)
 is \(\Diamond\exists \sfx\, \sfB\sfx\). Expressed semantically in
SQML: some world and some individual are such that, in that world,
that individual is Bergoglio's child. But that is to say no more
and no less than that some individual and some world are such that, in
that world, that individual is Bergoglio's child, \(\exists
\sfx\,\Diamond \sfB\sfx,\) i.e.,
 (4).



In its validation of **BF**, then, SQML underwrites in
general, and as a matter of logic, the possibilist's thesis that
*de dicto* modal truths like
 (2)
 asserting simply that there *could* be things that are thus
and so are *in fact* grounded in *de re* modal truths
about the modal properties of individuals, i.e., the properties they
have at some or all possible
 worlds.[45]
 If, as in the case of
 (2),
 that appears to commit us to things like merely possible human
beings, things that, for actualists anyway, intuitively do not in any
sense exist, things that in no sense *are*, so much the worse
for actualist intuitions; or so says the possibilist.



**BF** is not the only controversial logical truth of
SQML or even, perhaps, the most controversial one. Another is its
converse:



**CBF**:
\(\exists \nu \Diamond \varphi \to \Diamond\exists \nu
\varphi\)


That is, informally, if there is, in fact, something that
*could* satisfy a given description \(\varphi,\) then it is
possible that something satisfy that description. **CBF**
is valid for exactly the same reason that **BF** is: the
order of adjacent occurrences of \(\exists\) and \(\Diamond\) in a
formula does not alter its logical content.



To see why **CBF** is controversial, especially for
typical actualists, note that most all of us believe that there are
contingent beings, things that reality might just as well have lacked
as to have contained, things like you and me that simply might have
failed to be identical to anything:



**CB**:
\(\exists \sfx\, \Diamond \neg \exists \sfy\, \sfy=\sfx\)


However, **CBF** is incompatible with the existence of
contingent beings! For, as an instance of **CBF** we
have



**CBF**\*:
\(\exists \sfx\, \Diamond \neg \exists \sfy\, \sfy=\sfx \to
\Diamond \exists \sfx\, \neg\exists \sfy\, \sfy=\sfx\)


Hence, by **MP**, we can infer



:
\(\Diamond \exists \sfx\, \neg\exists \sfy\, \sfy=\sfx\)


which says that there could be something that is distinct from
everything (including, in particular, itself) and that, of course, is
logically impossible. Hence, **CB** is logically false in
SQML and so it is a logical truth of SQML that there are no contingent
beings, i.e., that, rather, everything is necessarily identical with
something:



**N**:
\(\forall \sfx\, \Box\exists \sfy\, \sfy=\sfx\)


Indeed, since logical truths in SQML are all necessary,
**N** is itself a necessary truth:



\(\Box\)**N**:
\(\Box\forall \sfx \Box\exists \sfy\, \sfy=\sfx\)


So SQML yields not only possibilism (given **BF** and
 (2)
 and its ilk) but *necessitism*, that is, the thesis that
everything there is and, indeed, everything there ever could be, is a
necessary
 being;[46]--you,
 me, the Eiffel Tower, Bergoglio's possible children, merely
possible members of exotic species that never in fact evolved, merely
possible suns that never formed, etc. Otherwise put: everything there
could possibly be already *is* and, moreover, could not have
failed to be.



Although it is a logical truth of SQML, necessitism is not
analytically entailed by possibilism. There is, in particular, nothing
in the idea of a mere *possibile* that demands its necessity,
nothing that rules out worlds from which it might be altogether
absent, worlds (in addition to those in which it is either concrete or
non-concrete) in which nothing is identical to it. This might lead one
to wonder whether SQML, by building necessitism into its logical
foundations, has (from a possibilist perspective) gotten the logical
cart before the philosophical horse. On reflection, however, it is
clear that necessitism is, not only a natural complement to
possibilism, but an essential component of it. For, as we've
seen, the central justification for possibilism is that it provides
truthmakers for modal propositions like
 (2);
 it grounds them in the modal properties of individuals. If the
purported truthmakers for those propositions could fail altogether to
be, if it *could be* that there are no such things, then, for
all we know, it might just as well be that there are *in fact*
no such things and, hence, that there are no truthmakers for
 (2)
 and its ilk after all; and with that, possibilism's central
philosophical justification collapses. Necessitism closes the door on
this prospect: both *possibilia* and actually existing things
alike are necessary
 beings.[47]



At first sight, of course, necessitism is a shocking philosophical
doctrine. For, most compellingly, the urgent intuition of our own
contingency--that at one time in the past we did not exist and
that at another in the not too distant future we will forever cease to
be--is a central element of our lived human experience. But by
necessitism, not only have we always existed and will so continue
forever into the future, we--no less than God and the number
17--could not have failed to be. The possibilist, however, will
respond that the worry here has badly conflated existence in the sense
of *being*, in the sense of mere identity with something, and
existence in the sense of *actuality*. More specifically, the
worry has confused two corresponding notions of contingency, viz.,
contingency as the possibility of absolute non-being, i.e.,



**Cont[?]**:
\(\textsf{Contingent}\_{\boldsymbol{\exists}}(\sfx) \eqdf
\Diamond\neg\exists \sfy\, \sfy=\sfx\)


and contingency as the possibility of non-actuality, i.e.,



**Cont**\(\_{\sfA!}\):
\(\textsf{Contingent}\_{\sfA!}(\sfx) \eqdf \Diamond\neg
\sfA!\sfx\)


As per SQML, says the possibilist, contingency[?]
is a logical
impossibility. However, they continue, contingency\(\sfA!\)
comports fully
with our lived experience and, in fact, is all that our experience
warrants: our sense of our own contingency is rooted in our possible,
indeed imminent, non-actuality and, hence, in the fact that, in the
not-too-distant future, we shall forever cease to be *actual*.
We shall thus forever cease to be conscious, to be embodied, to love
and be loved, etc.; nothing distinctive of our human existence will
survive. That something identical to each of us--a mere
*possibile*, a featureless point in logical space--remains
at worlds and times where we are non-actual offers no existential
solace. (See Williamson 2013: Ch. 1, esp section 3, 6, and 8 for
related reflections on necessitism.)



The actualist, of course, in denying possibilism, i.e., in denying
that there could be any contingently non-concrete objects, allows no
daylight between being, existence, and actuality: to be is to exist
and to exist to be actual. Hence, for the actualist,
contingency[?], the possibility of non-being, is the
only concept of contingency on the table. As contingency in this sense
is a logical impossibility according to SQML, the actualist clearly
needs an alternative quantified modal logic tuned to their
metaphysical sensibilities.



SQML thus throws
 the twofold challenge
 that possibilism presents to actualism into stark relief. The notion
of an intended SQML interpretation of an applied modal language, with
its single domain of actual and merely possible individuals and its
recursive account of truth at a world, vividly traces the semantic
dependence of complex propositions down to the individuals on which
their truth values are grounded. And the corresponding deductive
system *SQML* provides the possibilist with a clean, complete
framework in which to represent their reasoning. In the remaining
sections of this entry we will look at prominent actualist responses
to the two-fold possibilist challenge.

 
## 4. Actualist Responses to the Possibilist Challenge



There are any number of representatives of, and variations on,
actualism. For brevity, I will focus on several particularly important
accounts. We will begin with a close look at the extraordinarily
influential work of Saul Kripke. Although Kripke himself does not
appear to have been particularly motivated by any great commitment to
actualism, both his version of possible world semantics and its
corresponding deductive system capture important elements of the
actualist perspective.

 
### 4.1 Kripke Semantics and Its Logic



Given the controversial consequences of SQML, it seems clear that
actualists need an alternative quantified modal logic on which
**BF**, **CBF**, and \(\Box\textbf{N}\) fail
to be logically true; ideally, it will also have a sound and complete
deductive system in which, consequently, those principles are not
derivable as theorems. The system of Kripke 1963b satisfies both of
these desiderata and, hence, meets the second element of the
possibilist's two-fold challenge; as we will see, whether it
meets the first--a satisfying account of the truth conditions for
modal propositions like
 (2)--is
 a more delicate matter.

 
#### 4.1.1 Kripke Interpretations



Just as necessitism is not analytically entailed by possibilism, its
denial--i.e., that there at least could be contingent (i.e.,
henceforth, contingent[?]) beings--is not
analytically entailed by actualism. An actualist could consistently
maintain that, necessarily, everything is necessarily identical with
something and hence in particular, given their rejection of contingent
non-*concreta*, that, necessarily, every concrete thing is
necessarily identical to some concrete thing--put another way,
that, necessarily, if something is concrete at any time at all, it is
necessarily and eternally concrete.
 (Spinoza
 might be ascribed such a view, insofar as
 God-or-Nature
 is taken to be concrete and is, ultimately, the only concrete thing.)
For typical actualists, however, it is fundamental to their view that
there are in fact many contingent beings, many things that could have
failed to be identical with anything; reality could have altogether
lacked many things that just happen to exist. A natural (if, as
we'll see, not entirely unproblematic) way of expressing this is
to say that there are possible worlds where at least some things in
the actual world are simply absent; more generally, it is to say that
what exists--that is, for the actualist, what there *is*
in the broadest sense--*varies* from world to world. This
fundamental actualist intuition is at the heart of Kripke's
semantics for quantified modal languages.



Recall that an SQML interpretation \(\calM\) for a first-order modal
language \(\scrL\_\Box\) specifies nonempty sets \(D\) and
\(W\)--the "individuals" and "worlds" of
\(\calM\)--along with a distinguished element \(w^\ast\) of
\(W,\) the "actual" world of the interpretation; it then
assigns a denotation \(\tau^\calM \in D\) to each term \(\tau\) and an
extension \(\pi\_w^\calM \subseteq D^n\) to each *n*-place
predicate \(\pi\) at each world \(w\). A *Kripke
interpretation* \(\calK\) of \(\scrL\_\Box\) is exactly like an
SQML interpretation except for one modification that reflects the
fundamental actualist intuition noted above, viz., the addition of a
function \(\textit{dom}\) that assigns to each world \(w\) of
\(\calK\) a *subset* \(D\_w\) of \(D\)--intuitively, of
course, the individuals that *exist in* \(w\). No restrictions
are placed on the domain of any world; any set of individuals,
including the empty set, will do, although it is required that \(D =
\bigcup\{\textit{dom}(w):w \in W\},\) i.e., that \(D\) consists of
exactly the individuals that exist in some
 world.[48]



The definition of truth at a world in a Kripke interpretation
\(\calK\) is defined exactly as it is for an SQML interpretation
\(\calM\) except for the quantified clause, where the difference
between Kripke interpretations and SQML interpretations just noted
comes to the fore. Specifically, when a quantified formula
\(\forall\nu\varphi\) is evaluated at a world \(w,\) the quantifier
ranges only over \(\textit{dom}(w),\) the set of objects in the domain
of \(w\). Thus, the modal clause in the definition of truth at a world
is revised as follows:


* A universally quantified formula
\(\forall\nu\psi\) is true\(\_w^\calK\) if
and only if, for all individuals \(a \in \textit{dom}(w),\) \(\psi\)
is true\(\_w^{\calK[\frac{\nu}{a}]}\).



And, accordingly:


* An existentially quantified formula \(\exists\nu\psi\) is true\(\_w^\calK\)
if and only if, for some
individual \(a \in \textit{dom}(w),\) \(\psi\) is true\(\_w^{\calK[\frac{\nu}{a}]}\).



The definitions of truth, satisfiability and logical truth are
unchanged. Call the logic determined by Kripke semantics KQML.




**A Note on "Serious" Actualism**. KQML takes over the
semantics of predicates from SQML without modification: by the above
condition \(\pi\_w^\calK \subseteq D^n,\) the extension \(\pi\_w^\calK\)
that a Kripke interpretation \(\calK\) assigns to a predicate at a
world consists of arbitrary *n*-tuples of individuals in \(D.\)
However, as Kripke himself (1963b, p. 86, fn 1) notes,



> 
> 
> [i]t is natural to assume that [a predicate] should be *false*
> in a world...of all those individuals not existing in that
> world....
> 
> 
> 



Many actualists strongly agree, for the following reason: predicates
express properties and relations. Hence, if a (1-place) predicate
\(\pi\) is true of an individual \(a\) at a world \(w,\) it means that
\(a\) *exemplifies* the property that \(\pi\) expresses at
\(w\). But (these actualists continue) it is surely an undeniable
metaphysical principle--dubbed *serious actualism* by
Plantinga (1983)--that an object must *exist*, must be
identical with something, in order to exemplify properties; an object
cannot both be utterly absent from a world and yet have properties
there. To rule this prospect out in Kripke's model theory, then,
rather than allowing an *n*-place predicate's extension
at a world to contain arbitrary *n*-tuples of individuals, we
need to *restrict* \(\pi\)'s
extension at \(w\) to *n*-tuples of individuals *in*
\(w\). More formally put, we need to replace the offending condition
\(\pi\_w^\calK \subseteq D^n\) with the more felicitous condition
\(\pi\_w^\calK \subseteq \textit{dom}(w)^n.\)



However, others (notably Pollock (1985) and Fine (1985)) respond that,
while most properties and relations obviously entail
existence--*being a horse*, say, or *being taller
than*--it is far from clear that all do. Notably, if I fail
to exist at a world, then that in some sense clearly seems to
*characterize* me at that world and what else is
*characterization* than property exemplification? Thus (these
philosophers continue), it seems entirely reasonable to say that, at
worlds in which I fail to exist, I have the property
*non-existence*, as well as the complements of all the
distinctly human properties noted
above--*non-consciousness*, *non-embodiment*, etc.,
my nonexistence
 notwithstanding.[49]
 So for these philosophers, the condition \(\pi\_w^\calK \subseteq
D^n\) on the assignment of extensions to predicates is fine as it
stands.



Serious actualism--also known as *property actualism*
(Fine 1985), *the (modal) existence requirement* (Yagisawa
2005; Caplan 2007) and *the being constraint* (Williamson 2013:
SS4.1)--is a substantive logical and philosophical issue.
However, as it is for the most part a domestic dispute among
actualists, it is largely orthogonal to the possibilism-actualism
debate proper. Consequently, it will not be pursued here in any
greater depth. For further discussion in addition to the references
above see Plantinga 1985 (which contains replies to Pollock and Fine),
Salmon 1987, Menzel 1991 and 1993, Deutsch 1994, Bergmann 1996 and
1999, Hudson 1997, Stephanou 2007, Hanson 2018, and Jacinto
 2019.[50]



#### 4.1.2 KQML and the Controversial Logical Truths of SQML



All three of the controversial logical truths of
SQML--**BF**, **CBF**, and
\(\Box\textbf{N}\)--are invalid in KQML, that is, in KQML, they
are not logically true. The key in each case is KQML's
modification of the way that quantified formulas are evaluated at
worlds. As we saw in
 SS3.3
 above, the validity of **BF** in SQML--its truth in
every interpretation--depends essentially on the fact that, in
evaluating an existentially quantified formula \(\exists\nu\varphi\)
at an arbitrary possible world \(w,\) the initial occurrence of
\(\exists\nu\) in the formula ranges unrestrictedly over all
individuals. In KQML, by contrast, the range of the initial quantifier
\(\exists\nu\) is *restricted* to \(\textit{dom}(w),\) to the
individuals that exist in \(w\). And, because what exists can vary
from world to world, this renders **BF** invalid: from
the fact that some world and some individual *existing in that
world* are thus and so it certainly does not follow that some
individual *existing in the actual world* and some world are
thus and so. Notably, while in some worlds there are children of
Bergoglio, nothing here in the actual world is a child of Bergoglio in
any world, i.e., \(\Diamond\exists \sfx\, \sfB\sfx\) is true and
\(\exists \sfx\Diamond \sfB\sfx\) is false. Hence, the
**BF** instance



**BF**\*:
\(\Diamond\exists \sfx\, \sfB\sfx \to \exists \sfx \Diamond
\sfB\sfx\)


expressing the inference from
 (2)
 to
 (4)
 is false as
 well.[51]
 (See the preceding note for a more formal demonstration of the
invalidity of **BF**.)



As we also saw in
 SS3.3,
 **CBF** entails the principle **N** that
everything (i.e., everything there *happens* to be) is
necessarily identical to something. And, obviously, so too does the
full necessitism principle \(\Box\)**N**. But
**N** is clearly invalid in KQML: because world domains
can vary, an individual \(a\) in the actual world might not exist in
another world, i.e., it might be that nothing is identical to \(a\) in
some
 worlds.[52]
 Hence, since **N** is invalid in KQML, so are
**CBF** and
 \(\Box\)**N**.[53]


#### 4.1.3 *KQML*: Kripke's Deductive System for KQML



As noted in
 SS3.2,
 the deductive system *SQML* is sound and complete for
SQML--all and only the logical truths of SQML are provable in
*SQML*, including the three controversial principles
**BF**, **CBF**, and
\(\Box\)**N**. Since, as we just saw in the previous
section, those principles are all invalid in KQML, to formulate a
sound and complete deductive system of his own, Kripke had to modify
*SQML* rather severely to block the derivation of those
principles without also blocking any of KQML's valid
formulas.



Kripke's solution is nicely illustrated by means of a (somewhat
compressed) proof in *SQML* of
 **BF**\*
 (the structure of which will be shared by the proof of any
non-trivial instance of **BF**):




|  |  |  |
| --- | --- | --- |
| 1. | \(\forall \sfx \Box \neg \sfB\sfx \to \Box \neg \sfB\sfx\) | **Q2** |
| 2. | \(\Box(\forall \sfx \Box \neg \sfB\sfx \to \Box \neg
\sfB\sfx)\) | 1, **Nec** |
| 3. | \(\Diamond\forall \sfx \Box \neg \sfB\sfx \to \Diamond\Box \neg
\sfB\sfx\) | 2,
 *K*[54] |
| 4. | \(\Diamond\Box \neg \sfB\sfx \to \neg \sfB\sfx\) | **B*** |
| 5. | \(\Diamond\forall \sfx \Box \neg \sfB\sfx \to \neg
\sfB\sfx\) | 3, 4, *PL* |
| 6. | \(\forall \sfx(\Diamond\forall \sfx \Box \neg \sfB\sfx \to \neg
\sfB\sfx)\) | 5, **Gen** |
| 7. | \(\Diamond\forall \sfx \Box \neg \sfB\sfx \to \forall \sfx \neg
\sfB\sfx\) | 6, **Q1**, **Q3**, and
 *PL*[55] |
| 8. | \(\Box(\Diamond\forall \sfx \Box \neg \sfB\sfx \to \forall \sfx
\neg \sfB\sfx)\) | 7, **Nec** |
| 9. | \(\Box\Diamond\forall \sfx \Box \neg \sfB\sfx \to \Box\forall
\sfx \neg \sfB\sfx\) | 8, **K** and **MP** |
| 10. | \(\forall \sfx \Box \neg \sfB\sfx \to \Box\Diamond\forall \sfx
\Box \neg \sfB\sfx\) | **B** |
| 11. | \(\forall \sfx \Box \neg \sfB\sfx \to \Box\forall \sfx \neg
\sfB\sfx\) | 9, 10, *PL* |
| 12. | \(\Diamond\exists \sfx \sfB\sfx \to \exists \sfx \Diamond
\sfB\sfx\) | 11, \(\exists\)**Def**,
\(\Diamond\)**Def**, and *PL* |



It is clear where the problem lies: the universal instantiation schema
**Q2** is invalid in KQML! To see this in the particular
case of line 1, suppose that \(\forall \sfx \Box \neg \sfB\sfx\) is
true in (i.e., true at the "actual world" of) a given
Kripke interpretation \(\calK\) with individuals \(D\) and worlds
\(W\). Then, by
 the quantificational clause
 in the definition of truth at a world in Kripke's semantics,
everything that exists in the "actual world" \(w^\ast\) of
\(\calK\) is not in the extension of \(\sfB\) at any world \(u\) in
\(W,\) i.e., for all \(a \in \textit{dom}(w^\ast)\) and for all \(u
\in W,\) \(a \notin \sfB^\calK\_{u}\). Recall, however, that the value
\(\sfx^\calK\) assigned to \(\sfx\) can be anything in the set \(D\)
of individuals of \(\calK\); in particular, \(\sfx^\calK\) might not
exist in \(w^\ast\) and, hence, might well be in the extension of
\(\sfB\) at some other world of \(\calK,\) in which case \(\Box\neg
\sfB\sfx\) will be false in
 \(\calK.\)[56]



Here we begin to see the logical challenge that confronts the
actualist: a logically valid principle of SQML--in this case, a
standard principle of classical logic--is rendered invalid when
we attempt to revise our modal semantics so as to accommodate
actualist intuitions. The challenge is then how (or whether) to revise
the principle in question, and this in turn often requires a choice
between several competing possibilities, each of which may require
further revisions still. Kripke himself avoids several options that
might suggest themselves in light of the invalidity of
**Q2** in KQML. For instance, taking a lead from Prior
(1957: 33-35), an actualist might argue that
terms--individual constants and variables (when they occur
freely)--are directly referring expressions like proper names and
demonstratives and, hence, that an accurate model theoretic
representation of actualism should require that the value
\(\tau^\calK\) that \(\calK\) assigns to an arbitrary term \(\tau\)
should be restricted to the domain of the actual world \(w^\ast\) of
\(\calK\)--one cannot, after all, refer to individuals that
don't actually exist. This modification of KQML would indeed
preserve the validity of **Q2** but at the cost of
invalidating **Nec**--notably, it would render the
inference from line 1 to line 2 in the above proof invalid. For, while
line 1 might now be true at the actual world \(w^\ast\) of an
arbitrary interpretation \(\calK\) because we've stipulated that
\(\sfx^\calK\) exist in \(w^\ast,\) that very fact could well render
line 1 false at some other world \(u\): \(\forall \sfx \Box \neg
\sfB\sfx\) might be true at \(u\) but our actually existing individual
\(\sfx^\calK,\) although not in the extension of \(\sfB\) in the
actual world, might well be in its extension at \(u\). So, under the
proposed modification, the question for the actualist instead becomes
how to modify **Nec**--perhaps by limiting how it
applies to formulas involving individual constants and free variable
occurrences in some
 way.[57]



The most common actualist response to the invalidity of
**Q2**, rather than to try to preserve it by modifying
the semantics of KQML, is simply to replace it with its
 free logical
 counterpart (see, e.g., Fine 1978, Menzel 1991):



**FQ2**:
\(\forall\nu\varphi \to (\exists \nu\,\nu=\tau \to
\varphi^\nu\_\tau),\) where \(\tau\) is any term other than
\(\nu\)


That is, what is true of everything--i.e., everything that
*actually* exists--will be true of anything in particular
*if* it is actual. Unlike **Q2**,
**FQ2** is valid in KQML as it stands and, moreover,
every instance of it is necessary: what is true of everything in a
given world will be true of anything in particular *if* it is
identical with something in that world. Hence, **FQ2**
raises no problems for **Nec**. And, critically,
outfitted with **FQ2** instead of **Q2**, it
is no longer possible to prove
 **BF**.[58]



However, Kripke (1963b, p. 89, note 1) is loath to adopt a solution
that, like the ones just suggested, involves "revising
quantification theory or modal logic". So instead Kripke opts
for another that he borrows from Quine. In his account of
quantification, Quine (1951: SS15) points out that any occurrence
of a name in an ordinary logical truth is inessential to its truth:
what is said of the referent of the name could just as well be said of
anything. Thus, it is not in virtue of any distinctive properties of
God or Socrates that the following proposition is true:



(\(\*\))
If God created everything, then God created Socrates


or, more formally:



(\(\*\))
\(\forall \sfx\, \sfK\sfg\sfx \to \sfK\sfg\sfs\)


(\(\*\)), that is, would have been true no matter what
"God" and "Socrates" refer to. If anything,
then, names obscure the grounds of logical truth. Hence, the
discerning logical eye sees name occurrences in the likes of (\(\*\))
as implicitly quantified variables and, hence, sees the completely
general logical form that underlies
 it:[59]



(\(\*\*\))
\(\forall \sfz \forall \sfy(\forall \sfx\, \sfK\sfz\sfx \to
\sfK\sfz\sfy)\)


The same goes for free variable occurrences--semantically,
variables are essentially names, so free variable occurrences in
logical truths should also be seen as implicitly universally
quantified. Hence, properly expressed, the axioms of a
*logical* system, a system designed to have all and only
logical truths as its theorems, should be purely general and, hence,
should be purged of individual constants and free variable
occurrences. In particular, universal instantiation, properly
expressed, will replace the instantiated term \(\tau\) in the
consequent of
 **Q2**
 with a universally quantified variable:



**KQ2**:
\(\forall \nu'(\forall\nu\varphi \to \varphi^\nu\_{\nu'}),\) where
\(\nu'\) is substitutable for \(\nu\) in \(\varphi\)


Accordingly, line 1 of the proof of **BF**\* requires an
initial universal quantifier to bind the free occurrence of
\(\sfx\):



1.\*
\(\forall \sfx(\forall \sfx \Box \neg \sfB\sfx \to \Box \neg
\sfB\sfx)\)


Application of **Nec** to line 1\* now yields only



2.\*
\(\Box\forall \sfx(\forall \sfx \Box \neg \sfB\sfx \to \Box \neg
\sfB\sfx)\)


and the intended proof to **BF**\* stalls out. To continue
along the lines of the SQML proof above we would need to be able to
swap the initial necessity operator \(\Box\) in line 2\* with the
universal quantifier to infer



 
\(\forall \sfx\Box(\forall \sfx \Box \neg \sfB\sfx \to \Box \neg
\sfB\sfx)\)


That is, we would need to be able to prove the relevant instance
of



**CBF#**:
\(\Box\forall \nu \varphi \to \forall \nu \Box \varphi\)


But, as the label indicates, this is just an equivalent form of
**CBF** and, as Kripke (1963b, 88-9) notes,
attempting to prove it under the proposed revision
**KQ2** of **Q2** stalls out for the same
 reason.[60]



The necessitism principle \(\Box\)**N** meets a similar
fate. A standard proof of \(\Box\)**N** in SQML begins
with an instance of **Q2** and proceeds as follows:




|  |  |  |
| --- | --- | --- |
| 1. | \(\forall \sfy\,\neg \sfx=\sfy \to \neg \sfx=\sfx\) | **Q2** |
| 2. | \(\sfx=\sfx \to \exists \sfy\,\sfx=\sfy\) | 1, \(\exists\)**Def**, *PL* |
| 3. | \(\sfx=\sfx\) | **Id1** |
| 4. | \(\exists \sfy\,\sfx=\sfy\) | 2, 3, **MP** |
| 5. | \(\Box\exists \sfy\,\sfx=\sfy\) | 4, **Nec** |
| 6. | \(\Box\forall \sfx \Box\exists \sfy\, \sfx=\sfy\) | 5, **Gen**, **Nec** |



But with **KQ2** replacing **Q2**,
critically, we are unable to wedge the necessity operator in between
the two quantifiers; we can only get the likes of the following:




|  |  |  |
| --- | --- | --- |
| 1. | \(\forall \sfx(\forall \sfy\,\neg \sfx=\sfy \to \neg
\sfx=\sfx)\) | **KQ2** |
| 2. | \(\forall \sfx(\sfx=\sfx \to \exists \sfy\,\sfx=\sfy)\) | 1, \(\exists\)**Def**, *PL* |
| 3. | \(\forall \sfx\,\sfx=\sfx \to \forall \sfx\exists
\sfy\,\sfx=\sfy\) | 2, **Q1**, **MP** |
| 4. | \(\forall \sfx\,\sfx=\sfx\) | **Id1**, **Gen** |
| 5. | \(\forall \sfx\exists \sfy\,\sfx=\sfy\) | 3, 4, **MP** |
| 6. | \(\Box\forall \sfx\exists \sfy\,\sfx=\sfy\) | 5, **Nec** |



and line 6 is unproblematically valid in KQML: it is simply the
innocuous triviality that everything in a given world is identical to
something *in that world*--not, as
\(\Box\)**N** would have it, in *every* world.



Two further important modifications to the framework of SQML follow
from Kripke's Quinean conception of the purely general nature of
logical truth. First, individual constants have no place; since there
are no individual constants in any axioms, there are none in any
theorems. Hence, it is not possible to reason with them in
Kripke's system. Consequently, Kripke restricts his semantics to
languages with no individual
 constants.[61]
 Second, under the Quinean conception, the manner in which quantifiers
and modal operators are introduced into proofs needs serious revision.
For example, the proposition \(\forall \sfx(\sfF\sfx \to \sfF\sfx)\)
is valid in KQML. In *SQML*, it is proved by deriving the
tautology \(\sfF\sfx \to \sfF\sfx\) and applying the rule of
Generalization:




|  |  |  |
| --- | --- | --- |
| 1. | \(\sfF\sfx \to ((\sfG\sfx \to \sfF\sfx) \to \sfF\sfx)\) | **P1** |
| 2. | \((\sfF\sfx \to (\sfG\sfx \to \sfF\sfx)) \to (\sfF\sfx \to
\sfF\sfx)\) | 1, **P2**, **MP** |
| 3. | \(\sfF\sfx \to (\sfG\sfx \to \sfF\sfx)\) | **P1** |
| 4. | \(\sfF\sfx \to \sfF\sfx\) | 2, 3, **MP** |
| 5. | \(\forall \sfx(\sfF\sfx \to \sfF\sfx)\) | 4, **Gen** |



But axioms can't have free variables under the Quinean
conception and, hence, that proof is not available. Likewise, the
*de re* validity \(\forall \sfx\,\Box(\Box \sfF\sfx \to
\sfF\sfx)\) is proved in *SQML* by applying the rule of
Necessitation to the **T** instance \(\Box \sfF\sfx \to
\sfF\sfx\) and then generalizing:




|  |  |  |
| --- | --- | --- |
| 1. | \(\Box \sfF\sfx \to \sfF\sfx\) | **T** |
| 2. | \(\Box(\Box \sfF\sfx \to \sfF\sfx)\) | 1, **Nec** |
| 3. | \(\forall \sfx\, \Box(\Box \sfF\sfx \to \sfF\sfx)\) | 2, **Gen** |



But again, for the same reason, this proof is also unavailable.



Kripke's solution here--once again borrowing from Quine
(1951)--jettisons both **Gen** and
**Nec** and cleverly incorporates the desired effects of
both rules directly into his specification of the logical axioms of
his deductive system--which, of course, we will call
*KQML*. To express the solution clearly, say that a formula is
*closed* if it contains no free variable occurrences, and
define a *closure* of a formula \(\varphi\) to be any closed
formula resulting from prefixing a (possibly empty) string of
universal quantifiers and necessity operators, in any order, to
\(\varphi\). Then, given a language \(\scrL\_\Box\) without any
individual constants, any *closure* of any instance of any of
the following schemas is an axiom of *KQML*:


* *Propositional Axiom Schema*
  

**Taut**: \(\varphi,\) where \(\varphi\) is a
propositional
 tautology[62]
* *Modal Axiom Schemas*
  

**K**,
 **T**, and
 **5**
* *Quantificational Axiom Schemas*
  

**Q1**,
 **KQ2**, and
 **Q3**
* *Identity Schemas*
  

**Id1**,
 **Id2**



Should the serious actualism constraint be enforced, we add:


* *Serious Actualism Schema*
  

**SA**: \(\pi\nu\_1\ldots\nu\_n \to \exists \nu\,\nu =
\nu\_i,\) for \(1 \leq i \leq n\) and where \(\nu\) is a variable other
than \(\nu\_i\)



As noted, *KQML*'s sole rule of inference is
**MP**, Modus Ponens. Accordingly, a *proof*
*in KQML* is a finite sequence of formulas of \(\scrL\_\Box\)
such that each is either an axiom of *KQML* or follows from
preceding formulas in the sequence by **MP**.



Crucially, then, with regard to such KQML validities as \(\forall
\sfx(\sfF\sfx \to \sfF\sfx)\) and \(\forall \sfx\,\Box(\Box \sfF\sfx
\to \sfF\sfx)\) that are derivable in *SQML* by applications of
**Gen** and **Nec**, *instances* of
the above schemas can contain free variable occurrences. So, in
particular, \(\forall \sfx(\sfF\sfx \to \sfF\sfx)\) is a closure of
the propositional tautology \(\sfF\sfx \to \sfF\sfx,\) and \(\forall
\sfx\,\Box(\Box \sfF\sfx \to \sfF\sfx)\) is a closure of the
**T** instance \(\Box \sfF\sfx \to \sfF\sfx\) and, hence,
both validities are axioms (hence theorems) of *KQML*.



With these modifications in place, Kripke is able to demonstrate that
his deductive system *KQML* is sound and complete for closed
formulas relative to his semantics. Soundness, in particular, tells us
that no invalid formula is provable in the system. Hence, since
**BF**, **CBF**, and \(\Box\textbf{N}\) are
all invalid in KQML, soundness guarantees that they are all unprovable
in *KQML*.


#### 4.1.4 Is KQML a Genuinely Actualist Logic?



On the face of it, KQML provides the actualist with a powerful
alternative to SQML. However, one might well question its actualist
credentials. Specifically, despite the invalidity of the
actualistically objectionable principles **BF**,
**CBF**, and \(\Box\)**N**, it is
questionable whether KQML has escaped ontological commitment to
*possibilia*, for the following reason. KQML provides us with a
formal semantics for (constant-free) modal languages \(\scrL\_\Box\)
and, in particular, an account of how the truth value of a given
formula \(\varphi\) of \(\scrL\_\Box\) is determined in an
interpretation by the meanings assigned to its semantically
significant component parts, notably, the meanings of its predicates.
 As noted above
 in our exposition of SQML, truth-in-a-model is not the same as truth
*simpliciter*. However, given an
 applied modal language
 \(\scrL\_\Box,\) we were able to define a notion of modal truth
*simpliciter* for formulas of \(\scrL\_\Box\) in terms of truth
in an *intended* interpretation, an interpretation \(\calM\)
for \(\scrL\_\Box\) comprising the very things that the language is
intuitively understood to be "about". Thus, when \(\calM\)
is an intended SQML interpretation, the things it is about are the
honest-to-goodness actual and merely possible individuals in the
domain \(D\) of \(\calM\) and the honest-to-goodness possible worlds
of \(W\). In particular, if our applied language is one in which
\(\Diamond\exists \sfx\, \sfB\sfx\) is meant to express that Bergoglio
might have had children, \(D\) will include at least some of his
merely possible children and \(W\) will contain a world in which some
of them *are*, in fact, his children and hence concrete and,
hence, in which they are actual in the sense of
 **A!Def**.
 There is thus, more generally, for each world \(w \in W,\) the set
\(D\_w\) of things in \(D\) that are actual in \(w\). Note, however,
that, in light of this, we can transform \(\calM\) into a KQML
interpretation \(\calK\_\calM\) simply by defining a function \(\mathit
dom\) that, for each world \(w \in W,\) returns exactly the set
\(D\_w\). However, that is simply a formal modification that affects
the evaluation of quantified formulas at worlds; it does not alter
\(\calK\_\calM\)'s ontological
commitments, which are still exactly those of \(\calM\). But there
seems no other way to construct a notion of an *intended*
Kripke interpretation that will yield the right truth values for
propositions like
 (2).
 From this perspective, the invalidity of **BF**,
**CBF**, and \(\Box\)**N** in KQML--in
particular, their falsity in an intended Kripke interpretation
\(\calK\_\calM\)--results from a simple adjustment of the
semantical apparatus with no substantive change in the metaphysics. So
instead of a genuinely actualist alternative to the possibilist
commitments of SQML, KQML appears to remain no less steeped in
them--the denizens of other worlds that are not in the domain of
the actual world simply fail to be actual in the sense of
 **A!Def**.
 Of course, this metalinguistic fact cannot be *expressed* in
\(\scrL\_\Box\)--even if one adds the actuality predicate to
\(\scrL\_\Box,\) due to the quantifier restrictions in the semantics of
KQML, the assertion that there are things that aren't actual,
\(\exists \sfx\, \neg \sfA!\sfx,\) will be false in \(\calK\_\calM\) at
every world. But that does not make the possibilist commitments of
\(\calK\_\calM\)
 disappear.[63]



An option for the actualist here, perhaps, is simply to deny that
Kripke interpretations have any genuine metaphysical bite. Rather, the
role of Kripke's possible world semantics is simply to get the
intuitive validities right; soundness and completeness proofs for
*KQML* then demonstrate that the system preserves those
intuitive validities and, hence, that it is a trustworthy vehicle for
reasoning directly about the modal facts of the matter. The formal
semantics proper is just a useful device toward this end. However, on
the face of it, anyway, this instrumentalist take on Kripke's
semantics is an uncomfortable one for the actualist. Consider ordinary
Tarskian semantics for classical first-order logic FOL. Intuitively,
this semantics is more than just a formal instrument. Rather, an
intended interpretation for a given applied language \(\scrL\) shows
clearly how the semantic values of the relevant parts of a formula of
\(\scrL\)--the objects, properties, relations, etc. in the world
those parts signify--contribute to the actual truth value of the
sentence. The semantics thus provides insight into the
"word-world" connection that explains how it is that
sentences of natural language can express truth and falsity, how they
can carry good and bad information. The possibilist is able to
generalize this understanding of the semantics of first-order
languages directly to modal languages \(\scrL\_\Box\). The embarrassing
question for the actualist who would take the proposed instrumentalist
line on Kripke's semantics is: what distinguishes Kripkean
semantics from Tarskian? Why does the latter yield insight into the
word-world connection and not the former? Actualists owe us either an
explanation of how Kripke's semantics can be understood
non-instrumentally without entailing possibilism or a plausible
instrumentalist answer to the possibilist challenge. The former option
is explored in
 SS4.2,
 the latter in
 SS4.3
 and
 SS4.4.



### 4.2 Haecceitism



One of the best known responses to the possibilist challenge was
developed by Alvin Plantinga (1974, 1976). Plantinga's response
(more or less as fleshed out formally by Jager (1982)) requires no
significant modifications to KQML beyond explicitly adopting the
 serious actualism
 condition discussed at the end of
 SS4.1.1.[64]
 Rather, his response has almost entirely to do with the metaphysics
of intended interpretations, or what he calls *applied
semantics*. More specifically, he claims to provide an
actualistically acceptable ontology for constructing intended
interpretations for applied languages.


#### 4.2.1 Worlds and Essences



To appreciate Plantinga's account, note first that a challenge
to actualism that we have not focused on hitherto is the nature of
 possible worlds
 themselves. The main reason for this is simply that the basic
semantic argument for *possibilia* laid out in the Introduction
to this entry does not assume or require them. If, however, one takes
possible world semantics of some ilk seriously, so that possibility
and necessity are correlated in some manner with truth at some or all
possible worlds, then, insofar as *possibilia* are said to
*exist in* non-actual possible worlds, such worlds are
themselves reasonably classified as some variety of
*possibilia* themselves, some variety of merely possible,
non-existent object. Hence, because Plantinga takes possible world
semantics seriously, the first task he sets for himself is to define
possible worlds in an actualistically acceptable way.



Plantinga defines a possible world to be a *state of affairs*
of a certain sort. A state of affairs for Plantinga is an abstract,
fine-grained, proposition-like entity characteristically referred to
by sentential gerunds like *the earth's being smaller than
the*
*sun*.[65]
 Some states of affairs *obtain* and others do not: *the
earth's being smaller than the sun* obtains;
*seven's being the sum of three and five* does
 not.[66]
 Importantly, all states of affairs are necessary beings irrespective
of whether or not they obtain. A state of affairs \(s\) is
*possible* if it possibly obtains; \(s\) *includes*
another state of affairs \(s'\) if, necessarily, \(s\) obtains only if
\(s'\) does and \(s\) *excludes* \(s'\) if it is not possible
that both \(s\) and \(s'\) obtain; \(s\) is *maximal* if, for
any state of affairs \(s',\) \(s\) either includes or excludes \(s'\).
A *possible world*, then, is a maximal possible state of
affairs; and *the actual world* is the possible world that, in
fact, obtains. It is easy to show that, under those definitions, a
state of affairs is a possible world just in case it possibly includes
all and only the states of affairs that obtain. Since worlds are
states of affairs and, for Plantinga, states of affairs are all
actually existing abstract entities, their existence is consistent
with actualism, as desired.



The core of Plantinga's answer to the possibilist challenge is
the notion of an *(individual) essence*, an idea that traces
back clearly at least to
 Boethius.[67]
 Plantinga defines the
 *essential* properties
 of an object to be those properties that it couldn't possibly
have lacked without simply failing to exist
 altogether.[68]
 A bit more formally put: \(P\) is essential to \(a\) just in case,
necessarily, if \(a\) exists (i.e., for an actualist like Plantinga,
if something is identical to \(a\)), then \(a\) has \(P\).
Intuitively, the essential properties of an object are the ones that
make the object "what it is". For example, on at least
some conceptions of human persons, *being human* is essential
to Bergoglio while *being Catholic* isn't--there is
(on these conceptions) no possible world containing Bergoglio in which
he fails to be human but many in which he, say, becomes a Buddhist
monk or is a lifelong atheist. A property is an essence of an object
\(a,\) then, just in case (i) it is essential to \(a\) and (ii)
nothing *but* \(a\) could have exemplified it; and a property
is an essence *simpliciter* just in case it is possibly an
essence of something.



A critical, and distinctive, element of Plantinga's account is
that there are many unexemplified essences, i.e., essences that are
not, in fact, the essences *of* anything, specifically, those
he dubs *haecceities*. Haecceities are "purely
nonqualitative" properties like *being Plantinga*, or
perhaps, *being identical with Plantinga*, that do no more than
characterize an object \(a\) as *that very thing*. Pretty
clearly, haecceities are essences: Plantinga, for example, could not
have existed and failed to have the property *being Plantinga*
and it is not possible that anything else have that property. And,
importantly, *qua* abstract property, like all essences,
*being Plantinga* exists necessarily, whether exemplified or
not--although, of course, had it not been exemplified, it
wouldn't have had the name "being Plantinga", or any
name at all, for that matter.


#### 4.2.2 Intended Haecceitist Interpretations



As we saw in
 SS4.1.3
 above, the problem with KQML for actualists is that an intended
Kripke interpretation for an applied modal language \(\scrL\_\Box\)
appears to involve commitment to *possibilia* no less than
SQML--to get the truth values of propositions like
 (2)
 right, an intended Kripke interpretation \(\calK\) for \(\scrL\_\Box\)
will still have to include mere *possibilia* like
Bergoglio's merely possible children; indeed, but for the
addition of the domain function \(\textit{dom}\) on worlds tacked on
to invalidate the problematic validities of SQML, \(\calK\) will be
indistinguishable from an intended SQML interpretation of
\(\scrL\_\Box\). Plantinga's diagnosis, roughly put, is that
Kripke's semantics gets the *structure* of the modal
universe right but that it needs his worlds and haecceities to realize
that structure in an actualistically acceptable way. Specifically, an
intended *haecceitist* interpretation \(\calH\) for an applied
modal language \(\scrL\_\Box\) is a Kripke interpretation that
specifies sets \(D\) and \(W\) as usual, but \(W\) is a sufficiently
expansive set of Plantinga's maximal possible states of affairs
and \(D\) a corresponding set of haecceities. The domain function
\(\textit{dom}\) as usual will map each possible world \(w\) to a
subset of \(D,\) i.e., to a set of haecceities that Plantinga refers
to as the *essential domain* of \(w\). This is the set of
haecceities that are *exemplified* in \(w,\) where a haecceity
\(h\) is exemplified in \(w\) just in case \(w\) includes the state of
affairs *\(h\)'s being
exemplified*. Otherwise put, the essential domain
\(\textit{dom}(w)\) of \(w\) consists of those haecceities that would
have been exemplified if \(w\) had obtained. Since, as noted above,
haecceities are necessary beings that exist even if unexemplfied, they
can serve as actually existing "proxies" for the
individuals there *would have been* had \(w\) obtained (Bennett
2006). In this way, Plantinga's haecceitist interpretations can
*represent* the possibilist's non-actual worlds and their
merely possible inhabitants without incurring any possibilist
commitments.



Because an intended haecceitist interpretation \(\calH\) for an
applied language \(\scrL\_\Box\) is a Kripke interpretation (with the
serious actualism constraint enforced), every variable \(\nu\) is, as
usual, assigned a member \(\nu^\calH\) of the set \(D\) of individuals
of \(\calH,\) i.e., a haecceity, and every predicate \(\pi\) is
assigned a set \(\pi\_w^\calH\) of *n*-tuples of haecceities in
the essential domain of \(w,\) for each world \(w \in W\). Monadic
atomic formulas, in particular, are evaluated as usual: \(\pi\nu\) is
true\(\_w^\calH\) just in case \(\nu^\calH
\in \pi\_w^\calH\). However, although the *formal* truth
conditions in \(\calH\) for atomic formulas do not differ from those
defined for atomic formulas in an intended SQML interpretation
\(\calM,\) it is important to understand that, in these different
contexts, those formally identical truth conditions represent starkly
different metaphysical conditions. Specifically, in an intended SQML
interpretation \(\calM,\) that \(\nu^\calM \in \pi\_w^\calM\) indicates
that, at \(w,\) the (perhaps merely possible) object \(\nu^\calM\)
exemplifies the property \(P\_\pi\) expressed by \(\pi\). By contrast,
in an intended haecceitist interpretation \(\calH,\) that \(\nu^\calH
\in \pi\_w^\calH\) indicates, not that \(h\) *exemplifies*
\(P\_\pi\) at \(w,\) but that it is *coexemplified with*
\(P\_\pi\) at \(w\). Thus, for example, the truth condition for
 (2)--formalized
 as \(\Diamond\exists \sfx\, \sfB\sfx\)--in an intended
haecceitist interpretation \(\calH\) is obviously *not* that
there is a possible world in which some haecceity \(h\) is one of
Bergoglio's children--haecceities are properties and,
hence, necessarily non-concrete--but, rather, that



(5)
There is a possible world in which some haecceity \(h\) is
coexemplified with *being a child of Bergoglio*.


Likewise for *n*-place atomic formulas generally:
\(\rho\nu\_1\ldots\nu\_n\) is true at \(w\) just in case the haecceities
\(\nu\_1^\calH, \ldots, \nu\_n^\calH,\) respectively, are coexemplified
with the relation \(R\_\rho\) that \(\rho\) expresses in \(w\).
Plantinga's haecceitism thus delivers the same sort of
systematic, compositional account of the truth conditions for modal
propositions as possibilism but without the commitment to unactualized
*possibilia*.



It is important to be clear on the fact that coexemplification at a
world is primitive here, and not definable in terms of
exemplification. Of course, if a (monadic) atomic formula \(\pi\nu\)
is *in fact* true in an intended haecceitist interpretation
\(\calH,\) i.e., if it is true at the actual world \(w^\ast,\) then
there is an object that exemplifies both \(\nu^\calH\) and
\(P\_\pi\)--notwithstanding the fact that this implication is not
itself represented in \(\calH,\) since \(\textit{dom}(w^\ast)\) only
contains haecceities, not the objects that exemplify them. However, it
is critical not to generalize this implication and take the
coexemplification of two properties--notably, a haecceity \(h\)
and some property \(P\)--*at a world* \(w\) to imply that
there is something that exemplifies them at \(w\). For if that were
so, then the truth conditions for the likes of
 (2)
 would once again entail that there are *possibilia*, and
Plantinga could rightly be accused of being committed to them--he
simply *ignores* them in his haecceitist model theory. But the
correct implication is this: if \(\pi\nu\) is true\(\_{w}^\calH\),
so that \(\nu^\calH\) is
coexemplified with \(P\_\pi\) at \(w,\) then, *had* \(w\)
obtained, there *would have been* an individual that *would
have* exemplified both \(\nu^\calH\) and \(P\_\pi\). Importantly,
though, the fact that \(\nu^\calH\) is coexemplified with \(P\_\pi\) at
\(w\) stands on its own; it does not hold in virtue of the
corresponding counterfactual. Indeed, the dependency goes in the other
direction: that there could have been an individual with \(P\_\pi\)
(and would have been had \(w\) obtained) is true in virtue of the fact
that some (actually existing) haecceity is possibly coexemplified with
\(P\_\pi\).


#### 4.2.3 Modalism



Plantinga's haecceitist account illustrates an important point
about most forms of actualism that is sometimes a source of confusion,
namely, that most actualists are also *modalists*. That is,
most actualists take the English modal operators
"necessarily", "possibly", and the like,
collectively, to be primitive and, hence, not to be definable in terms
of non-modal notions. On the face of it, however, insofar as the
formal operators \(\Box\) and \(\Diamond\) are taken to symbolize
their English counterparts, defining the modal operators would appear
to be exactly what possible world semantics purports to do. For,
unlike
 the classical connectives and quantifiers,
 in basic possible world semantics the modal operators are not
interpreted *homophonically*. That is, they are not interpreted
with the very natural language operators they are intended to
represent--a necessitation \(\Box\psi,\) in particular, is not
defined to be true in an interpretation just in case \(\psi\) is
*necessarily* true in it. Rather, in basic possible world
semantics, the necessity operator is interpreted as a restricted
universal quantifier: a necessitation \(\Box\psi\) is true in an
interpretation \(\calM\) just in case \(\psi\) is true at *every
possible world* of \(\calM\). However, whether or not that proves
to be a *definition* of the necessity operator depends on what
one takes a possible world to be.
 David Lewis
 famously
 defined possible worlds
 to be (typically) large, scattered concrete objects similar in kind
but spatio-temporally unconnected to our own physical universe. Under
such an understanding of possible worlds, possible world semantics
arguably provides a genuine definition of the modal operators on which
the truth conditions assigned to modal formulas do not themselves
involve any modal
 notions.[69]
 As it is sometimes put, such a definition is
"eliminative" or "reductive"--modal
notions are eliminated in the semantics in favor of quantification
over (non-modal) worlds.



Plantinga's haecceitist semantics is decidedly not of this sort.
 Recall,
 in particular, that Plantinga defines a possible world to be a
maximal possible state of affairs. Spelling this definition out
explicitly in the truth condition for necessitations (under an
intended haecceitist interpretation \(\calH\)) then yields:



> 
> 
> \(\Box\psi\) is true if and only if \(\psi\) is true at all states of
> affairs \(w\) such that (i) for all states of affairs \(s,\) either,
> *necessarily*, \(w\) obtains only if \(s\) does or,
> *necessarily*, \(w\) obtains only if \(s\) doesn't, and
> (ii) *possibly*, \(w\) obtains.
> 
> 
> 



Clearly, the truth condition here interprets the modal operator
\(\Box\) homophonically and, hence, on pain of circularity, cannot be
taken to provide any kind of eliminative analysis of the modal
operator \(\Box.\) But, as we've detailed at length, such an
analysis is not Plantinga's goal. Rather, his account is meant
to show how the truth conditions for atomic formulas in an intended
interpretation serve to ground propositions like
 (2)
 in the modal properties of haecceities. (His truth conditions are
thus perhaps better thought of as *grounding conditions*.)
Given such an interpretation, then, the truth conditional clauses of
Plantinga's theory yield non-eliminative but philosophically
illuminating *equivalences* that reveal the connections between
the statements expressible in a basic modal language \(\scrL\_\Box\)
and the deeper metaphysical truths of his
 theory.[70]



Haecceities are thus the key to Plantinga's answers to both
prongs of possibilism's
 twofold challenge to actualism.
 First, Plantinga's haecceitist semantics delivers a
compositional theory of truth conditions that grounds general *de
dicto* modal propositions like
 (2)
 systematically in the modal properties of individuals of a certain
sort--specifically, coexemplification relations that haecceities
stand in with other properties at possible worlds. Second, his
haecceitist semantics enables him to adopt KQML (with the serious
actualism constraint) without modification and, hence, to have a
robust quantified modal logic that has no possibilist commitments.



### 4.3 Strict Actualism



As robust as the haecceitist solution is, many actualists find that it
grates against some very strong intuitions, especially about the
nature of properties. 
But there is a deeper issue dividing
the haecceitists from other actualists. To clarify, consider that many
propositions are *singular* in form. That is, unlike general
propositions like *all whales are mammals*, some propositions
are "directly about" specific individuals--for
example, the proposition *Marie Curie was a German citizen*.
Call the individuals a singular proposition is directly about the
*subjects* of the proposition. Singular propositions are
typically expressed by means of sentences involving names, pronouns,
indexicals, or other devices of direct reference. As we've seen,
possibilists believe that there are singular propositions whose
subjects are not actual, viz., propositions about mere
*possibilia*: for the possibilist, recall,
 (2)
 is grounded in singular propositions of the form *\(a\) is
Bergoglio's child*, for *possibilia* \(a\).
Similarly, haecceitists like Plantinga, although actualists in the
strict sense, also believe there are singular possibilities that are
in a derivative but clear sense directly about things that don't
exist, viz., those possibilities involving unexemplified haecceities
like *\(h\) is coexemplified with being Bergoglio's
child*. Say that a *strict* actualist is an actualist who
rejects the idea that there are, or even could be, singular
propositions that are directly about things that do not exist in
either the possibilist's or the haecceitist's sense. It
follows that, had some actually existing individual \(a\) failed to
exist, there would have been no singular propositions about \(a\); or,
as Prior puts it, there would have been no *facts* about \(a,\)
not even the fact that \(a\) fails to exist (see, e.g., Prior 1957:
48-49); or again: singular propositions *supervene on*
the individuals they are about. For the strict actualist, then,
necessarily, all propositions are either wholly general or, at most,
are directly about existing individuals only.



So understood, the strict actualist rejects the intuition that drives
both possibilism and haecceitism, viz., that *de dicto* modal
propositions like
 (2)
 are grounded ultimately in the modal properties of, and relations
among, individuals of some ilk. Rather, our illustrative statement
 (2)
 is, ultimately, brute; it is true simply because it is possible that
(or there is a world at which) something is Bergoglio's child,
full stop, as there is nothing to instantiate the quantifier; there is
nothing such that, possibly (or, at some world), *it* is
Bergoglio's child.



How then do things stand for the strict actualist with respect to the
first element of the possibilist's two-fold challenge? The
answer of course depends entirely on what one takes to be a
"systematically and philosophically satisfying" account of
the truth conditions for modal assertions like
 (2).
 Possibilists (and their haecceitist and eliminitivist counterparts)
clearly consider preservation of Tarski-style compositionality
(formalized in an appropriate notion of an *intended*
interpretation) to be essential for meeting this element of the
challenge: one must ultimately be able to ground the truth of complex
modal statements in the modal properties of (perhaps nonactual,
perhaps abstract) individuals. But it is not at all clear that the
strict actualist must agree. They might well rather simply reject this
demand and insist instead that
 (2)
 and its like stand on their own and do not need the sort of grounding
that possibilists insist upon. Since this is simply a consequence of
the rejection of *possibilia* and their actualist proxies, from
the strict actualist perspective, it is a philosophically satisfying
result. The apparent loss of Tarski-style compositionality is, for the
strict actualist, a price worth
 paying.[71]
 From this perspective, possible world semantics can with some
justification be seen as nothing more than a useful but ontologically
inert formal instrument, an evocative but ultimately dispensable
philosophical
 heuristic.[72]
 In that case, the only serious challenge that possibilism raises for
the strict actualist is the formulation of an actualistically kosher
quantified modal *deductive system* whose basic principles
intuitively reflect strict actualist sensibilities. We turn now to two
of the best known accounts. (We will follow common practice and refer
to a deductive system with or without a corresponding formal model
theory as a
 logic.[73])



#### 4.3.1 Prior's Quantified Modal Logic \(Q\)



It was Arthur Prior who, in 1956, first proved that
**BF** is a theorem of *SQML* (without identity).
Clearly aware of its possibilist and necessitist implications, Prior
sought to develop a logic consistent with strict actualism and, hence,
a logic in which **BF**, **CBF**, and
\(\Box\)**N** all fail. The result was his deductive
system \(Q\). The first installment--the propositional component
(Prior 1957: chs. IV and
 V[74])--arrived
 only a year later; quantification theory with identity was added a
decade later (Prior 1968b, 1968c).



Recall that the central intuition driving strict actualism is that,
necessarily, a singular proposition supervenes on the individuals it
is about--call these individuals the *subjects* of the
proposition--and, hence, that, necessarily, there is simply no
information, no singular propositions, about things that fail to
exist; if \(p\) is a singular proposition about \(a,\) then,
necessarily, \(p\) exists only if \(a\) does. Conversely, necessarily,
if all the individuals \(p\) is about exist, then so does \(p\). Prior
expressed this connection between singular propositions and their
subjects in terms of formulas and their constituent singular terms,
i.e., their constituent individual constants and free variables. And,
in formulating \(Q\), instead of the existence or nonexistence of
propositions, Prior typically spoke of the "statability"
or "unstatability" of the formulas that express
 them.[75]
 Specifically, let \(\varphi\) be a formula and \(p\_\varphi\) the
singular proposition \(\varphi\) expresses. Intuitively, then, for
Prior, \(\varphi\) is *statable*, or *formulable*,
*at* a world \(w\) if and only if \(p\_\varphi\) exists at \(w\)
and, hence, if and only if all of \(p\_\varphi\)'s
subjects exist at \(w\).
Thus, where \(a\_\tau\) is the denotation of a singular term
\(\tau\):



**S**:
A formula \(\varphi\) containing exactly the singular terms
\(\tau\_1,\ldots,\tau\_n\) is *statable* at a world \(w\) if and
only if \(a\_{\tau\_1},\ldots,a\_{\tau\_n}\) all exist at \(w\).


Hence, in particular, if any of the \(a\_{\tau\_i}\) fails to exist at
\(w,\) then \(\varphi\) is *unstatable* there.



So far, there is little for any strict actualist to take issue with in
Prior's framing of the issues here. However, Prior makes a
critical inference about unstatability that distinguishes his logic
starkly from other varieties of strict actualism (call it
Prior's *Gap Principle*):



**Gap**:
If a statement \(\varphi\) is unstatable at a world \(w,\) then
\(\varphi\) is neither true nor false at \(w.\)


The logical implications of **Gap** are dramatic. Note
first that the usual interdefinability of necessity and possibility
fails. To see this for the particular case of *SQML*'s
 \(\Diamond\textbf{Def}\),
 consider, say, the obvious truth that Bergoglio is not a subatomic
particle--say, a proton, \(\neg \sfP\sfb\). Since Bergoglio is a
contingent being, there are worlds where he doesn't exist.
Hence, there are worlds where \(\neg \sfP\sfb\) fails to be statable
and thus, by **Gap**, where it is neither true nor false
and thus in particular where it is not true. It follows that \(\neg
\sfP\sfb\) is not necessary, \(\neg\Box\neg \sfP\sfb.\) But,
obviously, \(\Diamond \sfP\sfb\) does not follow; assuming that
Bergoglio is essentially human, it is not possible that he is a
proton. So \(\Diamond\textbf{Def}\) will not do. By the same token, if
Bergoglio is essentially human, it is impossible that he fail to be
human, \(\neg\Diamond\neg \sfH\sfb\). For \(\neg \sfH\sfb\) is
obviously not true in worlds where he exists, and in worlds where he
doesn't, it is not statable and hence, by **Gap**
again, neither true nor false and thus again not true. But \(\Box
\sfH\sfb\) does not follow; for \(\sfH\sfb\) is also not statable,
hence not true, in Bergoglio-free worlds. So necessity cannot be
defined as usual in terms of possibility--impossible falsehood
does not imply necessary truth. Rather, necessity is impossible
falsehood *plus* necessary statability. Otherwise put: if a
statement \(\varphi\) couldn't be false, then in order for it to
be necessarily true, the proposition it expresses must necessarily
*exist*.



To capture this implication of **Gap** formally, and to
axiomatize the logic of strict actualism (under **Gap**)
generally, Prior makes basically two changes to the language
\(\scrL\_\Box\)--call this language \(\scrL\_{Q}\). First, for
reasons that will become evident shortly, he takes \(\Diamond\) rather
than \(\Box\) as a primitive operator. Secondly, he adds a new
sentential operator \(\text S\) for necessary statability
("*n*-statability") to \(\scrL\_\Box\). It is also
useful to think of impossible falsehood as a sort of *weak
necessity* signified by its own operator:



\([?]\)**Def**:
\([?]\varphi \eqdf \neg\Diamond\neg\varphi\)


Necessity proper--or *strong* necessity--is then
definable in \(\scrL\_Q\) as indicated in terms of weak necessity and
*n*-statability:



\(\Box\)**Def**\(\_Q\):
\(\Box\varphi \eqdf \rS \varphi \land [?]\varphi\)


*Axiom Schemas for Necessary Statability*



Prior's first axiom for *n*-statability is that it is not
a property a statement \(\varphi\) could lack if it could have it all;
otherwise put: if the proposition that \(\varphi\) expresses
*could* exist necessarily, then it *does* exist
necessarily:



**S1**:
\(\Diamond\rS \varphi \to \rS \varphi\)


Prior's next two axioms for *n*-statability express
principle **S** above. The first captures the
left-to-right direction that, intuitively, expresses the supervenience
of propositions on their subjects: a statement \(\varphi\) is
*n*-statable only if the things referred to in \(\varphi\) are
necessary beings. Prior could in fact express that the object
signified by an individual constant \(\sfa\) is necessary in the usual
fashion as \(\Box\exists \sfx\,\sfx=\sfa\) but, given
\(\Box\)**Def**\(\_Q\), this unpacks rather cumbersomely
to \(\rS \exists \sfx\,\sfx=\sfa \land [?]\exists
\sfx\,\sfx=\sfa\). However, as we will see, in the context of \(Q\)
this is equivalent simply to \(\rS \exists \sfx\,\sfx=\sfa.\) For
convenience, Prior shortens this to \(\rS \sfa\); more
 generally:[76]



**SDef**:
\(\rS \tau =\_{\textit df} \rS \exists\nu\,\nu=\tau,\) where
\(\nu\) is a variable distinct from \(\tau.\)


Now say that a term \(\tau\) occurring in a formula \(\varphi\) is
*free in* \(\varphi\) if \(\tau\) is either an individual
constant or a variable with a free occurrence in \(\varphi,\) and say
that \(\varphi\) is *singular* if some term is free in it and
*wholly general* otherwise. Given this, Prior axiomatizes the
left-to-right direction of principle **S** as
follows:



**S2**:
\(\rS \varphi \to \rS \tau,\) for terms \(\tau\) that are in free
in \(\varphi\).


That is, a singular assertion \(\varphi\) is *n*-statable only
if each individual that it names is a necessary being.



Prior's final axiom for *n*-statability is essentially
the converse: if all the individuals that \(\varphi\) names are
necessary, then \(\varphi\) is *n*-statable. A simple
convention enables us clearly to express this axiom (and others below)
schematically. Let \(\theta\) be any formula of \(\scrL\_{Q}\); then,
for any \(n \ge 0,\)



**SDef**\*:
\(\rS \tau\_1\ldots\tau\_n \to \theta =\_{\textit df} \rS \tau\_1 \to
(\ldots \to (\rS \tau\_n \to \theta)\ldots)\)


It is useful to note that, under this convention, when \(n \gt 0,\)
\(\rS \tau\_1\ldots\tau\_n \to \theta\) is equivalent to \((\rS
\tau\_1\,\land\,\ldots\,\land\,\rS \tau\_n) \to \theta,\) and that, when
\(n=0,\) \(\rS \tau\_1\ldots\tau\_n \to \theta\) is just \(\theta.\)[77]
 Given this, we can express Prior's final axiom for
*n*-statability as follows:



**S3**:
\(\rS \tau\_1\ldots\tau\_n \to \rS \varphi,\) where \(\tau\_1,
\ldots, \tau\_n\) are all of the terms that are free in
\(\varphi\)


Note in particular that it follows from
**S3**--specifically, from the case
\(n=0\)--that every wholly general formula is
*n*-statable.



By **S2** and **S3** together, then, a
formula \(\varphi\) is *n*-statable if and only if all of its
component free terms are. Or, more informally put: the proposition
that \(\varphi\) expresses exists necessarily if and only if all of
its subjects do.



Given the axioms for \(\text S\) we can lay out the rest of
Prior's system \(Q\). For its non-modal foundations, \(Q\) takes
every propositional tautology (not just its closures, as in
*KQML*) to be an axiom and follows *SQML* in adopting
standard classical axioms for quantification and identity. However,
the non-modal fragment of the system itself is not *quite*
classical. For reasons we will discuss below, it includes a modified
version of *modus ponens* that renders some classical
first-order validities unprovable.


* *Propositional Axiom Schema*
  

**Taut**
* *Quantificational Axiom Schemas*
  

**Q1**,
 **Q2**, and
 **Q3**
* *Identity Axiom Schemas*
  

**Id1**,
 **Id2**
* *Rules of Inference*

**MP**\(\_Q\):
\(\rS \tau\_1\ldots\tau\_n \to \psi\) follows from \(\varphi\) and
\(\varphi \to \psi,\) where \(\tau\_1, \ldots, \tau\_n\) are all of the
terms that are free in \(\varphi\) but not in \(\psi.\)
**Gen**:
\(\forall\nu\psi\) follows from \(\psi\)



\(Q\) also has its own version of
 **Gen**\*,
 which, recall, allows us to universally generalize on constants as if
they were free variables:


* **Gen**\(\astQ\):
Suppose \(\psi\) is a formula of \(\scrL\_{Q}\) that contains an
individual constant \(\kappa\) of \(\scrL\_{Q}\) and that \(\nu\) is a
variable that is substitutable for \(\kappa\) in \(\psi\). Then if
\(\psi\) is a theorem of \(Q\), so is
\(\forall\nu\psi^\kappa\_\nu.\)



\(Q\) preserves the classical assumption that all names denote
something: for any term \(\tau\) other than the variable \(\nu,\)
\(\exists \nu\,\nu =\tau\) is a theorem; in the special case where
\(\nu\) is \(\sfy\) and \(\tau\) is \(\sfa\):




|  |  |  |
| --- | --- | --- |
| 1. | \(\forall \sfy\,\neg \sfy=\sfa \to \neg \sfa=\sfa\) | **Q2** |
| 2. | \(\forall \sfy\,\neg \sfy=\sfa\ \to \neg \sfa=\sfa) \to (
\sfa=\sfa \to \neg\forall \sfy\,\neg \sfy=\sfa)\) | **Taut** |
| 3. | \(\sfa=\sfa \to \neg\forall \sfy\,\neg \sfy=\sfa\) | 1, 2, **MP**\(\_{Q}\) |
| 4. | \(\sfa=\sfa \to \exists \sfy\, \sfy=\sfa\) | 3, [?]**Def** |
| 5. | \(\sfa=\sfa\) | **Id1** |
| 6. | \(\exists \sfy\, \sfy=\sfa\) | 4, 5, **MP**\(\_{Q}\) |



However, because of the qualification in **MP**\(\_{Q}\),
the intuitively weaker classical theorem that something
exists--more exactly, that something is self-identical, \(\exists
\sfy\, \sfy=\sfy\)--is *not* provable. The usual proof in
classical predicate logic runs along the above lines:




|  |  |  |
| --- | --- | --- |
| 1. | \(\forall \sfy\,\neg \sfy=\sfy\ \to \neg \sfa=\sfa\) | **Q2** |
| 2. | \((\forall \sfy\,\neg \sfy=\sfy\ \to \neg \sfa=\sfa) \to
(\sfa=\sfa \to \neg\forall \sfy\,\neg \sfy=\sfy)\) | **Taut** |
| 3. | \(\sfa=\sfa \to \neg\forall \sfy\,\neg \sfy=\sfy\) | 1, 2, **MP**\(\_{Q}\) |
| 4. | \(\sfa=\sfa \to \exists \sfy\, \sfy=\sfy\) | 3, \(\exists\)**Def** |
| 5. | \(\sfa=\sfa\) | **Id1** |



In classical logic, of course, \(\exists \sfy\, \sfy=\sfy\) is
immediate from lines 4 and 5 by **MP**. However,
**MP**\(\_{Q}\) only yields




|  |  |  |
| --- | --- | --- |
| 6. | \(\rS \sfa \to \exists \sfy\, \sfy=\sfy\) | 4, 5, **MP**\(\_{Q}\) |



That is, in \(Q\), that something exists only follows from the
self-identity of some particular object *if* that object is a
necessary being, an object that exists in every possible world. We
will be able to appreciate Prior's motivations here (explained
 below)
 once *Q*'s underlying propositional modal logic is in
place.



\(Q\)'s propositional modal axiom schemas are structurally
identical to those of *SQML*, but with weak necessity in place
of strong
 necessity:[78]


* *Modal Axiom Schemas*

**K***Q*:
\([?](\varphi \to \psi ) \to ([?]\varphi \to [?]\psi
)\)
**T***Q*:
\([?]\varphi \to \varphi\)
**5***Q*:
\(\Diamond \varphi \to [?]\Diamond \varphi\)



Likewise its rule of necessitation; because some logical truths are
not *n*-statable and, hence, not true in every
world--notably, singular logical truths like *Prior, if a
logician, is a logician*, \(\sfL\sfp \to \sfL\sfp\)--we can
only infer that an arbitrary logical truth is weakly necessary, false
in no world:


* *Rule of Inference*

**Nec**\(\_{Q}\):
\([?]\psi\) follows from \(\psi.\)


#### 4.3.2 \(Q\), Contingency, and the Controversial Logical Truths of *SQML*



Prior touted \(Q\) as a "modal logic for contingent
beings" (1957: 50). It is therefore a rather awkward theorem of
\(Q\) that there are no contingent beings in the strong, actualist
sense of
 **Cont**\(\_{\exists}\):
 since, for any term \(\tau\) other than \(\sfy,\) \(\exists \sfy\,
\sfy=\tau\) is a theorem of \(Q\), by **Nec**\(\_{Q}\)
(and
 \([?]\)**Def**)
 it is weakly necessary, \(\neg\Diamond\neg\exists \sfy\,\sfy=\tau,\)
and the problematic theorem



\(\forall \sfx\, \neg\textsf{Contingent}\_{\exists}(\sfx)\)



follows directly by **Gen**\(\astQ\) and
**Cont**\(\_\exists\). However, Prior (1967: 150) argues
that there is still a robust way of expressing the intuitive
contingency of an object \(x,\) viz., that \(x\)'s
existence is not (strongly)
necessary, i.e. (where \(\sfx\) denotes \(x\)), that \(\exists \sfy\,
\sfy=\sfx\) is not true in all worlds: \(\neg\Box\exists \sfy\,
\sfy=\sfx\). But in the context of \(Q\) this is equivalent to saying
simply that \(\exists \sfy\, \sfy=\sfx\) is not *n*-statable,
\(\neg\rS \exists \sfy\, \sfy=\sfx\). Accordingly, given
 **SDef**,
 we have:



**Cont**\(\_Q\):
\(\textsf{Contingent}\_{Q}(\sfx) \eqdf \neg\rS \sfx\)


As noted above, for Prior, the prospect of logical truths that are not
necessarily statable renders the general principle of necessitation
unsound. Rather, strong necessity only follows for those logical
truths that are *n*-statable; this is usefully expressed as a
derived rule of
 inference:[79]



**Nec**\(\astQ\):
\(\rS \psi \to \Box\psi\) follows from \(\psi\).


Since wholly general formulas are all *n*-statable, another
important derived rule is immediate:



**Nec**\(^{\*\*}\_{Q}\):
\(\Box\psi\) follows from \(\psi,\) if \(\psi\) is wholly
general.


Of course, if *in fact* there is a (strongly) necessary
being--Allah, say--then, by **S3**, some
singular logical truths--e.g., *something exists if Allah
exists*, \(\exists \sfx\,\sfx=\sfa \to \exists
\sfx\,\sfx=\sfx\)--will be *n*-statable and, hence,
(strongly) necessary. But no such truths are *provably*
*n*-statable in \(Q\), as \(\rS \tau\) is not a theorem of
\(Q\), for any term \(\tau\) and, hence, by **S2**,
neither is \(\rS \varphi\) for any formula \(\varphi\) in which
\(\tau\) is free. Thus, not only are all wholly general theorems
provably necessary as per **Nec**\(^{\*\*}\_{Q},\) they are
the only provably necessary logical truths of \(Q\).



The unprovability of \(\rS \tau\) in \(Q\) and, hence, more generally,
\(Q\)'s inability to prove the existence of any necessary beings
is the key difference between \(Q\) and *SQML* and, more
specifically, it is what justifies the inapplicability of the full
necessitation principle **Nec** to formulas containing
free terms, since some of those terms might refer to contingent
beings. This is, in particular, the key to blocking the controversial
theorems of *SQML*, as their proofs all depend essentially on
such an application of
 **Nec**.[80]
 Indeed, \(Q\) is essentially just *SQML* shorn of its
necessitism. If, as Prior (1967: 155) observed, we restore it by
ruling out the prospect of contingent beings explicitly,



**N**\(\_Q\):
\(\forall \sfx\, \rS \sfx\)


\(Q\) simply collapses into
 *SQML*.[81]
 (The reader can quickly verify that **N**\(\_Q\) and the
necessitist principle \(\Box\)**N** are equivalent in
\(Q\).) More exactly put: for any formula \(\varphi\) of
\(\scrL\_\Box,\) \(\varphi\) is a theorem of *SQML* if and only
if it is a theorem of \(Q\) + **N**\(\_{Q}\) and hence if
and only if \(\forall \sfx\rS\sfx \to \varphi\) is a theorem of \(Q\).
In particular, both **BF** and **CBF** fall
out as entirely unproblematic theorems under the assumption of
necessitism:



**BF**\(\_Q\):
\(\vdash\_{Q}\forall \sfx\rS\sfx \to (\Diamond\exists \nu \varphi
\to \exists \nu \Diamond\varphi)\)
**CBF**\(\_Q\):
\(\vdash\_{Q}\forall \sfx\rS\sfx \to (\exists\nu\Diamond \varphi
\to \Diamond\exists\nu\varphi)\)


It is important to emphasize the profound philosophical difference
between Prior's \(Q\) and the necessitist's *SQML*
that is reflected in the preceding observations. As we've seen,
\(Q\) was deeply motivated by Prior's strict actualism; this is
most clearly seen in his introduction of the *n*-statability
operator \(\rS\) and the axioms
**S1**-**S3** expressing the necessary
ontological dependence of propositions on their subjects. But, in
stark contrast to the necessitist's *SQML*, Prior's
metaphysical predilections are not baked into \(Q\)--unlike the
necessitist, he has only made them *consistent* with his logic;
his axioms for \(\rS\) and his distinction between strong and weak
necessity only have purchase if **N**\(\_{Q}\) is
explicitly denied. For Prior, both necessitism and strict actualism
are substantive philosophical theses and, hence, the choice between
them--i.e., whether or not to adopt **N**\(\_{Q}\) as
a proper axiom of a philosophical *theory*--is left as a
matter for pure metaphysics, not logic, to decide.



This sentiment also lies behind Prior's restricted version
**MP**\(\_{Q}\) of **MP**. As noted above,
**MP**\(\_{Q}\) prevents the derivation of \(\exists
\sfy\,\sfy=\sfy\) from the likes of \(\exists \sfy\,\sfy=\sfa\). But
if \(\exists \sfy\,\sfy=\sfy\) *were* provable, then, since it
is wholly general, it would follow by
**Nec**\(^{\*\*}\_{Q}\) that it is strongly necessary,
\(\Box\exists \sfy\,\sfy=\sfy,\) that is, informally put, it would be
a theorem of \(Q\) that, in every possible world, there is at least
one thing. Again, though, that there couldn't be an empty world,
that it couldn't have been the case that nothing exists, is a
substantive metaphysical thesis that logic is better off leaving
undecided. Prior's restriction in **MP**\(\_{Q}\)
ensures that this is in fact the case in \(Q\).


### 4.4 Perspectivalism



Despite the fact that Prior was able to provide a notion of
contingency--contingency\(\_Q\)--for his logic \(Q\), it is
difficult to downplay the awkwardness of the fact that the most
natural expression of
contingency--contingency[?]--is unavailable
to him; it is, recall, a theorem of \(Q\) that, necessarily, there are
no contingent[?] beings--\(\Box\neg\exists \sfx\,
\Diamond\neg \exists \sfy\,\sfx=\sfy\). Thus, although seemingly
well-motivated by his strict actualist metaphysics, the fact that
Prior cannot consistently assert that he himself might not have
existed, that he himself is a contingent[?] being,
arguably throws strict actualism into doubt: as Deutsch (1990:
92-93) observes,



> 
> 
> ...surely there is a sense in which "Prior exists"
> might have been *false*; [but] there is no way to express this
> in Prior's system.
> 
> 
> 



Robert Adams (1974, 1981) developed an important and influential
strict actualist account of possible worlds that provides an intuitive
notion of modal truth on which Prior *is* a
contingent[?] being and which, more generally, provides
a strong justification for modifications that temper at least some of
the more severe elements of Prior's logic.



Because of the centrality of contingency[?] in
Adams' account, it will be useful to introduce a distinct
predicate \(\sfE!\) to express the property of being identical with
something:



**DefE!**:
\(\sfE!\tau \eqdf \exists \nu\,\nu=\tau,\) where \(\tau\) is any
term other than \(\nu\)


The contingency[?] of an object \(x\) is then simply
expressed as \(\Diamond\neg \sfE!\sfx\).



For actualists like Adams, of course, the property of being identical
with something is necessarily coextensive with the property of
existing and, hence, \(\sfE!\) is often called the *existence*
predicate. It is important to note the stark contrast between the
existence predicate \(\sfE!\) and the actuality predicate \(\sfA!\).
As we have seen, \(\sfA!\) is a non-logical predicate introduced in
the context of the possibilism-actualism debate to (purportedly)
signify a distinguished non-logical property: the property (however
understood) that, according to the possibilist, distinguishes the
likes of us (and *abstracta* like the numbers) from
*possibilia*. \(\sfE!,\) by contrast, is a purely logical
predicate that does not presuppose any particular metaphysical
baggage--actualists and possibilists alike (at least, those that
accept classical logic) will agree that, as a simple matter of logic,
necessarily, everything is identical to something, \(\Box\forall \sfx
\sfE!\sfx.\) However, given
 (as argued above)
 that possibilists are also committed to necessitism, \(\Box\forall
\sfx\Box \sfE!\sfx,\) they will disagree with the typical actualist
over whether or not there are any contingent[?] beings,
i.e., over whether or not \(\exists \sfx\, \Diamond\neg
\sfE!\sfx.\)


#### 4.4.1 World Stories



The most fundamental notion in Adams' account is that of a
*proposition*, by which he means an abstract entity that is
*expressed* by a declarative sentence and which can be either
true or false, depending on how the world is. (As in the discussion of
Prior, we will continue to use logical formulas ambiguously, sometimes
as names for themselves, sometimes as names for the propositions they
are meant to express, trusting context to disambiguate.) Particularly
important for Adams' account is the notion of a
*singular* proposition
 introduced in SS4.3,
 i.e., a proposition that "involves or refers to an individual
directly" (1981: 6). Crucially, as a strict actualist, Adams
follows Prior in holding that a singular proposition is ontologically
dependent on the individuals it is about--the
"subjects" of the proposition--and, hence, that,
necessarily, a singular proposition exists if and only if all of its
subjects exist. Beyond this, Adams does not provide a rigorous theory
of propositions but, rather, takes the notion to be sufficiently
well-understood for his purposes.



Adams' account centers around his notion of a *world
story*, his own "actualistic treatment of possible
worlds" (1981: 21). The intuitive idea is that a world story is
a complete description of things as they are or could be. Adams'
(1974) initially spells this idea out as follows. Say that a set \(S\)
of propositions is *maximal* if for any proposition \(p,\)
\(S\) contains either \(p\) or its negation \(\neg
 p,\)[82]
 and that \(S\) is *consistent* if it is possible that all the
members of \(S\) be (jointly) true; \(S,\) then, is a *world
story* if it is both maximal and consistent, and a proposition
\(p\) is *true in* a world story *w* just in case \(p\)
is a member of \(w\). And a world story is *true*, or
*obtains*, if all the propositions that are true in it are, in
fact, true. (Henceforth in this section we will use
"world" and "possible world" as synonyms for
"world story"; and by "the actual world" we
will mean the true world
 story.[83])



However, Adams came to realize that his 1974 definition did not
properly reflect strict actualist principles and subsequently (1981)
offered a more subtle
 definition.[84]
 According to strict actualism, no singular propositions are in any
sense "about" mere *possibilia* and, hence, there
are no such propositions in the actual world, i.e., the true world
story. For Adams, possible worlds generally ought to reflect this fact
about the actual world. That is, following Prior, a singular
proposition \(p\) about some (actually existing) individuals
\(a\_1,\ldots,a\_n\) should be true in a world \(w\) only if those
individuals--hence, \(p\) itself--would have existed if
\(w\) had obtained. As Prior would put it: a proposition can be true
in a world \(w\) only if is *statable* in \(w\). Thus, just as
there are in the actual world no singular propositions about
individuals that could have existed but do not, likewise, in every
other world \(w\) there should be no singular propositions about
(actual) individuals that would not have existed had \(w\) obtained.
To reflect this in his conception of worlds, Adams alters his
definition so that a set \(S\) of propositions is a world only if it
*could be* be maximal with respect to the *actually
existing* propositions that would exist if it obtained. As Adams
puts it:



> 
> 
> Intuitively, a world-story should be complete with respect to singular
> propositions about those actual individuals that would still be actual
> if all the propositions in the story were true, and should contain no
> singular propositions at all about those actual individuals that would
> not exist in that case. (1981: 22)
> 
> 
> 



A world, then, is a set of propositions that could be both maximal in
this sense and such that all of its members are true. Equivalently
put: a set of propositions is a world just in case, possibly, its
members are exactly the true propositions that *actually*
 exist.[85]
 It follows in particular that the proposition \(\neg \sfE!\sfa\) that
Adams doesn't exist is not true in any world story; for, by
strict actualism, since it is a singular proposition about Adams, it
couldn't be true without Adams existing--in which case, of
course, it would be false. Very much unlike Prior, however, it does
*not* follow for Adams that his non-existence is not possible,
\(\neg\Diamond\neg \sfE!\sfa\).


#### 4.4.2 Truth *at* a World



Adams' break with Prior on this point is semantic, not
metaphysical. Specifically, over and above the notion of truth
*in* a world defined above, Adams identifies a second notion of
truth relative to a world that he calls truth *at* a world. The
difference between the two is one of *perspective*, a
difference in what we might call the *evaluative standpoint*.
The propositions that are true *in* a world \(w,\) recall, are
those that are *members* of \(w\); hence, intuitively, they
would all have existed had \(w\) obtained. They are thus the
propositions that can be recognized as true from a standpoint
"within" \(w\)--since they would have existed if
\(w\) had obtained, an agent with sufficient cognitive powers (we can
imagine) that might have existed if \(w\) had obtained could have
grasped and evaluated them. However--and this is Adams'
crucial insight--from her perspective in \(w,\) that same agent
can also evaluate propositions in \(w\) with respect to *other*
worlds \(u\)--notably, for any individual \(x\) in \(w,\) she can
see that the proposition \(\neg \sfE!\sfx\) that \(x\) doesn't
exist rightly characterizes any \(x\)-free world \(u,\) the absence of
the proposition in question *in* \(u\) notwithstanding. The
proposition is thus, from the standpoint of \(w,\) said to be true
*at*, but not *in*, \(u\). As Adams, here in the actual
world, puts it with regard to the proposition \(\neg \sfE!\sfa\) that
he doesn't exist:



> 
> 
> A world story that includes no singular proposition about me ...
> represents my possible non-existence, not by including the proposition
> that I do not exist but simply by omitting me. That I would not exist
> if all the propositions it includes ... were true is not a fact
> internal to the world that it describes, but an observation that we
> make from our vantage point in the actual world, about the relation of
> that world story to an individual of the actual world. (1981: 22)
> 
> 
> 



If, therefore, we take a proposition to be possible if it is true
*at*, rather than *in*, a world, since the proposition
\(\neg \sfE!\sfa\) that Adams doesn't exist is true at some
worlds, his non-existence is possible after all, \(\Diamond\neg
\sfE!\sfa,\) contrary to Prior. This change in perspective thus paves
the way for a logic that is consistent with the existence of
contingent[?] beings. Recall also that Adams is a
 serious actualist:
 to have a property is to exist. Hence, at worlds where a given object
fails to exist, it has no properties. Thus, in particular, at a world
\(w\) where Prior fails to exist, he also fails to be a logician, that
is, it is *true at* \(w\) that he is not a logician, \(\neg
\sfL\sfp\). Hence, the conditional proposition that he is, if a
logician, a logician--\(\sfL\sfp \to \sfL\sfp\)--is also
true at \(w\) and so true at *all* worlds, not just those in
which he exists. It is therefore not just
 weakly necessary,
 i.e., unable to be false, \(\neg\Diamond\neg(\sfL\sfp \to
\sfL\sfp),\) but strongly necessary, \(\Box(\sfL\sfp \to \sfL\sfp).\)
By taking truth *at*, rather than *in*, a world to be
our guiding semantic intuition, then, it appears that, without
compromising the metaphysics of strict actualism, we can restore at
least some of the familiar logical connections between (im)possibility
and necessity that were lost in Prior's \(Q\) and, in
particular, restore the modal status of at least some logical truths
to full necessity.



Two things are worth mentioning here. First, it is important to note
that, while most philosophers will welcome these results, there is
still a clear *choice* that is being made here. Adams has
identified two intuitive notions of truth with respect to a world
\(w\) within a strict actualist metaphysics that, as we've just
seen, appear to yield dramatically different logical truths. Both,
however, are coherent. Prior built his logic \(Q\) on the one; Adams
proposes to build one on the other--at least, in part (see
 SS4.4.4
 below).



Second, it needs to be emphasized that these are *intuitive*
semantic notions only. For not only are there well-known, formidable
challenges to the notion of a world story that he does not
 address,[86]
 because Adams is a strict actualist, a formal, compositional
semantics is simply unavailable to him, for the reasons noted in the
 introduction to SS4.3.
 Hence, Adams' talk of propositions being true in or at possible
worlds cannot themselves be considered literally true but, as with
similar talk from Prior, must simply be viewed as an evocative
heuristic whose primary purpose is to motivate the basic logical
principles underlying his variety of strict actualism.


#### 4.4.3 A Quantified Modal Logic for Perspectivalists



Unlike Prior, Adams does not develop a rigorous deductive system.
Rather, in his 1981, he simply identifies a number of informal
semantic principles (labeled C1-C9). However, a clear set of
axioms and rules requiring only a bit of supplementation can in fact
be derived from those
 principles.[87]
 We will call the resulting system *A*.



Adams assumes classical propositional logic (principle (C3)), which
can be incorporated into *A* via the basic propositional
schemas **P1**-**P3** and the rule
**MP** of *modus ponens*. For the rest, it is
illuminating to start with the axioms of *SQML*--albeit
with both modal operators as primitive--and then focus on
additions and modifications required by Adams' informal
principles.



Adams' first modally significant principle (C2) (1981: 23) is
one expressing
 serious actualism,
 the principle that an individual cannot have a property or stand in a
relation without existing. For reasons that will become clear below,
we will use a somewhat more general schema than schema
 **SA**
 introduced in the context of Kripke's system
 *KQML*:[88]



**GSA**:
\(\pi\tau\_1\ldots\tau\_n \to \sfE!\tau\_i,\) for terms \(\tau\_1,
\ldots, \tau\_n\) (\(1 \leq i \leq n\))


Thus, in particular, for any property \(F,\) by **GSA**
and **Nec** we have that, necessarily, Adams has \(F\)
only if he exists: \(\Box(\sfF\sfa \to \sfE!\sfa)\). The problem here
is that Adams (reasonably) considers identity to be a relation;
identity statements are thus simply a variety of atomic formula and,
hence, it is also an instance of **GSA** that Adams is
self-identical only if he exists, \(\sfa=\sfa \to \sfE!\sfa\). As
Adams accepts the usual
 laws of identity
**Id1** and **Id2**, we appear to have the
following simple argument to Adams' necessary existence:




|  |  |  |  |
| --- | --- | --- | --- |
| 1. | \(\sfa=\sfa\) |  | **Id1** |
| 2. | \(\sfa=\sfa \to \sfE!\sfa\) |  | **GSA** |
| 3. | \(\sfE!\sfa\) |  | 1, 2 **MP** |
| 4. | \(\Box \sfE!\sfa\) |  | 3 **Nec** |



However, if identity is a relation, then, given serious actualism, one
is not self-identical at worlds in which one fails to exist. This
might suggest restricting necessitation to theorems provable without
any instance of **Id1**, which would block not only the
inference from line 3 to line 4 but also the
 proof of the more general necessitist principle
 \(\Box\)**N** that, necessarily, everything necessarily
exists, \(\Box\forall \sfx \Box\exists \sfy\, \sfy=\sfx\). However,
that would still not be enough for, as we saw above, necessitism also
follows immediately from the instance
 **CBF**\*
 of the converse Barcan formula, whose
 proof
 involves no instance of **Id1**.



Adams (1981: 25) lays the blame for such problems on the classical
quantification schema **Q2**. Following Fine (1978:
SS3), he adopts the solution mentioned above in our discussion of
*KQML*, namely, replacing **Q2** with its free
logical counterpart (which we rewrite here with \(\sfE!\)):



**FQ2**:
\(\forall\nu\varphi \to (\sfE!\tau \to \varphi^\nu\_\tau),\) where
\(\tau\) is any term that is substitutable for \(\nu\) in
\(\varphi\)


It is a simple exercise to show that, with only **FQ2**
at one's disposal instead of **Q2**, one can only
prove that **CBF** holds (innocuously) under the
assumption that everything exists necessarily, which of course the
typical actualist roundly rejects:



**CBF*A***:
\(\vdash\_{A} \forall \sfx \Box \sfE!\sfx \to (\exists\nu\Diamond
\varphi \to \Diamond\exists\nu\varphi)\)


\(\Box \sfE!\sfa\) and \(\Box\)**N** can then be blocked
simply by replacing **Id1** with its generalization:



\(\boldsymbol{\forall}\)**Id1**:
\(\forall\nu\,\nu = \nu\)


which, together with **FQ2**, only allows one to prove
modally innocuous *conditional* identities of the form
\(\sfE!\tau \to \tau=\tau.\)



In general, the strategy of giving up the classical schema
**Q2** in favor of its free counterpart *solely*
to make room for contingent beings in one's quantified modal
logic is a bit fraught for the actualist. For, according to actualism,
there are no *possibilia* and, hence, constants and free
variables cannot but denote actually existing things. Consequently,
instances of both **Q2** and **Id1** are
still logical truths; they are simply logical truths of the actual
world that fail at some possible worlds. Ideally, then, as Adams puts
(1981: 30), in a proper quantified modal logic, they should be
"contingent theorems", that is, provable but not subject
to necessitation. This idea can in fact be preserved in *A*
simply by adding a schema expressing the actualist view that all terms
denote existing things:



**E!**\(\_{A}\):
\(\sfE!\tau,\) for all terms \(\tau\)


and modifying necessitation accordingly:



**Nec**\*:
\(\Box\psi\) follows from \(\psi,\) so long as \(\psi\) is
provable without any instance of **E!**\(\_{A}\).


Given **E!***A* and a bit of
propositional logic, all instances of **Q2** follow
immediately from **FQ2**, and all instances of
**Id1** follow from **FQ2** and

\(\boldsymbol{\forall}\)**Id1**.[89]
 But, because of the restriction in **Nec**\*, their
necessitations do not.


##### Perspectivalism and *De Re* Modality



As we've seen, Adams' notion of truth at a world promises
to restore at least some of what was arguably lost in Prior's
\(Q\), notably the logical equivalence between necessary truth and
impossible falsehood:



\(\Box\Diamond\):
\(\Box\varphi \leftrightarrow \neg\Diamond\neg\varphi\)


But the gains are rather marginal, given the profound implications of
Adams' take on perspectivalism for propositional modal logic.
Matters here turn on Adams' understanding of what it is for a
*modal* proposition \(\Diamond\varphi\) or \(\Box\varphi\) to
be true at a possible world \(w\). Just as in the actual world, from a
standpoint within another possible world \(w,\) a proposition
\(\varphi\) that exists in \(w\) can be true *at* worlds where
it doesn't exist. \(\Diamond\varphi\)/\(\Box\varphi\) will thus
be true from a standpoint in \(w\) if, from that perspective,
\(\varphi\) is true at some/all worlds. But if \(\varphi\) does
*not* exist in \(w\)--if, for example, \(\varphi\) is
\(\neg \sfE!\sfa\) and \(w\) is Adams-free--then it is not
available to be evaluated from a standpoint within \(w\) and, hence,
from that standpoint, is neither true nor false at any world and,
hence, neither possible nor necessary. For Adams, this means that,
from *our* standpoint in the actual world, both
\(\neg\Diamond\varphi\) and \(\neg\Box\varphi\) are true at \(w\):



> 
> 
> [F]rom an actualist point of view...there *are* no
> possibilities or necessities *de re* about non-actual
> individuals. So if I were not an actual individual there would be none
> about me. The singular propositions that I exist and that I do not
> exist would not exist to have the logical properties, or enter into
> the relations with some or all world-stories, by virtue of which my
> existence or non-existence would be possible or necessary. I therefore
> say that "\(\Diamond\)(I exist)," "\(\Diamond\)~(I
> exist)," "\(\Box\)(I exist)," and "\(\Box\)~(I
> exist)" are all false, and their negations true, at worlds in
> which I do not exist. Neither my existence nor my non-existence would
> be possible or necessary if I did not exist. (1981: 29)
> 
> 
> 



Thus, according to Adams, *de re* modal propositions are
existence entailing: an (actually existing) modal proposition
\(\Box\varphi\) or \(\Diamond\varphi\) can be true at a world only if
it exists there and, hence, only if its subjects do as well. This idea
can be axiomatized by a modalized version of the serious actualism
principle **GSA** (see Adams' principles (C6) and
 (C7)[90]):



**MSA**:
\(\triangle\varphi \to \sfE!\tau,\) where \(\triangle\) is either
\(\Box\) or \(\Diamond\) and \(\tau\) is free in \(\varphi\).


An immediate consequence of **MSA** is to render many
instances of \(\Box\Diamond\) contingent. For, at an Adams-free world
\(w,\) \(\neg \sfE!\sfa,\) for example, is true so, by
**MSA** both \(\neg\Diamond\neg \sfE!\sfa\) and
\(\neg\Box\neg \sfE!\sfa\) are true at \(w\). So, given
**MSA**, it appears that the equivalence between
necessary truth and impossible falsehood holds at a world only for the
propositions that exist there, i.e., in Prior's terminology, the
propositions that are
 *statable*
 there. Because, for the strict actualist, necessarily, a proposition
exists if and only if all of its subjects exist, no new sentential
operator is needed to express propositional existence; a simple
counterpart to **SDef**\* will do:



**E!Def**\*:
\(\sfE!\varphi \to \theta =\_{\textit df} \sf\sfE!\tau\_1 \to
(\ldots \to (\sf\sfE!\tau\_n \to \theta)\ldots),\) where
\(\tau\_1,\ldots,\tau\_n\) are all of the terms that occur free in
\(\varphi\)


That is, \(\sfE!\varphi \to \theta\) says that \(\theta\) holds if all
of \(\varphi\)'s subjects
exist--hence, for the strict actualist, if the proposition
(expressed by) \(\varphi\) itself exists. Note that, when \(\varphi\)
is wholly general, \(n=0\) and so \(\sfE!\varphi \to \theta\) is just
\(\theta\).



Given **E!Def**\*, the observation above--that the
equivalence of necessity and impossible falsehood at a world only
holds for the propositions that exist there--can now be expressed
axiomatically in the schema:



\(\Box\Diamond\_A\):
\(\sfE!\varphi \to (\Box\varphi \leftrightarrow
\neg\Diamond\neg\varphi)\)


Much like the instances of **Q2**, then, because
\(\sfE!\tau\) is an axiom of *A* for any term \(\tau,\)
singular instances of \(\Box\Diamond\) fall out as contingent theorems
of *A*. By the observation above, wholly general instances of
\(\Box\Diamond\_A\) are simply instances of \(\Box\Diamond\) and,
hence, their necessitations are provable.


##### Implications for Necessitation, **4**, **B**, and **5**



**MSA**'s impact reverberates throughout the modal
propositional base of Adams' logic. Recall that the
truth-in/truth-at distinction appeared to have restored the necessity
of singular logical truths like *Prior, if a logician, is a
logician*, \(\sfL\sfp \to \sfL\sfp\). The addition of
**MSA**, however, leads quickly to necessitism, as it
becomes possible to reason to Prior's existence, \(\sfE!\sfp,\)
without any instances of **E!***A*:




|  |  |  |
| --- | --- | --- |
| 1. | \(\sfL\sfp \to \sfL\sfp\) | *PL* |
| 2. | \(\Box(\sfL\sfp \to \sfL\sfp)\) | 1, **Nec**\* |
| 3. | \(\Box(\sfL\sfp \to \sfL\sfp) \to \sfE!\sfp\) | **MSA** |
| 4. | \(\sfE!\sfp\) | 2,3, **MP** |
| 5. | \(\Box \sfE!\sfp\) | 4, **Nec**\* |
| 6. | \(\Box\forall \sfx\Box \sfE!\sfx\) | 5, **Gen**\*, **Nec**\* |



Clearly, as Adams (1981: 30) notes, a "suitable
restriction" on necessitation (beyond **Nec**\*) is
called for. Adams himself does not specify, but it is clear what it
must be. Singular logical truths like \(\sfL\sfp \to \sfL\sfp\) are
indeed true at all worlds and, hence, necessary--but only
contingently so; for, by **MSA**, \(\Box(\sfL\sfp \to
\sfL\sfp)\) is false at Prior-free worlds. More generally, just as the
equivalence of necessity and impossible falsehood only holds at a
world for the propositions that exist there, the necessity of a
proposition at a world will only hold for the propositions that exist
there. **Nec**\* thus requires a qualification similar to
\(\Box\Diamond\_A\)'s:



**Nec***A*:
\(\sfE!\psi \to \Box\psi\) follows from \(\psi,\) so long as
\(\psi\) is provable without any instance of
**E!***A*.


\(\sfL\sfp \to \sfL\sfp\) is now still provably necessary:




|  |  |  |
| --- | --- | --- |
| 1. | \(\sfL\sfp \to \sfL\sfp\) | *PL* |
| 2. | \(\sfE!\sfp \to \Box(\sfL\sfp \to \sfL\sfp)\) | 1, **Nec***A* |
| 3. | \(\sfE!\sfp\) | **E!***A* |
| 4. | \(\Box(\sfL\sfp \to \sfL\sfp)\) | 2, 3, **MP** |



But, because \(\Box(\sfL\sfp \to \sfL\sfp)\) depends essentially on
**E!***A*, *its* necessitation
is not provable. And if one soldiers on in an attempt to replicate the
reasoning in the above proof to derive necessitism:




|  |  |  |
| --- | --- | --- |
| 5. | \(\Box(\sfL\sfp \to \sfL\sfp) \to \sfE!\sfp\) | **MSA** |
| 6. | \(\sfE!\sfp\) | 4,5, **MP** |



one can proceed no further, as the proof of \(\sfE!\sfp\) in line 6
depends on \(\sfE!\sfp\) *qua* instance of
**E!***A* in line 3.



The modal schemas
 **K**
 and
 **T**,
 with their simple modalities, are unaffected by **MSA**
but not so for the schemas
 **4**,
 **B**, and
 **5**,
 all three of which involve nested modalities. The example of
\(\Box(\sfL\sfp \to \sfL\sfp)\) above motivating
**Nec***A* has already shown that
singular necessities might not themselves be necessary and, hence,
serves as well to illustrate the invalidity of **4**:
\(\sfL\sfp \to \sfL\sfp\) is true at all possible worlds, and hence,
necessary, \(\Box(\sfL\sfp \to \sfL\sfp)\); but \(\Box(\sfL\sfp \to
\sfL\sfp)\) fails to be true at Prior-free worlds and, hence, is not
itself necessary, i.e., \(\neg\Box\Box(\sfL\sfp \to \sfL\sfp)\).
**B**'s invalidity is illustrated by any contingent
singular truth; that Prior is a logician, say, \(\sfL\sfp\). By
**MSA**, \(\sfL\sfp\) is not possible at Prior-free
worlds, and, hence, it is not necessarily possible, \(\neg\Box\Diamond
\sfL\sfp\). And, of course, since it is also in fact possible that
Adams is a philosopher, \(\Diamond \sfL\sfp,\) the same example shows
that **5** is invalid.



The natural fix for Adams, of course, is to qualify the nested
modalities in each case with the condition that the proposition in
question
 exists:[91]



**4***A*:
\(\Box \varphi \to \Box(\sfE!\varphi \to \Box \varphi)\)
**B***A*:
\(\varphi \to \Box(\sfE!\varphi \to \Diamond\varphi)\)
**5***A*:
\(\Diamond\varphi \to \Box(\sfE!\varphi \to
\Diamond\varphi)\)


As with *KQML*, only **T** and
**5***A* are needed for *A*;
**4***A* and
**B***A* are derivable from them. And,
similar to what was observed with *Q*, it is easy to show that,
if one extends *A* with the thesis of necessitism by replacing
**E!***A* with its necessitation,



\(\Box\textbf{E!}\_A\):
\(\Box \sfE!\tau,\) for all terms \(\tau\)


the system simply collapses into *SQML*.



Summarizing, then: Adams' distinction between truth *in*
a world and truth *at* a world motivated a number of intuitive
semantic principles that, when formalized, yield a
"perspectivalist" modal logic *A*:


* *Propositional Axiom Schemas*
  

**P1**, **P2**, **P2**
* *Modal Axiom Schemas*
  

**K**,
 **T**, **5***A*,
\(\Box\Diamond\_A\)
* *Serious Actualist Axiom Schemas*
  

**GSA**,
 **MSA**
* *Quantificational Axiom Schemas*
  

**Q1**,
 **FQ2**,
 **Q3**
* *Existence Axiom Schema*
  

**E!***A*
* *Identity Axiom Schemas*
  

**[?]Id1**,
 **Id2**
* *Rules of Inference*
  

**MP**, **Gen**,
**Nec***A*



As noted, *A* restores a number of intuitively desirable
logical features that were lost in Prior's \(Q\), in particular,
the logical equivalence of necessity and impossible falsehood, and the
necessity of singular logical truths. Within modal contexts, however,
properties similar to \(Q\) return, albeit in a milder form: where
\(Q\) requires the necessary existence of a proposition in order for
it to be possible or necessary in a modal context, *A* requires
only *de facto* existence. The final approach that will be
examined here argues that the strict actualist needn't even
concede that much.


#### 4.4.4 Perspectivalism without **MSA**



A number of philosophers who are at least sympathetic to strict
actualism have pointed out that, contrary to Adams, strict actualism
does not entail
 **MSA**--in
 effect, the principle that a singular modal propositions cannot be
true without existing--and, hence, that the still rather severe
restrictions of *A* are unnecessary (Menzel 1990, 1993; Turner
2005; Einheuser 2012; Mitchell-Yellin & Nelson 2016; Masterman
forthcoming). Moreover, they suggest that Adams' own distinction
between truth in a world and truth at a world provides the intuitive
semantic foundation for this claim. Thus Menzel (1993: 132) notes
that, just as we can consider a negated proposition to be true
*at* a world where it doesn't exist by considering it
from our perspective in the actual world where it does, we can do the
same for modal propositions. Turner spells out the idea as follows
regarding an arbitrary *de re* modal proposition \(\Diamond
\sfP\sfa,\) where \(a\) is an actually existing object that, in fact,
has the property \(P\):



> 
> 
> Return to the "picture thinking" that truth at a world is
> supposed to capture. We are standing outside of a world, looking into
> it, and using the propositions, objects, properties, and relations of
> our own world to describe what we see. It makes sense to think that
> which predications of \(a\) are true at a world is determined solely
> by things going on in that world--how could facts from other
> worlds ever get into the picture? But we tend to think that modal
> truths are not made true solely by what is going on in any one world
> but by what goes on in the entire space of possible worlds.
> Furthermore, on the model of "standing outside" of a world
> looking into it, it is not implausible to think that we should be able
> to "see" the entire space of possible worlds. We can say,
> of a non-\(a\) world \(W,\) that at that world it is possible that
> \(a\) is \(P,\) precisely because, standing outside of \(W,\) we can
> see other worlds - worlds where \(a\) exists and is \(P\).
> (2005: 205)
> 
> 
> 



On this take, then, Adams fails to fully embrace his distinction
between truth in a world and truth at a world. He evaluates negated
propositions at worlds where they fail to exist from our perspective
in the actual world, but does not extend the idea to modal
propositions; instead, to evaluate a modal proposition
\(\Box\varphi\)/\(\Diamond\varphi\) at a world \(w,\) he switches back
to an internal perspective that requires the existence of \(\varphi\)
in \(w,\) i.e., the evaluative perspective that Prior adopted across
the board. It is unsurprising, then, that Adams' logic
*A* restores only some of the logical truths that were lost in
Prior's \(Q\). But, as Turner notes, a broader embrace of
Adams' perspectivalism would have us evaluate *all*
propositions from our standpoint in the actual world. So
"situated", a modal proposition
\(\Box\varphi\)/\(\Diamond\varphi\) can then be considered true at a
world \(w\) simply if \(\varphi\) is (from our standpoint, not
\(w\))'s) true at all/some worlds. As in the shift from Prior to
Adams, then, there is no change to the metaphysics of strict actualism
but only a shift in the underlying (intuitive)
 semantics.[92]



The logical implications of maintaining this fixed perspective in the
actual world are quite dramatic. First and foremost, it undermines the
motivation for **MSA** entirely and, with it, the
arguments against the validity of the principles \(\Box\Diamond\),
**4**,
**B**, **5**, and the rule of necessitation
**Nec**\*. All of those can therefore be embraced without
the existence qualifications of \(\Box\Diamond\_A\),
**4**\(\_A\),
**B**\(\_A\),
**5**\(\_A\),
and \(\textbf{Nec}\_A\).
Substituting the unqualified
principles and the rule
 **Nec**\*
 in their place results in what might be called a *fully*
perspectivalist modal logic *A*\* for the strict actualist that
restores everything that was lost in Prior's *Q*, and
does so without the expressive restrictions of Kripke's
*KQML*.


* *Propositional Axiom Schemas*
  

**P1**, **P2**, **P2**
* *Modal Axiom Schemas*
  

**K**,
 **T**, **5**, \(\Box\Diamond\)
* *Serious Actualist Axiom Schema*
  

**GSA**
* *Quantificational Axiom Schemas*
  

**Q1**,
 **FQ2**,
 **Q3**
* *Existence Axiom Schema*
  

**E!***A*
* *Identity Axiom Schemas*
  

**[?]Id1**,
 **Id2**
* *Rules of Inference*
  

**MP**, **Gen**,
 **Nec**\*



*A*\* (under the name "*A*") is proved sound
and complete in Menzel 1991.