## 1. Possible Worlds and Modal Logic



Although 'possible world' has been part of the
philosophical lexicon at least since Leibniz, the notion became firmly
entrenched in contemporary philosophy with the development of
*possible world semantics* for the languages of propositional and
first-order modal logic. In addition to the usual sentence operators
of
 classical logic
 such as 'and' ('[?]'), 'or'
('[?]'), 'not' ('!'),
'if...then' ('-'), and, in the
first-order case, the quantifiers 'all'
('[?]') and 'some'
('[?]'), these languages contain operators intended to
represent the modal adverbs 'necessarily'
('#') and 'possibly'
('*'). Although a prominent aspect of logic in both
Aristotle's work and the work of many medieval philosophers, modal
logic was largely ignored from the modern period to the mid-20th
century. And even though a variety of modal deductive systems had in
fact been rigorously developed in the early 20th century, notably by
Lewis and Langford (1932), there was for the languages of those
systems nothing comparable to the elegant semantics that Tarski had
provided for the languages of classical first-order logic.
Consequently, there was no rigorous account of what it means for a
sentence in those languages to be *true* and, hence, no account
of the critical semantic notions of validity and logical consequence
to underwrite the corresponding deductive notions of theoremhood and
provability. A concomitant philosophical consequence of this void in
modal logic was a deep skepticism, voiced most prominently by Quine,
toward any appeal to modal notions in metaphysics generally, notably,
the notion of an essential property. (See Quine 1953 and 1956, and the
appendix to Plantinga 1974.) The purpose of the following two
subsections is to provide a simple and largely ahistorical overview of
how possible world semantics fills this void; the final subsection
presents two important applications of the semantics. (Readers
familiar with basic possible world semantics can skip to SS2 with
no significant loss of continuity.) 


### 1.1 Extensionality Lost



Since the middle ages at least, philosophers have recognized a
semantical distinction between *extension* and
*intension*. The extension of a denoting expression, or
*term*, such as a name or a definite description is its
referent, the thing that it refers to; the extension of a predicate is
the set of things it applies to; and the extension of a sentence is
its truth value. By contrast, the intension of an expression is
something rather less definite -- its *sense*, or
*meaning*, the semantical aspect of the expression that
determines its extension. For purposes here, let us say that a
*logic* is a formal language together with a semantic theory for
the language, that is, a theory that provides rigorous definitions of
truth, validity, and logical consequence for the
 language.[2]
 A logic is *extensional* if the truth value of every sentence
of the logic is determined entirely by its form and the extensions of
its component sentences, predicates, and terms. An extensional logic
will thus typically feature a variety of valid *substitutivity
principles*. A substitutivity principle says that, if two
expressions are coextensional, that is, if they have the same
extension, then (subject perhaps to some reasonable conditions) either
can be substituted for the other in any sentence *salva
veritate*, that is, without altering the original sentence's truth
value. In an
 *intensional* logic,
 the truth values of some sentences are determined by something over
and above their forms and the extensions of their components and, as a
consequence, at least one classical substitutivity principle is
typically rendered invalid. 



Extensionality is a well known and generally cherished feature of
classical propositional and predicate logic. Modal logic, by contrast,
is intensional. To illustrate: the substitutivity principle for
sentences tells us that sentences with the same truth value can be
substituted for one another *salva veritate*. So suppose that
John's only pets are two dogs, Algol and BASIC, say, and consider two
simple sentences and their formalizations (the predicates in question
indicating the obvious English counterparts): 




|  |  |  |
| --- | --- | --- |
|  |  | All John's dogs are mammals: [?]*x*(*Dx* -
*Mx*).  |




|  |  |  |
| --- | --- | --- |
|  |  | All John's pets are mammals: [?]*x*(*Px* -
*Mx*) |



As both sentences are true, they have the same extension. Hence, in
accordance with the classical substitutivity principle for sentences,
we can replace the occurrence of (1) with (2) in the false sentence





|  |  |  |
| --- | --- | --- |
|  |  | Not all John's dogs are mammals: ![?]*x*(*Dx*
- *Mx*)  |



and the result is the equally false sentence 




|  |  |  |
| --- | --- | --- |
|  |  | Not all John's pets are mammals: ![?]*x*(*Px*
- *Mx*).  |



However, when we make the same substitution in the true sentence 




|  |  |  |
| --- | --- | --- |
|  |  | Necessarily, all John's dogs are mammals:
#[?]*x*(*Dx* - *Mx*), |



the result is the sentence 





|  |  |  |
| --- | --- | --- |
|  |  | Necessarily, all John's pets are mammals:
#[?]*x*(*Px* - *Mx*),  |



which is intuitively false, as John surely could have had a
non-mammalian pet. In a modal logic that accurately represents the
logic of the necessity operator, therefore, the substitutivity
principle for sentences will have to fail. 



The same example illustrates that the substitutivity principle for
predicates will have to fail in modal logic as well. For, according to
our example, the predicates '*D*' and
'*P*' that are true of John's dogs and of John's
pets, respectively, are coextensional, i.e.,
[?]*x*(*Dx* - *Px*). However, while
substituting the latter predicate for the former in (3) results in a
sentence with the same truth value, the same substitution in (5) does
not. 



Modal logic, therefore, is intensional: in general, the truth value of
a sentence is determined by something over and above its form and the
extensions of its components. Absent a rigorous semantic theory to
identify the source of its intensionality and to systematize
intuitions about modal truth, validity, and logical consequence, there
was little hope for the widespread acceptance of modal logic. 


### 1.2 Extensionality Regained



The idea of possible worlds raised the prospect of extensional
respectability for modal logic, not by rendering modal logic itself
extensional, but by endowing it with an extensional semantic theory
-- one whose own logical foundation is that of classical
predicate logic and, hence, one on which possibility and necessity can
ultimately be understood along classical Tarskian lines. Specifically,
in *possible world semantics*, the modal operators are
interpreted as *quantifiers* over possible worlds, as expressed
informally in the following two general principles: 





|  |  |  |
| --- | --- | --- |
| **Nec** |  |  A sentence of the form [?]Necessarily,
ph[?]
([?][?]ph[?]) is true if and only
if ph is true in every possible
 world.[3] |





|  |  |  |
| --- | --- | --- |
| **Poss** |  |  A sentence of the form [?]Possibly,
ph[?]
([?]*ph[?]) is true if and only
if ph is true in some possible world.  |



Given this, the failures of the classical substitutivity principles
can be traced to the fact that modal operators, so interpreted,
introduce contexts that require subtler notions of meaning for
sentences and their component parts than are provided in classical
logic; in particular, a subtler notion (to be clarified shortly) is
required for predicates than that of the set of things they happen to
apply to. 




**Tarskian Semantics.** Standard model theoretic semantics for the
languages of predicate logic deriving from the work of Tarski (1933,
1944) is the paradigmatic semantic theory for extensional logics.
Given a standard first-order language L, a Tarskian
*interpretation* **I** *for* L specifies a set
**D** for the quantifiers of L to range over (typically, some
set of things that L has been designed to describe) and assigns,
to each term (constant or variable) t of L, a referent
**a**t [?] **D** and, to each *n*-place
predicate p of L, an appropriate extension
**E**p -- a truth value (TRUE or FALSE) if
*n* = 0, a subset of **D** if *n* = 1, and a set of
*n*-tuples of members of **D** if *n* > 1. Given these
assignments, sentences are evaluated as true under the interpretation
**I** -- true**I**, for short -- according
to a more or less familiar set of clauses. To facilitate the
definition, let **I**[n/**a**] be the interpretation that
assigns the individual **a** to the variable n and is otherwise
exactly like **I**. Then we have: 


* An atomic sentence
[?]pt1...t*n*[?]
(of L) is *true***I** if and only if
	+ *n* = 0 (i.e., p is a sentence letter) and the extension
	of p is the truth value TRUE; or
	+ *n* = 1 and **a**t1 is in the
	extension of p; or
	+ *n* > 1 and <**a**t1,
	..., **a**t*n*>  is in the
	extension of p.
* A negation [?]!ps[?] is
true**I** if and only if ps is not
true**I**.
* A material conditional[?]ps -
th[?] is true**I** iff, if ps is
true**I**, then th is true**I**.
* A universally quantified sentence
[?][?]nps[?] is
true**I** if and only if, for all individuals **a**
[?] **D**, ps is
 true**I**[n/**a**].[4]



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





|  |  |  |
| --- | --- | --- |
|  |  |  [?]nph =*def*
![?]n!ph  |



it follows that: 


* An existentially quantified sentence
[?][?]nps[?] is is
true**I** if and only if, for some individual **a**
[?] **D**, ps is true**I**[n/**a**].



It is easy to verify that, in each of the above cases, replacing one
coextensional term, predicate, or sentence for another has no effect
on the truth values rendered by the above clauses, thus guaranteeing
the validity of the classical substitutivity principles and, hence,
the extensionality of first-order logic with a Tarskian semantics.




**From Tarskian to Possible World Semantics.** The truth
conditional clauses for the three logical operators directly reflect
the meanings of the natural language expressions they symbolize:
'!' means *not*; '-' means
*if...then*; '[?]' means *all*. It is
easy to see, however, that we cannot expect to add an equally simple
clause for sentences containing an operator that symbolizes necessity.
For a Tarskian interpretation *fixes* the domain of
quantification and the extensions of all the predicates. Pretty
clearly, however, to capture necessity and possibility, one must be
able to consider alternative "possible" domains of
quantification and alternative "possible" extensions for
predicates as well. For, intuitively, under different circumstances,
fewer, more, or other things might have existed and things that
actually exist might, in those circumstances, have had very different
properties. (6), for example, is false because John could have had
non-mammalian pets: a canary, say, or a turtle, or, under
*very* different circumstances, a dragon. A bit more formally
put: Both the domain of quantification and the extension of the
predicate '*P*' could, in some sense or other, have
been different. 



Possible world semantics, of course, uses the concept of a possible
world to give substance to the idea of alternative extensions and
alternative domains of quantification. (Possible world semantics can
be traced most clearly back to the work of Carnap (1947), its basic
development culminating in the work of Hintikka (1957, 1961), Bayart
(1958, 1959), and Kripke (1959, 1963a, 1963b).[5])
 Similar to Tarskian semantics, a possible world interpretation
**M** of a modal language L specifies a nonempty set
**D**, although thought of now as the set of "possible
individuals" of **M**. Also as in Tarskian semantics,
**M** assigns each term t of L a referent
**a**t in
 **D**.[6]
 Additionally however, **M** specifies a set **W**, the set of
"possible worlds" of **M**, one of which is designated
its "actual world", and each world **w** in **W** is
assigned its own domain of quantification, **d**(**w**) [?]
**D**, intuitively, the set of individuals that exist in
 **w**.[7]
 To capture the idea of both the actual and possible extensions of a
predicate, **M** assigns to each *n*-place predicate p a
function **M**p -- the *intension* of
p -- that, for each possible world **w**, returns the
*ex*tension **M**p(**w**) of p
*at* **w**: a truth value, if *n* = 0; a set of
individuals, if *n* = 1; and a set of *n*-tuples of
individuals, if *n* >
 1.[8]
 We can thus rigorously define a "possible extension" of a
predicate p to be any of its ***w**-extensions*
**M**p(**w**), for any world **w**. 



The Tarskian truth conditions above are now generalized by
relativizing them to worlds as follows: for any possible world
**w** (the *world of evaluation*): 


* An atomic sentence
[?]pt1...t*n*[?]
(of L) is *true***M** *at* **w** if
and only if:
	+ *n* = 0 and the **w**-extension of p is the truth
	value TRUE; or
	+ *n* = 1 and *a*t1 is in the
	**w**-extension of p; or
	+ *n* > 1 and
	<**a**t1,...,
	**a**t*n*>  is in the
	**w**-extension of p.
* A negation [?]!ps[?] is
true**M** at **w** if and only ps is not
true**M** in **w**.
* A material
conditional[?]ps-th[?] is
true**M** at **w** iff, if ps is
true**M** at **w**, then th is
true**M** at **w**.
* A quantified sentence
[?][?]nps[?] is
true**M** at **w** if and only if, for all
individuals **a** that exist in **w**, ps is
true**M**[n/**a**].



And to these, of course, is added the critical modal case that
explicitly interprets the modal operator to be a quantifier over
worlds, as we'd initially anticipated informally in our principle
 **Nec**:
 


* A necessitation [?][?]ps[?]
is true**M** at **w** if and only if, for all
possible worlds **u** of **M**, ps is true**M**
at
 **u**.[9]




A sentence ph is *false***M** at **w** just
in case it is not true**M** at **w**, and ph is
said to be *true***M** just in case ph is
true**M** at the actual world of **M**.



On the assumption that there is a (nonempty) set of all possible
worlds and a set of all possible individuals, we can define
"objective" notions of truth at a world and of truth
*simpliciter*, that is, notions that are not simply relative to
formal, mathematical interpretations but, rather, correspond to
objective reality in all its modal glory. Let L be a modal
language whose names and predicates represent those in some fragment
of ordinary language (as in our examples (5) and (6) above). Say that
**M** is the "intended" interpretation of L if
(i) its set **W** of "possible worlds" is in fact the
set of all possible worlds, (ii) its designated "actual
world" is in fact the actual world, (iii) its set **D** of
"possible individuals" is in fact the set of all possible
individuals, and (iv) the referents assigned to the names of L
and the intensions assigned to the predicates of L are the ones
they in fact have. Then, where **M** is the intended interpretation
of L, we can say that a sentence ph of L is *true
at* a possible world **w** just in case ph is
true**M** at **w**, and that ph is *true*
just in case it is true**M** at the actual world.
(Falsity at **w** and falsity, *simpliciter*, are defined
accordingly.) Under the assumption in question, then, the modal clause
above takes on pretty much the exact form of our informal principle
 **Nec**.



Call the above *basic* possible world semantics. Spelling out the
truth conditions for (6) (relative to the intended interpretation of
its language), basic possible world semantics tells us that (6) is
true if and only if 




|  |  |  |
| --- | --- | --- |
|  |  |  For all possible worlds **w**,
'[?]*x*(*Px* - *Mx*)' is true at
**w**.  |



And by unpacking (8) in terms of the quantificational, material
conditional, and atomic clauses above we have that (6) is true if and
only if 




|  |  |  |
| --- | --- | --- |
|  |  |  For all possible worlds **w**, and for all possible
individuals **a** that exist in **w**, if **a** is in the
**w**-extension of '*P*' then **a** is in the
**w**-extension of '*M*'.  |



Since we are evaluating (6) with regard to the intended interpretation
of its language, the **w**-extension of '*P*' that
is returned by its intension, for any world **w**, is the (perhaps
empty) set of John's pets in **w** and that of
'*M*' is the set of mammals in **w**. Hence, if
**w** is a world where John has a pet canary -- COBOL, say
-- COBOL is in the **w**-extension of '*P*'
but not that of '*M*' , i.e.,
'[?]*x*(*Px* - *Mx*)' is false at
**w** and, hence, by the truth condition (9), (6) is false at the
actual world -- that is, (6) is false *simpliciter*, as it
should be. 



Note that interpreting modal operators as quantifiers over possible
worlds provides a nice theoretical justification for the usual
definition of the possibility operator in terms of necessity,
specifically: 





|  |  |  |
| --- | --- | --- |
|  |  | [?]*ph[?] =*def*
[?]![?]!ph[?].  |



That is, a sentence is possible just in case its negation isn't
necessary. Since, semantically speaking, the necessity operator is
*literally* a universal quantifier, the definition corresponds
exactly to the definition
 (7)
 of the existential quantifier. For, unpacking the right side of
definition (10) according to the negation and necessitation clauses
above (and invoking the definitions of truth and truth at a world
*simpliciter*), we have: 




|  |  |  |
| --- | --- | --- |
|  |  | [?]*ph[?] is true iff it
is not the case that, for all possible worlds **w**, ph is not
true at **w**.  |



Clearly, however, if it is not the case that ph fails to be true at
all possible worlds, then it must be true at some world; hence: 




|  |  |  |
| --- | --- | --- |
|  |  | [?]*ph[?] is true iff,
for some possible world **w**, ph is true at **w**.  |



And that corresponds exactly to our intuitive truth condition
 **Poss**
 above. Thus, spelling out the negation
'!#[?]*x*(*Px* - *Mx*)'
of our false sentence
 (6)
 above in accordance with definition (10) (and the standard definition
of conjunction [?]), we have: 





|  |  |  |
| --- | --- | --- |
|  |  | Possibly, one of John's pets is not a mammal:
*[?]*x*(*Px* [?] !*Mx*),  |



for which (12) and the possible world truth conditions for quantified,
Boolean, and atomic sentences yield the correct truth condition: 





|  |  |  |
| --- | --- | --- |
|  |  |  There is a possible world **w** and an individual **a**
existing in **w** that is in the **w**-extension of
'*P*' but not that of '*M*',  |



that is, less stuffily, there is a possible world in which, among
John's pets, at least one is not a mammal. 



**Summary: Intensionality and Possible Worlds.** Analyzed in terms
of possible world semantics, then, the general failure of classical
substitutivity principles in modal logic is due, not to an irreducibly
intensional element in the meanings of the modal operators, but rather
to a sort of mismatch between the surface syntax of those operators
and their semantics: syntactically, they are unary sentence operators
like negation; but semantically, they are, quite literally,
quantifiers. Their syntactic similarity to negation suggests that,
like negation, the truth values of
[?]#ph[?] and
[?]*ph[?], insofar as they are
determinable at all, must be determined by the truth value of ph.
That they are not (in general) so determined leads to the distinctive
substitutivity failures noted above. The possible worlds analysis of
the modal operators as quantifiers over worlds reveals that the unary
syntactic form of the modal operators obscures a semantically relevant
parameter. When the modal operators are interpreted as quantifiers,
this parameter becomes explicit and the reason underlying the failure
of extensionality in modal logic becomes clear: That the truth values
of [?]#ph[?] and
[?]*ph[?] are not in general
determined by the truth value of ph at the world of evaluation is,
semantically speaking, nothing more than the fact that the truth
values of '[?]*xFx*' and
'[?]*xFx*' are not in general determined by the
truth value of '*Fx*', for any particular value of
'*x*'. Possible world semantics, therefore,
*explains* the intensionality of modal logic by revealing that
the syntax of the modal operators prevents an adequate expression of
the meanings of the sentences in which they occur. Spelled out as
possible world truth conditions, those meanings can be expressed in a
wholly extensional fashion.
 (For
 a more formal exposition of this point, see the supplemental article
 The Extensionality of Possible World Semantics.)
 


### 1.3 Two Applications: The Analysis of Intensions and the *De Re* / *De Dicto* Distinction



As noted, the focus of the present article is on the metaphysics of
possible worlds rather than applications. Of course, the semantics of
modal languages is itself an application, but one that is of singular
importance, both for historical reasons and because most applications
are in fact themselves applications of (often extended or modified
versions of) the semantical apparatus. Two particularly important
examples are the analysis of intensions and a concomitant explication
of the *de re*/*de dicto*
 distinction.[10]




**The Analysis of Intensions.** As much a barrier to the acceptance
of modal logic as intensionality itself was the need to appeal to
intensions *per se* -- properties, relations,
propositions, and the like -- in semantical explanations.
Intensional entities have of course featured prominently in the
history of philosophy since Plato and, in particular, have played
natural explanatory roles in the analysis of
 intentional attitudes
 like
 belief
 and
 mental content.
 For all their prominence and importance, however, the nature of these
entities has often been obscure and controversial and, indeed, as a
consequence, they were easily dismissed as ill-understood and
metaphysically suspect "creatures of darkness" (Quine
1956, 180) by the naturalistically oriented philosophers of the early-
to mid-20th century. It is a virtue of possible world semantics that
it yields rigorous *definitions* for intensional entities. More
specifically, as described above, possible world semantics assigns to
each *n*-place predicate p a certain function
**I**p -- p's intension -- that, for
each possible world **w**, returns the extension
**I**p(**w**) of p at **w**. We can define
an intension *per se*, independent of any language, to be any
such function on worlds. More specifically: 


* A *proposition* is any function from worlds to truth
values.
* A *property* is any function from worlds to sets of
individuals.
* An **n*-place relation* (*n* > 1) is any
function from worlds to sets of *n*-tuples of individuals.



The adequacy of this analysis is a matter of lively debate that
focuses chiefly upon whether or not intensions, so defined, are too
"coarse-grained" to serve their intended purposes. (See,
e.g., Stalnaker 1987 and 2012 for a strong defense of the analysis.)
However, Lewis (1986, SS1.5) argues that, even if the above
analysis fails for certain purposes, it does not follow that
intensions cannot be analyzed in terms of possible worlds, but only
that more subtle constructions might be required. This reply appears
to side-step the objections from granularity while preserving the
great advantage of the possible worlds analysis of intensions, viz.,
the rigorous definability of these philosophically significant
notions. 



**The *De Re* / *De Dicto* Distinction.** A particularly
rich application of the possible world analysis of intensions concerns
the analysis of the venerable distinction between *de re* and
*de dicto*
 modality.[11]
 Among the strongest modal intuitions is that the possession of a
property has a modal character -- that things exemplify, or fail
to exemplify, some properties *necessarily*, or
*essentially*, and others only *accidentally*. Thus, for
example, intuitively, John's dog Algol is a pet accidentally; under
less fortunate circumstances, she might have been, say, a stray that
no one ever adopted. But she is a dog essentially; she couldn't have
been a flower, a musical performance, a crocodile or any other kind of
thing. 



Spelling out this understanding in terms of worlds and the preceding
analysis of intensions, we can say that an individual **a** has a
property **F** essentially if **a** has **F** in every world
in which it exists, that is, if, for all worlds **w** in which
**a** exists, **a** [?] **F**(**w**). Likewise,
**a** has **F** accidentally if **a** has **F** in the
actual world @ but lacks it in some other world, that is, if **a**
[?] **F**(@) but, for some world **w** in which **a**
exists, **a** [?] **F**(**w**). Thus, let
'*G*' and '*T*' symbolize 'is
a dog' and 'is someone's pet', respectively; then,
where '*E!x*' is short for
'[?]*y*(*x*=*y*)' (and, hence,
expresses that *x* exists), we have: 





|  |  |  |
| --- | --- | --- |
|  |  |  Algol is a dog essentially: #(*E!a* -
*Ga*)  |





|  |  |  |
| --- | --- | --- |
|  |  |  Algol is a pet accidentally: *Ta* [?] *(*E!a*
[?] !*Ta*)  |



More generally, sentences like (15) and (16) in which properties are
ascribed to a specific individual in a modal context -- signaled
formally by the occurrence of a name or the free occurrence of a
variable in the scope of a modal operator -- are said to exhibit
modality *de
 re*[12]
 (modality *of the thing*). Modal sentences that do not, like






|  |  |  |
| --- | --- | --- |
|  |  |  Necessarily, all dogs are mammals:
#[?]*x*(*Gx* - *Mx*)  |



are said to exhibit modality *de dicto* (roughly, modality *of
the* *proposition*). Possible
world semantics provides an illuminating analysis of the key
difference between the two: The truth conditions for both modalities
involve a commitment to possible worlds; however, the truth conditions
for sentences exhibiting modality *de re* involve in addition a
commitment to the meaningfulness of
 *transworld identity*,
 the thesis that, necessarily, every individual (typically, at any
rate) exists and exemplifies (often very different) properties in many
different possible worlds. More specifically, basic possible world
semantics yields intuitively correct truth values for sentences of the
latter sort by (i) permitting world domains to overlap and (ii)
assigning intensions to predicates, thereby, in effect, relativizing
predicate extensions to worlds. In this way, one and the same
individual can be in the extension of a given predicate at all worlds
in which they exist, at some such worlds only, or at none at all. (For
further discussion, see the entry on
 essential vs. accidental properties.)
 


## 2. Three Philosophical Conceptions of Possible Worlds



The power and appeal of basic possible world semantics is undeniable.
In addition to providing a clear, extensional formal semantics for a
formerly somewhat opaque, intensional notion, cashing possibility as
truth in some possible world and necessity as truth in every such
world seems to tap into very deep intuitions about the nature of
modality and the meaning of our modal discourse. Unfortunately, the
semantics leaves the most interesting -- and difficult --
philosophical questions largely unanswered. Two arise with particular
force: 




|  |  |  |
| --- | --- | --- |
| **QW** |  |  What, exactly, is a possible world?  |



And, given **QW**: 




|  |  |  |
| --- | --- | --- |
| **QE** |  |  What is it for something to exist in a possible world?  |



In this section we will concern ourselves with, broadly speaking, the
three most prominent philosophical approaches to these
 questions.[13]



### 2.1 Concretism



Recall the informal picture that we began with: a world is, so to say,
the "limit" of a series of increasingly more inclusive
situations. Fleshed out philosophical accounts of this informal idea
generally spring from rather different intuitions about what one takes
the "situations" in the informal picture to be. A
particularly powerful intuition is that situations are simply
structured collections of physical objects: the immediate situation of
our initial example above, for instance, consists of, among other
things, the objects in Anne's office -- notably Anne herself, her
desk and her computer, with her seated at the former and typing on the
latter -- and at least some of the things in the next room
-- notably, her husband and the phone he is talking on. On this
view, for one situation **s** to include another **r** is simply
for **r** to be a (perhaps rather complex and distributed) physical
part of **s**. The actual world, then, as the limit of a series of
increasingly more inclusive situations in this sense, is simply the
entire physical universe: all the things that are some spatiotemporal
distance from the objects in some arbitrary initial situation,
structured as they in fact are; and other possible worlds are things
of exactly the same sort. Call this the *concretist* intuition,
as possible worlds are understood to be concrete physical situations
of a special sort. 


#### 2.1.1 Concrete Worlds and Existence Therein



The originator and, by far, the best known proponent of concretism is
 David Lewis.
 For Lewis and, as noted, concretists generally, the actual world is
the concrete physical universe as it is, stretched out in space-time.
As he rather poetically expresses it (1986, 1): 



> 
> The world we live in is a very inclusive thing....There is nothing so
> far away from us as not to be part of our world. Anything at any
> distance is to be included. Likewise the world is inclusive in time.
> No long-gone ancient Romans, no long-gone pterodactyls, no long-gone
> primordial clouds of plasma are too far in the past, nor are the dead
> dark stars too far in the future, to be part of this same
> world....[N]othing is so alien in kind as not to be part of our world,
> provided only that it does exist at some distance and direction from
> here, or at some time before or after or simultaneous with now.
> 



The actual world provides us with our most salient example of what a
possible world is. But, for the concretist, other possible worlds are
no different in kind from the actual world (*ibid.*, 2): 



> 
> There are countless other worlds, other very inclusive things. Our
> world consists of us and all our surroundings, however, remote in time
> and space; just as it is one big thing having lesser things as parts,
> so likewise do other worlds have lesser other-worldly things as parts.
> 



It is clear that spatiotemporal relations play a critical role in
Lewis's conception. However, it is important to note that Lewis
understands such relations in a very broad and flexible way so as to
allow, in particular, for the possibility of spirits and other
entities that are typically thought of as non-spatial; so long as they
are located in time, Lewis writes, "that is good enough"
(*ibid.*, 73). So with this caveat, let us say that that an
object **a** is *connected* if any two of its parts bear
some spatiotemporal relation to each
 other,[14]
 and that **a** is *maximal* if none of its parts is
spatiotemporally related to anything that is not also one of its
parts. Then we have the following concretist answers to our questions:





|  |  |  |
| --- | --- | --- |
| **AW1** |  | **w** is a possible world =*def*
**w** is a maximal connected
 object.[15] |



And, hence, to exist in a world is simply to be a part of it: 





|  |  |  |
| --- | --- | --- |
| **AE1** |  | Individual **a** exists in world **w** =*def*
**a** is a part of **w**.
 |



It follows from **AW1** (and reasonable assumptions) that distinct
worlds do not overlap, spatiotemporally; that no spatiotemporal part
of one world is part of
 another.[16]
 Moreover, given
 Lewis's counterfactual analysis of causation,
 it follows from this that objects in distinct worlds bear no causal
relations to one another; nothing that occurs in one world has any
causal impact on anything that occurs in any other world. 


#### 2.1.2 Actuality



Critically, for Lewis, worlds and their denizens do not differ in the
*manner* in which they exist. The actual world does not enjoy a
kind of privileged existence that sets it apart from other worlds.
Rather, what makes the actual world actual is simply that it is
*our* world, the world that we happen to inhabit. Other worlds
and their inhabitants exist just as robustly as we do, and in
precisely the same sense; all worlds and all of their denizens are
equally
 *real*.[17]
 A significant semantic corollary of this thesis for Lewis is that the
word 'actual' in the phrase 'the actual world'
does not indicate any special *property* of the actual world that
distinguishes it from all other worlds; likewise, an assertion of the
form '**a** is actual' does not indicate any special
property of the individual **a** that distinguishes it from the
objects existing in other worlds. Rather, 'actual' is
simply an *indexical* whose extension is determined by the
context of utterance. Thus, the referent of 'the actual
world' in a given utterance is simply the world of the speaker,
just as the referent of an utterance of 'the present
moment' is the moment of the utterance; likewise, an utterance
of the form '**a** is actual' indicates only that
**a** shares the same world as the speaker. The speaker thereby
ascribes no special property to **a** but, essentially, expresses
no more than when she utters '**a** is *here*',
understood in the broadest possible sense. By the same token, when we
speak of non-actual *possibilia* -- Lewis's preferred label
for the denizens of possible worlds -- we simply pick out those
objects that are *not* here in the broadest sense. In the mouth
of an other-worldly metaphysician, we here are all among the
non-actual *possibilia* of which she speaks in her lectures on
*de re* modality. 


#### 2.1.3 Modal Reductionism, Counterparts, and the Analysis of Intensions



**Modal Reductionism and Counterparts.** Lewis parted ways
dramatically with his mentor
 W. V. O. Quine
 on modality. Quine (1960, SS41) stands in a long line of
philosophers dating back at least to
 David Hume
 who are skeptical, at best, of the idea that modality is an objective
feature of reality and, consequently, who question whether modal
assertions in general can be objectively true or false, or even
coherent. Lewis, by contrast, wholly embraces the objectivity of
modality and the coherence of our modal discourse. What he denies,
however, is that modality is a fundamentally *irreducible*
feature of the world. Lewis, that is, is a *modal
reductionist*. For Lewis, modal notions are not primitive. Rather,
truth conditions for modal sentences can be given in terms of worlds
and their parts; and worlds themselves, Lewis claims, are defined
entirely in non-modal terms. The earliest presentation of Lewis's
theory of modality (Lewis 1968) -- reflecting Quine's
 method of regimentation
 -- offers, rather than a possible world semantics, a scheme for
*translating* sentences in the language of modal predicate logic
into sentences of ordinary first-order logic in which the modal
operators are replaced by explicit quantifiers over
 worlds.[18]
 The mature account of Lewis 1986 is much more semantic in
orientation: it avoids any talk of translation and offers instead a
(somewhat informal) account of concretist possible world truth
conditions for a variety of modal assertions. Nonetheless, it is
useful to express the logical forms of these truth conditions
explicitly in terms of worlds, existence in a world (in the sense of
**AE1**, of course), and the *counterpart* relation, which
will be discussed shortly: 




|  |  |  |
| --- | --- | --- |
| *Wx*: |  | *x* is a *world* |
| *Ixy*: |  | *x* *exists in* world *y* |
| *Cxy*: |  | *x* is a *counterpart* of *y* |



For sentences like
 (17)
 involving only *de dicto* modalities, Lewis's truth conditions
are similar in form to the truth conditions generated by the modal
clauses of basic possible world semantics; specifically, for (17):






|  |  |  |
| --- | --- | --- |
|  |  |  For every world **w**, every individual **x** in **w**
that is a dog is a mammal: [?]*w*(*Ww* -
[?]*x*(*Ixw* - (*Gx* - *Mx*))).
 |



As in possible world semantics, the modal operators
'#' and '*' "turn
into" quantifiers over worlds in concretist truth conditions
(1986, 5). Also as in possible world semantics, a quantifier (in
effect) ranging over individuals that occurs in the scope of a
quantifier (in effect) ranging over worlds --
'[?]*x*' and '[?]*w*',
respectively, in (18) -- is, for each value **w** of the bound
world variable, restricted to the objects existing in **w**.
However, unlike possible world semantics, predicates are not to be
thought of as having different extensions at different worlds. Rather,
for Lewis, each (*n*-place) predicate has a single extension that
can contain (*n*-tuples of) objects across many different worlds
-- intuitively, all of the objects that have the property (or
*n*-tuples of objects that stand in the relation) expressed by
the predicate across all possible worlds. Thus, in particular, the
predicate '*G*' picks out, not just this-worldly dogs
but other-worldly canines as well. Likewise, the pet predicate
'*T*' picks out both actual and other-worldly pets.
Such a move is not feasible in basic possible world semantics, which
is designed for a metaphysics in which one and the same individual can
exemplify a given property in some worlds in which they exist but not
others. Hence, a typical predicate will be true of an individual with
respect to some worlds and false of it with respect to others. But,
for Lewis, as we've seen, distinct possible worlds do not overlap and,
hence, objects are worldbound, thereby eliminating the need to
relativize predicate extensions to worlds. 



However, this very feature of Lewis's account -- worldboundedness
-- might appear to threaten its coherence. For example, since
Algol is in fact a pet, given worldboundedness and the definition
 **AE1**
 of existence in a world **w**, we have: 





|  |  |  |
| --- | --- | --- |
|  |  |  There is no world **w** such that Algol exists in **w**
and fails to be someone's pet: ![?]*w*(*Iaw* [?]
!*Ta*),  |



But, according to Lewis's analysis, the modal operators
'#' and '*', semantically, are
quantifiers over worlds. Hence, (19) might appear to be exactly the
concretist truth condition for the denial of (the right conjunct of)
 (16),
 i.e., it might appear that, on Lewis's analysis, Algol is not a pet
accidentally but essentially; likewise, more generally, any individual
and any intuitively accidental property of that individual. 



In fact, Lewis whole-heartedly accepts that things have accidental
properties and, indeed, would accept that
 (16)
 is robustly true. His explanation involves one of the most
interesting and provocative elements of his theory: the doctrine of
*counterparts*. Roughly, an object **y** in a world
**w**2 is a counterpart of an object **x** in
**w**1 if **y** resembles **x** and nothing else
in **w**2 resembles **x** more than
 **y**.[19]
 Each object is thus its own (not necessarily unique) counterpart in
the world it inhabits but will typically differ in important ways from
its other-wordly counterparts. A typical other-worldly counterpart of
Algol, for example, might resemble her very closely up to some point
in her history -- a point, say, after which she continued to live
out her life as a stray instead of being brought home by our kindly
dog-lover John. Hence, sentences making *de re* assertions about
what *Algol* might have done or what *she* could or could
not have been are unpacked, semantically, as sentences about her
*counterparts* in other possible worlds. Thus, when we analyze
(16) accordingly, we have the entirely unproblematic concretist truth
condition: 





|  |  |  |
| --- | --- | --- |
|  |  |  Algol is a pet, but there is a world in which exists a
counterpart of hers that is not:

*Ta* [?] [?]*w*(*Ww* [?]
[?]*x*(*Ixw* [?] *Cxa* [?] !*Tx*)).
 |



Ascriptions of essential properties, as in
 (15),
 are likewise unpacked in terms of counterparts: to say that Algol is
a dog essentially is to say that 





|  |  |  |
| --- | --- | --- |
|  |  |  All of Algol's counterparts in any world are dogs:

[?]*w*(*Ww* - [?]*x*((*Ixw* [?]
*Cxa*) - *Gx*)).  |



**The Analysis of Intensions.** Lewis's possible world truth
conditions are expressed in classical non-modal logic and, hence, they
are to be interpreted by means of standard Tarskian semantics. Thus,
*n*-place predicates p are assigned extensions
**E**p -- in particular, for 1-place predicates,
sets of individuals -- as their semantic values, as described in
the exposition in
 SS1.2
 above. However, given worldboundedness and the fact that predicate
extensions are drawn not simply from the actual world but from all
possible worlds, these extensions are able to serve as
*in*tensions in Lewis's theory. As in basic possible world
semantics, intensional entities in general can be defined in terms of
the basic ontology of the theory independent of the linguistic roles
they can play as the intensions of predicates. And because individuals
are worldbound, Lewis is able to simplify the definitions given in
 SS1.3
 by defining intensions as sets rather than functions: 


* A *proposition* is any set of worlds.
* A *property* is any set of individuals.
* An **n*-place relation* (*n* > 1) is any set
of *n*-tuples of
 individuals.[20]



Thus, on this analysis, a proposition **p** is *true in* a
world **w** just in case **w** [?] **p** and an individual
**a** has a property **P** just in case **a** [?]
**P**. (Note that propositions are thus simply properties of worlds
on these definitions.) **a** has **P** *accidentally*
just in case **a** [?] **P** but **b** [?] **P**
for some other-worldly counterpart of **b** of **a**; and
**a** has **P** *essentially* if **b** [?] **P**
for every counterpart **b** of **a**. 



In Lewis's theory of modality, then, modal operators are understood
semantically to be quantifiers over concrete worlds, predicates denote
intensions understood as sets of (*n*-tuples of) parts of those
worlds, and sentences involving *de re* modalities are understood
in terms of counterparts. To the extent that these notions are free of
modality, Lewis has arguably reduced modal notions to non-modal. 


#### 2.1.4 Plenitude and Recombination



That Lewis's truth conditions for modal statements are themselves free
of modality and, hence, that his theory counts as a genuine reduction
of modal notions to non-modal is not terribly controversial (albeit
not undisputed -- see Lycan 1991, 224-27; Divers and Melia
2002, 22-24). Significantly more controversial, and perhaps far
more critical to the project, is whether or not his account is
*complete*, that is, whether or not, for all modal statements
ph, (i) if ph is intuitively true, then its Lewisian truth
condition holds (ii) if ph is intuitively false, then its Lewisian
truth condition
 fails.[21]
 The challenge to Lewis, then, is that his account can be considered
*successful* only if it is complete in this sense. 



The chief question Lewis faces in this regard is whether there are
*enough* worlds to do the job. The truth condition
 (20)
 for the intuitively true
 (16)
 says that there exists a possible world in which a counterpart of
Algol is no one's pet. By virtue of what in Lewis's theory does such a
world exist? The ideal answer for Lewis would be that some principle
in his theory guarantees a *plenitude* of worlds, a maximally
abundant array of worlds that leaves "no gaps in logical space;
no vacancies where a world might have been, but isn't" (Lewis
1986, 86). From this it would follow that the worlds required by the
concretist truth condition for any intuitive modal truth exist. Toward
this end, Lewis initially considers the evocative principle: 




|  |  |  |
| --- | --- | --- |
| **Ways** |  | Absolutely every way that a world could be is a way that some
world is.  |



Since, in particular, a world satisfying
 (20)
 seems quite obviously to be a way a world could be, by **Ways**
such a world exists. But there is a fatal flaw here: Lewis himself
(1973, 84) identifies *ways* that a world could be with worlds
themselves. So understood, **Ways** collapses into the triviality
that every world is identical to some
 world.[22]




Lewis finds a replacement for **Ways** in a principle of
*recombination* whereby "patching together parts of
different possible worlds yields another possible world" (1986,
87-88). The principle has two aspects. The first is the
principle that "anything can coexist with anything". For
"if there could be a dragon, and there could be a
unicorn," Lewis writes, "but there couldn't be a dragon
and a unicorn side by side, that would be ... a failure of
plenitude" (*ibid*., 88). Given that individuals are
worldbound, however, the principle is expressed more rigorously (and
more generally) in terms of other-worldly *duplicates*: 




|  |  |  |
| --- | --- | --- |
| **R1** |  | For any (finite or infinite) number of objects
**a**1, **a**2, ..., there is a world
containing any number of duplicates of each of those objects in any
spatiotemporal arrangement (size and shape permitting).  |



The second aspect of the principle expresses "the Humean denial
of necessary connections" (*ibid*., 87), that is, the idea
that anything can *fail* to coexist with anything else. For
"if there could be a talking head contiguous to the rest of a
living human body, but there couldn't be a talking head separate from
the rest of a human body, that too would be a failure of
plenitude" (*ibid*). To express this a bit more rigorously,
say that objects **a**1, **a**2, ..., are
*independent of* objects **b**1,
**b**2, ..., if no sum of any parts of the former are
parts or duplicates of any sum of any parts of the latter and vice
versa; then we have: 




|  |  |  |
| --- | --- | --- |
| **R2** |  | For any world **w** any (finite or infinite number of)
objects **a**1, **a**2, ..., in **w**
and any objects **b**1, **b**2, ..., in
**w** that are independent of **a**1,
**a**2, ..., there is a world containing duplicates of
**a**1, **a**2, ..., and no duplicates of
**b**1, **b**2, ... .  |



Worlds that satisfy the concretist truth conditions for workaday
possibilities like
 (16)
 are easily conceived as consisting of duplicates of relevant parts of
the actual world -- suitably organized to retain their actual
properties, or not, as needed. Hence, the existence of such worlds
does indeed appear to follow from the existence of the actual world by
recombination. Worlds containing talking donkeys, exotic species
resulting from a wholly different evolutionary history, worlds with
silicon-based life forms, and so on present a bigger challenge to the
view. Nonetheless, it is not entirely implausible to think such worlds
exist given suitable duplication and reorganization of microphysical
 objects.[23]




Whether recombination completely captures our modal intuitions
regarding plenitude is still a matter of some
 dispute.[24]
 However, even if it doesn't, it is less than clear whether this
counts against the success of Lewis's reductionist project. For, as a
realist about worlds, Lewis does not seem to be under any obligation
to "derive" plenitude from more fundamental principles.
Hence, there is no obvious reason why he cannot respond to charges of
incompleteness by saying that it is simply a presupposition of his
theory that logical space has no gaps, that there are always enough
worlds to satisfy the concretist truth condition for any intuitive
modal
 truth.[25]
 So understood, the role of recombination for a realist about worlds
like Lewis is something like the role of such axioms as powerset and
replacement for a realist about sets: given some sets, these
principles provide us with a detailed -- but always less than
complete -- characterization of what further sets there are.
Their role, therefore, is to give us insight into the richness and
diversity of set theoretic space, not a complete mechanism for proving
which particular sets do or do not exist. Likewise recombination
vis-a-vis worlds and logical space. 


#### 2.1.5 A Brief Assessment of Concretism



Lewis's theory is particularly commendable for its striking
originality and ingenuity and for the simple and straightforward
answers **AW1** and **AE1** that it provides to our two
questions **QW** and **QE** above. Furthermore, because worlds
are (plausibly) defined entirely in nonmodal terms, the truth
conditions provided by Lewis's translation scheme themselves appear to
be free of any implicit modality. Hence, unlike many other popular
accounts of possible worlds (notably, the abstractionist accounts
discussed in the following section), Lewis's promises to provide a
genuine *analysis* of the modal operators. 



Perhaps the biggest -- if not the most philosophically
sophisticated -- challenge to Lewis's theory is "the
incredulous stare", i.e., less colorfully put, the fact that its
ontology is wildly at variance with common sense. Lewis faces this
objection head on: His theory of worlds, he acknowledges,
"*does* disagree, to an extreme extent, with firm common
sense opinion about what there is" (1986, 133). However, Lewis
argues that no other theory explains so much so economically. With
worlds in one's philosophical toolkit, one is able to provide elegant
explanations of a wide variety of metaphysical, semantical, and
intentional phenomena. As high as the intuitive cost is, Lewis (135)
concludes, the existence of worlds "ought to be accepted as
true. The theoretical benefits are worth it." 



Additional discussion of, and objections to, concretism can be found
in the supplemental document
 Further Problems for Concretism.
 


### 2.2 Abstractionism




A rather different set of intuitions about situations is that they are
*abstract* entities of a certain sort: They are *states*
or *conditions*, of varying detail and complexity, that a
concrete world could be in -- they are *ways* that things,
as a whole, could
 be.[26]
 Thus, returning to our original example, one very simple way things
could be is for our philosopher Anne to be in her office. We can now
imagine, as in our example, further detail being successively added to
that description to yield more complex ways things could be: Anne
working at her desk in her office; music being in the background; her
husband being on the phone in the next room; her neighbor mowing the
lawn next door; and so on. Roughly speaking, then, a possible world
for an abstractionist is the *limit* of such a
"process" of consistently extending and adding detail to
some initial state of the world; it is a *total* way things
could be, a consistent state of the world that settles every
possibility; a consistent state to which no further detail could be
added without rendering it inconsistent. 


#### 2.2.1 Abstract Possible Worlds and Existence Therein



To give the notion of a state, or condition, of the world a little
more metaphysical substance, abstractionists typically appeal to more
traditional ontological categories. Thus, for example, that things
could be in the simple state described above might be spelled out in
one of the following ways: 


* The *proposition* **that Anne is in her office and at her
desk** is possibly true.
* The *set of propositions* {**that Anne is in her
office**, **that Anne is at her desk**} is such that, possibly,
all of its members are true.
* The *property* **being such that Anne is in her office
and at her desk** is possibly exemplified (by "things as a
whole").



Possible worlds are then defined as special cases of the type of
entity in question that are in some relevant sense *total*.
Adams (1974), for example, defines possible worlds to be consistent
sets of propositions that are total in the sense of containing, for
every proposition **p**, either **p** or its negation; Fine
(1977), fleshing out ideas of Prior, defines a possible world to be a
consistent proposition **w** that is total in the sense that, for
every proposition **p**, **w** entails either **p** or its
negation. For purposes here, however, we will sketch the fundamentals
of the abstractionist view in terms of *states of affairs*,
following the basic features of the account developed by Plantinga
(1974, 1976), an account that, in the literature, frequently serves as
a particularly trenchant abstractionist counterpoint to Lewis's
 concretism.[27]




States of affairs (SOAs) are abstract, intensional entities typically
signified by sentential gerundives like "Algol's being John's
pet" and "There being more than ten solar planets".
Importantly, SOAs constitute a primitive ontological category for the
abstractionist; they are not defined in terms of possible worlds in
the manner that propositions are in
 SS1.3.
 Just as some propositions are true and others are not, some SOAs are
*actual* and others are
 not.[28]
 Note, then, that to say an SOA is non-actual is *not* to say
that it does not actually *exist*. It is simply to say that it
is not, in fact, a condition, or state, that the concrete world is
actually in. However, because '\_\_\_\_ is actual' is often
used simply to mean '\_\_\_\_ exists', there is considerable
potential for confusion here. So, henceforth, to express that an SOA
is actual we will usually say that it *obtains*. 



An SOA is said to be *possible* (*necessary*,
*impossible*) insofar as it is possible (necessary, impossible)
that it obtain. One SOA **s** is said to *include* another
**t** if, necessarily, **s** obtains only if **t** does;
**s** *precludes* **t** if, necessarily, **s** obtains
only if **t** doesn't. So, for example, **Algol's being John's
pet** includes **Algol's being someone's pet** and precludes
**there being no pets**. Thus, on the abstractionist's
understanding of a situation as a state or condition of the physical
world rather than a concrete, structured piece of it, the inclusion of
one situation in another is a purely *logical* relation, not a
 mereological
 one. Finally, say that an SOA **s** is *total* if, for every
SOA **t**, **s** either includes or precludes **t**.
(Abstractionists often use 'maximal' instead of
'total', but we have already introduced this term in the
context of concretism.) Abstractionist possible worlds are now
definable straightaway: 




|  |  |  |
| --- | --- | --- |
| **AW2** |  | **w** is a possible world =*def*
**w** is an SOA that is both
possible and
 total.[29] |



It is easy to see that this definition covers the more intuitive
characterizations of abstract possible worlds above: they are
consistent -- i.e., possible -- states of the world that
settle every possibility, consistent states to which no further detail
could be added without rendering them inconsistent. Note also that,
for the abstractionist, as for the concretist, the actual world is no
different in kind from any other possible world; all possible worlds
exist, and in precisely the same sense as the actual world. The actual
world is simply the total possible SOA that, in fact, obtains. And
non-actual worlds are simply those total possible SOAs that do not.




What of existence in such worlds? As we've seen, on Lewis's account,
to exist in a concrete world **w** is literally to exist *in
**w***, that is, within the spatiotemporal boundaries of
**w**. Clearly, because SOAs are abstract, individuals cannot exist
in abstractionist worlds in anything like the same literal,
mereological sense. Accordingly, the abstractionist defines existence
in a world simply to be a special case of the inclusion relation: 




|  |  |  |
| --- | --- | --- |
| **AE2** |  | Individual **a** *exists in* possible world **w**
=*def* **w** includes
****a**'s existing**.  |



Unlike concretism, then, abstractionism does not entail that
individuals are worldbound; there is no inconsistency whatever in the
idea that many distinct worlds can include the existence of one and
the same individual. Indeed, typically, abstractionists are staunchly
committed to transworld identity and hold that most any given
individual exists in many possible worlds and, moreover, that
contingent individuals, at least, can exemplify very different
properties from world to world. Abstractionists, therefore, have no
need to appeal to counterparts to understand *de re* modalities
and can therefore accept the truth conditions for such modalities
given by basic possible world semantics (spelled out, of course, in
terms of their definitions **AW2** and **AE2**). In particular,
they can take the standard possible world truth condition for, e.g.,
the right conjunct of
 (16)
 at face value: '*(*E!a* [?] !*Ta*)'
is true on the abstractionist's approach if and only if there is is a
world in which Algol herself, rather than some counterpart of hers,
exists but fails to be anyone's pet. 


#### 2.2.2 Irreducible Modality and Intensional Entities



It is important to note that the possible worlds of abstractionism do
not yield a reductive analysis of modality. The reason for this is
clear: abstract possible worlds are defined in irreducibly modal terms
-- a possible world is an SOA that (among other things)
*possibly* obtains; or a set of propositions such that it is
*possible* that all of its members are true; or a property that
is *possibly* exemplified; and so on. Hence, unpacked in terms
of the abstractionist's definitions, the possible world truth
conditions for modal propositions are themselves irreducibly modal.
For example, when we unpack Plantinga's definition of a possible world
in the semantic clause for sentences of the form
[?][?]ps[?] in order to derive
the truth condition for
 (17),
 '#[?]*x*(*Gx* - *Mx*)', we
end up with this: 





|  |  |  |
| --- | --- | --- |
|  |  |  For all SOAs **w**, if (i) *possibly*, **w** obtains
and (ii) for all SOAs **s**, either (a) *necessarily*,
**w** obtains only if **s** does or (b) *necessarily*,
**w** obtains only if **s** doesn't, then,
'[?]*x*(*Gx* - *Mx*)' is true at
**w**.  |



If we now unpack the modal operators in (22) using the corresponding
truth conditional clauses of standard possible world semantics, the
result will contain further world quantifiers. And spelling out those
world quantifiers in turn using Plantinga's definition will
re-introduce those same modal operators yet again.



More generally, and a bit more exactly, put: As noted above, the
logical framework of basic possible world semantics is classical
predicate logic. The logical framework of abstractionism is modal
predicate logic. Hence, if possible world semantics is supplemented
with abstractionist definitions of possible worlds, then the logical
framework of possible world semantics becomes modal predicate logic as
well and, as a consequence, the extensionality of the semantics is
lost once again. (This point is expressed somewhat more formally in
the supplemental document
 The Intensionality of Abstractionist Possible World Semantics.)
 Since, as noted above, the central motivation for possible world
semantics was to deliver an extensional semantics for modal languages,
any motivation for abstractionism as a semantic theory is arguably
 undermined.[30]



However, it is not entirely clear that this observation constitutes an
objection to abstractionism. For abstractionists can argue that the
goal of their analysis is the converse of the reductionist's goal: The
reductionist wants to understand modality in terms of worlds; the
abstractionist, by contrast, wants to understand worlds in terms of
modality. That is, abstractionists can argue that we begin with a
primitive notion of modality and, typically upon a certain amount of
philosophical reflection, we subsequently discover an intimate
connection to the notion of a possible world, as revealed in the
principles
 **Nec**
 and
 **Poss**.
 The analysis that abstractionists provide is designed to make this
connection explicit, ideally, in such a way that **Nec** and
**Poss** fall out as theorems of their theory (see, e.g., Plantinga
1985 and Menzel and Zalta 2014). 



Hand in glove with the irreducible nature of modality is the nature of
intensional entities. Concretists define intensional entities in terms
of worlds, as described in
 SS2.1.3.
 Abstractionists, by contrast, define worlds in terms of intensional
entities. This divergence in their choice of ontological primitives
reflects, not only their differing stances toward modality, but also
an important methodological difference with regard to metaphysical
inquiry. The concretist is far more pragmatic; notions of
*property*, *relation*, *proposition*, and the
like play certain roles in our theorizing and are subject to a
"jumble of conflicting *desiderata*" (Lewis 1986,
54). Within a given theory, any entities that can play those roles
fruitfully for the purposes at hand are justifiably identified with
those notions -- regardless of how well they comport with
pre-theoretic intuitions. Thus, Lewis finds it to be a strength of his
position that he is able to adopt the set theoretic definitions in
SS2.1.3. By contrast, at least some abstractionists --
Plantinga (1987) perhaps most notably -- believe that we have
intuitive, pre-theoretic knowledge of intensional entities that
precludes their being identified with set theoretic constructions of
any
 sort.[31]
 (See Stalnaker 1976 for a particularly illuminating discussion of the
contrast between concretism and abstractionism with respect to the
treatment intensional entities.) 


#### 2.2.3 Actuality and Actualism



As was noted in
 SS2.1.2,
 for the concretist, there is no special property of the actual world
-- *actuality* -- that distinguishes it, in any
absolute sense, from all of the others; it is simply the world that
*we* inhabit. For abstractionists, however, actuality
*is* a special property that distinguishes exactly one possible
world from all others -- the actual world is the only world that
happens to *obtain*; it is the one and only way things could be
that is the way things as a whole, in fact, *are*. However, for
most abstractionists, the distinctiveness of the actual world does not
lie simply in its actuality but in its ontological comprehensiveness:
the actual world encompasses all that there is. In a word: most
abstractionists are *actualists*. 



Actualism
 is the thesis that everything that there is, everything that has
*being* in any sense, is actual. In terms of possible worlds:
Everything that exists in any world exists in the actual
 world.[32]
 Possibilism, by contrast, is the denial of actualism; it is the
thesis that there are *mere possibilia*, i.e., things that are
not actual, things that exist in other possible worlds but fail to
exist in the actual world. Concretists are obviously not actualists
(on their understanding of 'actual', at any
 rate).[33]
 Indeed, for the concretist, since individuals are worldbound,
everything that exists in any nonactual possible world is distinct
from everything in the actual world. However, although possibilism and
abstractionism are entirely compatible -- Zalta (1983), for
example, embraces both positions -- abstractionists *tend*
to be actualists. The reason for this is clear: Basic possible world
semantics appears to be committed to possibilism and abstractionism
promises a way of avoiding that commitment. 



The specter of possibilism first arises with regard to
*non-actual* possible worlds, which would seem by definition to
be prime examples of mere *possibilia*. However, we have just
seen that the abstractionist can avoid this apparent commitment to
possibilism by defining possible worlds to be SOAs of a certain sort.
So defined, non-actual worlds, i.e., worlds that fail to obtain, can
still actually exist. Hence, the commitment of basic possible world
semantics to non-actual worlds does not in itself threaten the
actualist's ontological scruples. 




However, the specter of possibilism is not so easily exorcised. For
non-actual worlds are not the only, or even the most compelling,
examples of mere *possibilia* that seem to emerge out of basic
possible world semantics. For instance, it is quite reasonable to
think that evolution could have taken a very different course (or, if
you like, that God could have made very different creative choices)
and that there could have been individuals -- call them
*Exotics* -- that are biologically very different from all
actually existing individuals; so different, in fact, that no actually
existing thing could possibly have been an Exotic. According to basic
possible world semantics, the sentence 'There could have been
Exotics' or, more formally, 





|  |  |  |
| --- | --- | --- |
|  |  |  *[?]*xEx* |



is true just in case there is a world in which
'[?]*xEx*' is true, i.e., when all is said and
done, just in case: 





|  |  |  |
| --- | --- | --- |
|  |  |  There is a possible world **w** and an individual **a**
in **w** such that **a** is an Exotic in **w**,  |



which, a bit less formally, is simply to say that 




|  |  |  |
| --- | --- | --- |
|  |  |  Some individual is an Exotic in some possible world.  |



However, since no actually existing thing could have been an Exotic,
anything that is an Exotic in some possible world cannot be among the
things that exist in the actual world. Thus, the truth conditions that
basic possible world semantics assigns to some of our intuitive modal
beliefs appear to entail that there are non-actual individuals as well
as non-actual possible worlds. Defining possible worlds as SOAs
provided a way for the actualist to embrace non-actual worlds without
compromising her actualism. But how is the actualist to understand the
apparent commitment to non-actual *individuals* in such truth
conditions as (25)? 



Answers that have been given to this question represent a rather deep
divide between actualist abstractionists. On the one hand,
"trace" actualists introduce actually existing entities
into their ontologies that can play the role of mere *possibilia*
in (25) and its like. Trace actualists come in two varieties:
*new* actualists and *haecceitists*. New actualists like
Linsky and Zalta (1996) and Williamson (1998, 2000, 2013) argue that,
in fact, all individuals are actually existing, necessary beings but
not all of them are necessarily *concrete*. Some concrete
individuals -- those traditionally (mis-)categorized
as contingent beings -- are
only contingently concrete. Likewise, some non-concrete individuals
-- those, like possible Exotics, traditionally (mis-)categorized
as contingently non-actual mere
*possibilia* -- are only contingently
 non-concrete.[34]




This novel take on modal metaphysics allows the new actualist to
reinterpret possible world semantics so as to avoid possibilism.
Notably, the domain **d**(**w**) of a world **w** is
understood not as the set of things that exist in **w** -- for
all individuals exist in all worlds -- but the set of things that
are *concrete* in
 **w**.[35]
 Hence, for the new actualist, the correct truth condition for
 (23)
 is: 





|  |  |  |
| --- | --- | --- |
|  |  |  There is a possible world **w** and an individual **a**
that is (i) concrete in **w** and (ii) an Exotic in **w**.  |



On the other hand, haecceitists like Plantinga introduce special
properties -- haecceities -- to similar ends. The haecceity
of an individual **a** is the property of being *that very
individual*, the property **being **a****. A property is a
*haecceity*, then, just in case it is possible that it is the
haecceity of some
 individual.[36]
 It is a necessary truth that everything has a haecceity. More
importantly, for haecceitists, haecceities are necessary beings. Thus,
not only is it the case that, had any particular individual **a**
failed to exist, its haecceity **ha** would still have
existed, it is also the case that, for any "merely
possible" individual **a**, there is an actually existing
haecceity that would have been **a**'s haecceity had **a**
existed. More generally (and more carefully) put: Necessarily, for any
individual **a**, (i) **a** has a haecceity **h** and (ii)
necessarily, **h** exists. 



Like the new actualists, then, the metaphysics of the haecceitists
enables them to systematically reinterpret possible world semantics in
such a way that the truth conditions of modal discourse are expressed
solely in term of actually existing entities of some sort rather than
actual and non-actual individuals. More specifically, for the
haecceitist, the domain **d**(**w**) of a world **w** is
taken to be the set of haecceities that are *exemplified* in
**w**, that is, the set of haecceities **h** such that **w**
includes ****h**'s being exemplified**. Likewise, the
**w**-extension of a (1-place) predicate p is taken to be a
*set* of haecceities -- intuitively, those haecceities
that are coexemplified in **w** with the property expressed by
p. So reinterpreted, the truth condition for
 (23)
 is: 





|  |  |  |
| --- | --- | --- |
|  |  |  There is a possible world **w** and a haecceity **h**
that is (i) exemplified in **w** and (ii) coexemplified with the
property **being an Exotic** in **w**.  |



By contrast, "no-trace", or *strict*, actualists
like Prior (1957), Adams (1981), and Fitch (1996) hew closely to the
intuition that, had a contingent individual **a** failed to exist,
there would have been absolutely no trace, no metaphysical vestige, of
**a** -- neither **a** itself in some non-concrete state
nor any abstract proxy for **a**. Hence, unlike trace actualism,
there are no such vestiges in the actual world of objects that are not
actual but only could have been. The logical consequences for no-trace
actualists, however, appear to be severe; at the least they cannot
provide a standard
 compositional
 semantics for modal languages, according to which (roughly) the
meaning of a sentence is determined by its logical form and the
meanings of its semantically significant constituents. In particular,
if there is nothing to play the role of a "possible
Exotic", nothing that is, or represents, an Exotic in some other
possible world -- a mere *possibile*, a contingently
non-concrete individual, an unexemplified haecceity -- then the
strict actualist cannot provide standard, compositional truth
conditions for quantified propositions like
 (23)
 that yield the intuitively correct truth value. For, understood
compositionally,
 (23)
 is true if and only if '[?]*xEx*' is true at
some world **w**. And that, in turn, is true at **w** if and
only if '*Ex*' is true at **w** for some value of
'*x*'. But, as just noted, for the strict actualist,
there is no such value of '*x*'. Hence, for the
strict actualist, '*Ex*' is false at **w** for all
values of '*x*' and, hence,
 (23)
 is false as well. (These issues are explored in much greater detail
in the entry on the
 possibilism-actualism debate.)
 


#### 2.2.4 A Brief Assessment of Abstractionism



Like concretism, abstractionism provides a reasonably clear and
intuitive account of what worlds are and what it is to exist in them,
albeit from a decidedly different perspective. Although, as noted in
 SS2.2.2,
 the fact that modality is a primitive in abstractionist definitions
of possible worlds arguably compromises its ability to provide
semantically illuminating truth conditions for the modal operators,
those definitions can be taken to illuminate the connection between
our basic modality concepts and the evocative notion of a possible
world that serves as such a powerful conceptual tool for constructing
philosophical arguments and for analyzing and developing solutions to
philosophical problems. In this regard, particularly noteworthy are:
Plantinga's (1974) influential work on
 the ontological argument
 and the free will defense against
 the problem of evil;
 Adams' (1974, 1981) work on
 actualism
 and actuality; and Stalnaker's (1968, 1987) work on
 counterfactual conditionals
 and
 mental content.
 



A number of important objections have been voiced in regard to
abstractionism. Some of these are addressed in the document
 Problems with Abstractionism.
 


### 2.3 Combinatorialism



As its name might suggest, our third approach --
*combinatorialism* -- takes possible worlds to be
recombinations, or rearrangements, of certain metaphysical simples.
Both the nature of simples and the nature of recombination vary from
theory to theory. Quine (1968) and Cresswell (1972), for example,
suggest taking simples to be space-time points (modeled, perhaps, as
triples of real numbers) and worlds themselves to be arbitrary sets of
such points, each set thought of intuitively as a way that matter
could be distributed throughout space-time. (A world *w*, so
construed, then, is *actual* just in case a space-time point
*p* is a member of *w* if and only if *p* is occupied
by matter.) Alternatively, some philosophers define states a world
could be in, and possible worlds themselves, simply to be maximally
consistent sets of
 sentences[37]
 in an expressively rich language --
"recombinations", certainly, of the sentences of the
language. (Lewis refers to this view as *linguistic
 ersatzism*.[38])
 However, the predominant version of combinatorialism finds its
origins in Russell's (1918/1919) theory of
 logical atomism
 and Wittgenstein's (1921, 1922, 1974) short but enormously
influential
 *Tractatus Logico-Philosophicus*.
 A suggestive paper by Skyrms (1981) spelling out some of the ideas in
the *Tractatus*, in turn, inspired a rich and sophisticated
account that is developed and defended in great detail in an important
series of books and articles by D. M. Armstrong (1978a, 1978b, 1986a,
1989, 1997, 2004b, 2004c). In this section, we present a somewhat
simplified version of combinatorialism that draws primarily upon
Armstrong's work. Unless otherwise noted, this is what we shall mean
by 'combinatorialism' for the remainder. 


#### 2.3.1 The Basic Ontology of Combinatorialism



Wittgenstein famously asserted that the world is the totality of
*facts*, not of things (*ibid.*, SS1.1). The
combinatorialist spells out Wittgenstein's aphorism explicitly in
terms of an ontology of objects (a.k.a., particulars), universals
(a.k.a., properties and relations), and facts. Facts are either atomic
or molecular. Every atomic fact -- *Sachverhalt*, in the
language of the *Tractatus* -- is "constituted"
by an *n*-place relation (= property, for *n*=1) and
*n* objects that *stand in*, or *exemplify*, that
relation. Thus, for example, suppose that John is 1.8 meters tall.
Then, in addition to John and the property **being 1.8 meters
tall**, there is for the combinatorialist the atomic fact of John's
exemplifying that property. More generally, atomic facts exist
according to the following principle: 





|  |  |  |
| --- | --- | --- |
| **AF** |  |  Objects **a**1, ..., **a***n*
exemplify *n*-place relation **R** iff there is the fact
**a1, ..., a*n*'s exemplifying R**
([**R**,**a**1,...,**a***n*], for
short).  |



Say that the **a***i* are the *constituent
objects* of the fact in question and **R** its *constituent
universal*, and that **R** and the **a***i*
all *exist in*
[**R**,**a**1,...,**a***n*]. 



A fact is *monadic* if its constituent universal is a property.
A *molecular* fact **f** is a conjunction of atomic facts.
Its constituent objects and universals are exactly those of its
conjuncts and an entity exists in **f** just in case it exists in
one of its conjuncts. (For simplicity, we stipulate that an atomic
fact has (only) itself as a conjunct and, hence, is
"trivially" molecular.) One fact **f** *includes*
another **g** if every conjunct of **g** is a conjunct of
**f**. (Note, importantly, that inclusion, so defined, is quite
different from the homonymous notion defined in the discussion of
abstractionism above -- most notably, combinatorial inclusion is
not a *modal* notion.) For purposes below, say that an object
**a** is a *bare particular in* a molecular fact **f** if
there is no monadic conjunct of **f** of which **a** is the
constituent object, no conjunct of the form **a exemplifies F**,
for some property **F**. **a** is a *bare particular* if
it is bare in every molecular fact. Intuitively, of course, a bare
particular is an unpropertied object. 



There is no upper bound on the "size" of a molecular fact
and no restriction on which atomic facts can form a conjunction; for
any atomic facts at all, there is a molecular fact whose conjuncts are
exactly those facts. As a first cut, then, we can spell out
Wittgenstein's characterization of the (actual) world as the totality
of facts by defining the world to be the largest molecular fact, the
molecular fact that includes all of the atomic
 facts.[39]




Although objects and universals are typically included along with
facts in the basic ontology of combinatorialism, facts are typically
considered more fundamental. Indeed, taking his queue from the
Tractarian thesis that the world consists of facts, not things,
Armstrong (1986a, 577) argues that facts alone are ontologically basic
and that objects and universals are simply "aspects of,
abstractions from" facts. Accordingly, he calls the object
constituent of a fact of the form [**P**,**a**] a
"thin" particular, an object "considered in
abstraction from all its [intrinsic] properties" (1993, 433);
and where **N** is the conjunction of "all the non-relational
properties of that particular (which would presumably include
**P**), the atomic fact **a's exemplifying N** itself is the
corresponding "thick" particular " (*ibid*.,
434 -- we will occasionally use italics to distinguish a thin
particular *a* from the corresponding thick particular **a**).
Though not all combinatorialists of every stripe buy into Armstrong's
"factualist" metaphysics (Bricker 2006), they do generally
agree that facts are more fundamental, at least to the extent that
both the notion of a bare particular, i.e., an object exemplifying no
properties, and that of an unexemplified property are considered
incoherent; insofar as they exist at all, the existence of both
particulars and universals depends on their "occurring" in
some fact or other. Whatever their exact ontological status, it is an
important combinatorialist thesis that exactly *what* objects
and universals exist is ultimately a matter for natural science, not
metaphysics, to decide. 



Objects can be either simple or complex. An object is *simple* if
it has no proper parts, and *complex* otherwise. Like objects,
universals too divide into simple and complex. A universal is simple
if it has no other universal as a constituent, and complex otherwise.
Complex universals accordingly come in two varieties: conjunctive
-- the constituents of which are simply its conjuncts -- and
structural. A structural universal **U** is one that is exemplified
by a complex object **O**, and its constituents are universals
(distinct from **U**) exemplified by simple parts of **O** that
are relevant to **O**'s being an instance of
 **U**.[40]
 It is important to note that, for Armstrong, the constituency
relation is not the mereological parthood relation. Rather, complex
universals (hence also complex facts of which they are constituents)
enjoy a "non-mereological mode of composition" (1997,
119-123) that, in particular, allows for a richer conception of
their
 structure.[41]
 (An assumption of our simplified account here will be that both the
*proper part of* relation and the constituency relation are
well-founded. It follows that (i) there is no
 *gunk*,
 i.e., that every complex object is composed, ultimately, entirely of
simples and (ii) complex universals -- hence the complex facts in
which they are exemplified -- are ultimately
"grounded" in simple facts, i.e., that they cannot be
infinitely decomposed into further complex
 universals/facts.[42])
 



To illustrate the basic idea: in Figure 1, the left-hand diagram
depicts a water molecule **W** comprising an oxygen atom **o**
and two hydrogen atoms **h1** and **h2**.
For the combinatorialist, "thick" particulars like the
molecule itself as well as its constituent atoms are themselves facts:
**o** is the fact [**O**,*o*] in which the universal
**oxygen** (**O**) is exemplified by a thin particular
 *o*;[43]
 likewise **h1** and **h2**. **W** in
turn comprises those monadic facts and the relational facts
[**B**,**o**,**h**1],
[**B**,**o**,**h**2] wherein the covalent bonding
relation **B** holds between the oxygen atom and the two hydrogen
atoms. The structural universal **Water** itself, then, shares this
structure -- it is, so to say, an *isomorph* consisting of
the monadic universals **O** and **H** and the binary relation
**B**, structured as indicated in the right-hand diagram of Figure
 1.[44]







![diagram of structure of water](Water.png)




Figure 1: A Water Molecule **W** and the Structural Universal
**Water** 

 

#### 2.3.2 States of Affairs and Recombination



It should be clear from
 principle **AF**
 that all atomic facts *hold*; that is, all of them reflect
actual exemplification relations. Obviously, however, possibility
encompasses more than what is actual, that is, there are
*possible* facts as well as actual facts; the world's
universals might have been exemplified by its objects very
differently. If they had -- if the world's objects and universals
had combined in a very different way -- there would have been a
very different set of atomic facts and, hence, a very different world.




To spell out the idea of a possible fact, the combinatorialist
introduces the more general notion of an atomic *(combinatorial)
state of affairs*, that is, an entity that simply has the
*form* of an atomic fact -- *n* objects exemplifying
an *n*-place relation -- but without any requirement that
the exemplification relation in question actually holds between them.
More exactly: 




|  |  |  |
| --- | --- | --- |
| **AS** |  |  For any objects **a**1, ...,
**a***n* and any *n*-place relation **R**,
there is an atomic (combinatorial) state of affairs
**a**1, ..., **a***n*'s exemplifying
**R** (again,
[**R**,**a**1,...,**a***n*], for
short).  |



Thus, even if the two hydrogen atoms **h**1 and
**h**2 in a water molecule do not in fact stand in the
covalent bonding relation **B**, there is nonetheless the
(non-factual) state of affairs
[**B**,**h**1,**h**2]. 



Combinatorialism takes facts to be literal, structured parts of the
physical world. This suggests that a non-factual state of affairs
-- a merely *possible* fact -- must be part of a
merely possible physical world. This idea is at odds with the strong,
scientifically-grounded form of actualism that typically motivates
combinatorialism. Two options are available: The combinatorialist can
follow the (actualist) abstractionists and define states of affairs to
be philosophical or mathematical constructs consisting only of actual
objects, properties, relations, and facts. For example, the state of
affairs
[**R**,**a**1,...,**a***n*] can
simply be identified with the ordered *n*-tuple
<**R**,**a**1,...,**a***n*> .
So long as the combinatorialist is willing to adopt the additional
metaphysical or set theoretic machinery, this sort of approach offers
a way of introducing non-factual states of affairs that does not
involve any untoward ontological commitments to merely possible
entities. Alternatively, following Armstrong (1989, 46-51; 1997,
172-4), the combinatorialist can refuse to grant non-factual
states of affairs any genuine ontological status and adopt a form of
  modal fictionalism
 that nonetheless permits one to speak *as if* such states of
affairs exist. The exposition to follow will remain largely neutral
between these options. 



Constituency for states of affairs is understood as for facts.
Additionally, analogous to molecular facts, there are molecular states
of affairs -- conjunctions of atomic states of affairs. Inclusion
between states of affairs is understood exactly as it is between facts
and being a *bare particular* in a molecular state of affairs
**s** is understood as for facts: **a** is a bare particular in
**s** if there is no monadic conjunct of **s** of the form
****a** exemplifies F**. The notion of recombination is now
definable straightaway: 





|  |  |  |
| --- | --- | --- |
|  |  | **s** is a *recombination* of a molecular state of
affairs **f** =*def*
**s** is a molecular state of affairs whose constituent objects and
constituent universals are exactly those of **f**. **s** is a
*non-trivial* recombination of **f** if it does not include
the same states of affairs as **f**.  |



Very roughly then, a *possible world* will be a certain sort of
recombination of (some portion of) the *actual* world, the
molecular fact that includes all of the atomic facts. This idea will
be refined in the following sections. 


#### 2.3.3 Structural States of Affairs and Supervenience



Say that a state of affairs is *structural* if it is atomic and
its constituent universal is structural or it is molecular and
includes a structural state of affairs; and say that it is
*simple* otherwise. The difference between structural and simple
universals and states of affairs is particularly significant with
regard to the important concept of *supervenience* (Armstrong
1989, Ch
 8).[45]
 Entity or entities **S** supervene on entity or entities **R**
if and only if the existence of **R** necessitates that of **S**
(*ibid.*, 103). (Necessitation here is, of course, ultimately to
be spelled out in terms of combinatorial possible worlds.)
Non-structural states of affairs supervene directly on their atomic
 conjuncts.[46]
 However, things are not in general quite so straightforward for
structural states of affairs. For, although structural states of
affairs are ultimately constituted entirely by simple states of
affairs, unlike non-structural states of affairs, structural states of
affairs typically supervene on more than the totality of their
constituents. For, in many cases, whether or not a structural fact
exists depends not only on the existence of its constituent facts but
also on the *absence* of certain others (Armstrong 1997, 34ff).
For example, as noted in our example above, our water molecule
**W** comprises two further facts in which two hydrogen atoms
**h**1 and **h**2 both stand in the
covalent binding relation with an oxygen atom **o**. However, if
**o** were to bind with a further hydrogen atom
**h**3, then, despite the fact that the constituent
facts of **W** would still hold, **W** would not be water; there
would be no such fact as **W's being
 water**.[47]
 Rather, **W** would exist only as a complex part of a hydronium
ion; the new binding [**B**,**o**,**h**3] would,
so to say, "spoil" the instantiation of **Water**.
Thus, more generally, whether or not a structural state of affairs
**S** exists in a possible world typically requires something over
and above its constituent states of affairs being "welded
together" in the right sort of way (Armstrong, 1997, 36); it
requires also that there be no relevant "spoilers" for
 **S**.[48]
 Armstrong draws directly on the initial passages of the
 *Tractatus*[49]
 for the necessary apparatus: a structural state of affairs **S**
in any possible world **w**, supervenes, not simply on its
constituent atomic states of affairs but on a certain
*higher-order* state of affairs **T****w**,
namely, the state of affairs that the (first-order) atomic states of
affairs of **w** are *all the (first-order) atomic states of
affairs* and, hence, that **w** includes no spoilers for
**S**. Armstrong (*ibid.*, 35, 134-5, 196-201)
calls **Tw** the *totality* state of affairs for
the atomic states of affairs of
 **w**.[50]



#### 2.3.4 Combinatorial Possible Worlds and Existence Therein



The idea of possibility being rooted in arbitrary recombinations of
the actual world, rearrangements of its objects and universals, is
intuitively appealing. Clearly, however, not just any such
recombination can count as a possible world. Some states of affairs
are intuitively impossible -- [**being an elephant**,
**e**], where **e** is an individual electron, say -- and
some pairs of states of affairs, while individually possible, are not
*com*possible -- the states of affairs [**having 1kg
mass**, **a**] and [**having 2kg mass**, **a**] for a given
object **a**, or, for a given mereological sum **m** of simples,
the states of affairs [**being a baboon**, **m**] and [**being
a hoolock**, **m**]. But nothing that has been said rules out the
existence of recombinations of the actual world -- rearrangements
of its objects and universals -- that include such states of
affairs. Obviously, however, such recombinations cannot be thought to
represent genuinely possible worlds. Of course, like the
abstractionist, the combinatorialist could simply stipulate as part of
the definition that all legitimate recombinations must be genuinely
*possible* states of affairs of a certain sort, genuinely
*possible* recombinations. But this will not do. For, like
concretism, combinatorialism purports to be a *reductive*
account of modality, an account of possible worlds that does not
depend ultimately on modal notions (see Armstrong 1989,
 33).[51]




Here the distinction between simple and structural states of affairs
together with the combinatorialist's strong notion of supervenience
come to the fore. For, given that structural facts supervene on simple
facts and the actual totality fact **T**@, the actual
world can be defined more parsimoniously as the molecular fact that
includes all the *simple* atomic facts and the totality fact
**T**@. And at the level of simples, there are no
limitations whatever on recombination (Wittgenstein 1921,
2.062-2.063); hence, any recombination of simple objects and
universals is by definition considered possible. Thus Armstrong
(1986a, 579): 




> 
> The simple individuals, properties, and relations may be combined in
> *all* ways to yield possible [simple] atomic states of affairs,
> provided only that the form of atomic facts is respected. That is the
> combinatorial idea.
> 



Worlds, in particular, can be defined as special cases of such
recombinations, together with appropriate totality facts. To state
this, we need a condition that ensures the existence of a unique
actual world: 





|  |  |  |
| --- | --- | --- |
|  |  |  States of affairs **s** and **t** are identical iff they
include exactly the same states of affairs.  |



Given this, we have: 





|  |  |  |
| --- | --- | --- |
|  |  | The (combinatorial) actual world =*def*
the fact @ that includes exactly all
the simple atomic facts and the totality state of affairs
**T**@ for the conjunction of those facts.  |




|  |  |  |
| --- | --- | --- |
| **AW3** |  | **w** is a (combinatorial) possible world =*def*
**w** is a recombination of the
simple atomic facts of the actual world conjoined with the totality
fact **Tw** for that
 recombination.[52] |



Armstrong's ontological commitments are notoriously rather slippery
but, given **AW3**, a reasonably complete notion of existence in a
world is forthcoming. First, let us note that, for Armstrong, the
"combinatorial idea" yields a substantial metaphysical
thesis, as well, viz., the *ontological free lunch* (1986,
12ff), i.e., the thesis that "[w]hat supervenes is no addition
of being"; that "whatever supervenes ... is not something
ontologically additional to the subvenient entity or entities."
Hence, for Armstrong, it appears that *simple* states of
affairs and their constituents exist most fundamentally and that the
existence of more complex entities is in a certain sense
*derivative*. Thus: 





|  |  |  |
| --- | --- | --- |
|  |  |  Entity **a** exists *fundamentally* in
(combinatorial) possible world **w** =*def*
(i) **a** is a simple state of
affairs that **w** includes or (ii) **a** is a constituent or
conjunct of an entity that exists fundamentally in **w**.  |



Given this, existence in a world generally -- both fundamental
and derivative -- both for simples and
 (*first-order*[53])
 non-simples alike, is definable as follows: 




|  |  |  |
| --- | --- | --- |
| **AE3** |  | Entity **a** exists in (combinatorial) possible world
**w** =*def* either (i)
**a** exists fundamentally in **w** or (ii) **a** supervenes
on entities that exist in **w**.  |



Semantics receives rather short shrift in Armstrong's version of
combinatorialism -- at least, semantics in the model theoretic
sense of
 SS1.2
 -- but, as it has played an important role in our discussion of
concretism and abstractionism, we note briefly how the ontology of
combinatorialism might be taken to populate a possible world
interpretation of the language of modal predicate logic. Specifically,
we can take the range of the modal operators -- understood,
semantically, as quantifiers -- to be all of the combinatorial
possible worlds in the sense of **AW3**. The domain
**d**(**w**) of each world **w** is the set of all simple and
complex objects that exist in **w** according to **AE3** and the
**w**-extension **I**p(**w**) of a predicate
p expressing a simple or complex universal **R** is the set of
all *n*-tuples, <**a**1, ...,
**a***n*>  such that the atomic fact
[**R**,**a**1,...,**a***n*] exists
in **w**. 


#### 2.3.5 Analytic and Emergent Modalities; Essential Properties



There are, then, for the combinatorialist no intrinsically modal
phenomena; there are just all of the various worlds that exist on
unrestricted combinatorial grounds alone. Ultimately, all genuine
possibilities, simple or not, are just states of affairs that exist in
these combinatorial worlds in the sense of **AE3**. However, it is
not immediately as clear how to understand many intuitive
*necessities*/*impossibilities* involving complex
structural universals, for example, the impossibilities noted in the
previous section, viz., that something simultaneously have a mass of
both 1kg and 2kg or simultaneously be both a baboon and a hoolock.
Likewise, it is not entirely clear how combinatorialism accounts for
intuitive facts about essential properties, such as that our water
molecule **W** is essentially water or that Algol is essentially a
dog. Combinatorialists argue that such modal facts can nevertheless be
explained in terms that require no appeal to primitive modal features
of the world (Armstrong 2004b, 15). 



**Analytic Modalities.**  Armstrong argues that many
intuitive modal facts -- notably, the impossibility of an object
exemplifying more than one determinate of the same determinable
-- can be understood ultimately as logical, or analytic,
modalities that are grounded in meaning rather than any primitive
modal features of reality. For example, intuitively it is impossible
that an object simultaneously exemplify the structural properties
**having 2kg mass** and **having 1kg mass**. The combinatorial
reason for this (cf. Armstrong 1989, 79) is that, for an object
**a** to exemplify the former property is simply for there to be a
division of **a** into two wholly distinct parts, both of which
exemplify the latter property. Moreover, this division into parts is
entirely arbitrary, that is, for *any* part **a**1
of **a** exemplifying **having 1kg mass**, there is a (unique)
part **a**2 of **a** wholly distinct from
**a**1 that also exemplifies that property. It follows
that, if our 2kg object **a** itself *also* exemplifies
**having 1kg mass**, then, as **a** is a part of itself, there
must be a 1kg part of **a** that is wholly distinct from **a**.
And that is analytically false, false "solely by virtue of the
meaning we attach to" the word 'part' (*ibid*.,
 80).[54]





**Emergent Modalities.**   Combinatorialism purports to
explain a further class of intuitive modal facts as features that
simply "emerge" from facts about structural
 properties.[55]
 The discussion of structural states of affairs and supervenience
above provides an example. Let us suppose the actual world
**w**1 includes our water molecule **W** from Figure
1 plus a further hydrogen atom **h**3. In this world,
only **h**1 and **h**2 bind to **o**.
Hence, this world includes the state of affairs **W's being water**
but not the state of affairs **I's being hydronium** in which
**o**, **h**1, **h**2, and
**h**3 are so bonded as to constitute a hydronium ion
**I**. Conversely, however, given the unrestricted nature of
recombination, there is a world **w**2 that includes
**W** structured as it actually is in **w**1 but
which also includes the spoiler
[**B**,**o**,**h**3] -- where **o** and
**h**3 bond -- and, hence, the structural state of
affairs **I's being hydronium**. Thus, the absence of
[**B**,**o**,**h**3] in **w**1
enables the emergence of **W's being water** and precludes **I's
being hydronium** whilst its presence in **w2**
enables the emergence of the latter but precludes the former. As a
consequence, it is impossible that the states of affairs **W's being
water** and **I's being hydronium**
 coexist.[56]





![diagram of structure of water](Water-and-Hydronium.png)

Figure 2: **W**'s being water and (given a bond between **o**
and **h**3) **I**'s being hydronium 

 


Although more dramatic, large-scale examples of incompatible states of
affairs -- such as a thing's being simultaneously both a baboon
and a hoolock -- might be vastly more complex, there is no
obvious reason why their impossibility could not have the same sort of
combinatorial explanation. 



**Essential Properties.**   It follows from the
unrestricted nature of recombination that, for any simple object
**a** and simple universal **P**, **a** recombines with
**P** in some worlds and fails to recombine with **P** in
others. Generalizing from this fact, it follows that no simple or sum
of simples has any simple universal or conjunction of simple
universals essentially. It also follows that no such object has any
structural property essentially. For assume **o** is such an object
and that it exemplifies a structural property **P**. Since **P**
is structural, it supervenes on some set of simple states of affairs.
But by the nature of recombination, there are combinatorial worlds in
which those states of affairs do not exist and, hence, in which
**P** doesn't but **o** -- being a simple or a sum of
simples -- does. 




 Thick particulars like our water molecule **W** don't fare much
better because of the possibility of spoilers. For Armstrong (1997,
35), **W** is simply the conjunction of its constituent states of
affairs. As we've just seen, however, in the presence of spoilers,
that conjunction would exist -- hence, **W** would exist
-- without being **Water**. Hence, it would seem that at least
some properties that, intuitively, are essential to their bearers turn
out not to be for the combinatorialist. The problem is compounded by
the fact that some intuitively non-essential properties of some thick
particulars are arguably essential for the combinatorialist. The shape
properties of a thick particular **A**, for example, would seem to
be a function of its constituent states of affairs. Moreover, the
exemplification of such properties are not obviously subject to
spoilers the way that natural kind properties like **Water** are.
Hence, as **A** is identical to the conjunction of its constituent
states of affairs, it would seem that it will have the same shape in
any world in which it exists, i.e., it will have that shape
essentially. 



That said, combinatorialism can arguably provide a reasonably robust
analysis of intuitions about the essential properties of ordinary
thick particulars like dogs or persons. Such objects can be taken to
be temporal successions of sums of simples and each sum in the
succession as its temporal parts. Sums in the same rough temporal
neighborhood are composed of roughly the same simples and are
structured in roughly the same way. Similarities between such objects
across worlds in turn determine counterpart relations. Following
Lewis, the essential properties of such objects can then be identified
with those properties exemplified by (all of the temporal parts of)
all of its counterparts in every world in which it exists (Armstrong
1997, 99-103,
 169).[57]



#### 2.3.6 Fewer Things and Other Things: Modified Combinatorialism




 Since a possible world is a recombination of the actual world and
every recombination includes states of affairs involving every simple
individual and every simple universal, by **AE3**, every simple
entity exists in every world. Hence, there could not have been fewer
of them; nor could there have been simples other than the ones there
actually are. In this section, we address this issue and the issue of
contingent existence generally in combinatorialism. 



**Fewer things.**  Combinatorialism as it stands has no
problem accounting for the general intuition that there could have
been fewer things. We have already noted in
 SS2.3.3
 and again in
 SS2.3.5
 how our water molecule **W**, as such, might not have existed.
More generally, given the unrestricted nature of recombination, for
any **a** involving a structural fact **S**, there are
recombinations of the actual world wherein either (a) some of the
relations among **a**'s constituents that are critical to
**S**'s structure fail to be exemplified by those constituents, or
(b) there are further states of affairs included by those
recombinations that act as spoilers for **S**. Consequently, the
combinatorialist seems to have no difficulty explaining how there
might have been fewer water molecules, humans, etc. 



Intuitively, however, there isn't anything in the idea of a simple
that suggests that simples are necessary beings -- especially if,
as combinatorialists generally agree, simples are physical things of
some sort and simple universals are properties of, and relations
among, those things. For there is nothing in the nature of a simple
object to suggest that any given simple had to have existed. Likewise,
there is nothing in the nature of a simple universal to suggest it had
to have been exemplified and, hence, on the combinatorialist's own
conception of universals, that it had to exist. Otherwise put, as
simples exist only insofar as they are constituents of facts, there
seems no reason why there couldn't have been a very small number of
facts, indeed, just a single simple, atomic, monadic fact and, hence,
a lone simple object and a lone simple universal. 



In fact, however, **AW3** can be easily modified to accommodate
these intuitions without doing any serious violence to
combinatorialist intuitions. Specifically, the combinatorialist can
admit "contracted" worlds in which fewer simples exist by
allowing any recombination of *any* simple fact -- that
is, equivalently, by allowing any state of affairs -- to count as
a possible world: 





|  |  |  |
| --- | --- | --- |
| **AW3'** |  | **w** is a (combinatorial) possible world =*def*
**w** is a recombination of some
simple fact **f** conjoined with the totality state of affairs
**Tw** for that recombination. |



**AE3** requires no modification, as it was defined with sufficient
generality above. Under **AW3'**, however, **AE3** entails
that all entities alike -- objects and universals, simple and
structural -- are contingent and, indeed, that every simple
object is the sole constituent of some combinatorial possible world.




**Other things.**  Intuitively, not only could there have
been fewer things, there could have been more things or, more
generally, things *other* than those that actually exist. As
above, combinatorialism as it stands seems able to account for many
instances of this intuition: Figure 2 illustrates how a non-actual
hydronium ion **I** might exist in another world. Likewise, there
seems no reason to deny, e.g., that there are rearrangements **w**
of the actual world's simples wherein exist all of the human beings
that actually exist (at, say, 0000GMT 1 January 2013) and more besides
that are composed of simples that, in fact, constitute things other
than human beings (Armstrong 1997,
 165).[58]
 Combinatorialism also seems able to account for the possibility of
conjunctive and structural universals that are simply rearrangements
of actual simples. It is not implausible to think that such
recombinations can give rise to, say, exotic biological kinds that
have no actual instances (Armstrong 1989, 55-56). Thus, in
particular, combinatorialism seems quite able to provide the truth
condition
 (24)
 for
 (23)
 and, hence, can account for some possibilities involving
"missing" universals that, intuitively, ought to be
possible. 



However, it is far from clear that such possibilities exhaust the
modal intuition that other things could have existed. Notably,
intuitively, there could have been different *simple*
universals distinct from any that actually exist -- different
fundamental properties of simples, for example. Likewise for simple
objects. Either way, there seems to be nothing in the idea of a simple
object or simple universal that suggests there couldn't have been
simples other than, or in addition to, the simples there are in fact.
But **AW3**' does not allow for this; the simples of every
possible world are a subset of the actual simples and there is no
obvious way of modifying the principle to accommodate the intuition.
Nor is there any obvious way of modifying the principle to accommodate
the intuition in
 question.[59]




The combinatorialist could of course abandon actualism and include
merely possible simples into her ontology. Again, she could follow the
new actualists and draw a division between actually concrete and
non-actual, possibly concrete simples; or she could introduce
Plantinga-style haecceities to go proxy for merely possible simples.
But all of these options would be badly out of step with the strong,
naturalist motivations for combinatorialism: There is but the one
physical world comprising all of the facts; recombinations of (at
least some of) those facts -- arbitrary rearrangements of their
simple objects and universals -- determine the possible worlds.
Mere *possibilia*, merely possible non-*concretia*, and
non-qualitative haecceities have no real place in that picture. 



The "purest" option for the combinatorialist is simply to
brazen it out and argue that the actual simples are, in fact, all the
simples there could be (Armstrong 1989, 54ff; Driggers 2011,
56-61). A more robust option suggested by Skyrms (1981) makes
some headway against the problem by introducing an
"outer", or "second-grade" realm of
possibility, but at the cost of moving beyond the basic intuitions of
combinatorialism (Armstrong 1989, 60; 1997, 165-167). Finally,
Sider (2005, 681) suggests that combinatorialists who (like Armstrong)
are modal fictionalists can deal with the problem of missing entities
simply by appealing to yet more fictionalism: As the combinatorialist
fiction *already* includes non-actual states of affairs with
actually existing constituents, there seems no reason not to extend
the fiction to include non-actual states of affairs whose constituents
include non-actual particulars and universals. Fictionalism itself,
however, leaves the combinatorialist with the deep problems detailed
by Kim (1986), Lycan (1993), and Rosen
 (1993).[60]



#### 2.3.7 A Brief Assessment of Combinatorialism



As with concretism and abstractionism, combinatorialism provides
reasonably clear definitions of possible worlds and existence in a
world and is noteworthy for its attempt to avoid what might be thought
of as the metaphysical excesses of the two competing views. In
contrast to concretism, combinatorialism is staunchly actualist:
instead of an infinity of alternative physical universes, each with
its own unique inhabitants existing as robustly as the inhabitants of
the actual world, the worlds of combinatorialism are simply
rearrangements of the universals and particulars of the actual world;
and commitment even to them might be avoided if some version of
fictionalism is tenable. Likewise, in contrast to abstractionism's
rather rich and unrestrained ontology of SOAs, combinatorialism's
states of affairs are comparatively modest. Moreover, in contrast to
nearly all versions of abstractionism, combinatorialism shares with
concretism the virtue of a reductive theory of modality: Modal
statements, ultimately, are true or false in virtue of how things
stand with respect to worlds that are themselves defined in non-modal
terms. 



Combinatorialism's ontological modesty, however, is also a weakness.
For, unlike, the two competing approaches, there are modal intuitions
that the combinatorialist is not easily able to account for, notably,
the intuition that there could have been other things. Additional
difficulties are discussed in the supplemental document
 Further Problems for Combinatorialism.