Propositional contingentism and possible worlds

Abstract

Propositional contingentism is the view that what propositions there are is a contingent matter—certain propositions ontologically depend on objects which themselves only contingently exist. Possible worlds are, loosely, complete ways the world could have been. That is to say, the ways in which everything in its totality could have been. Propositional contingentists make use of possible worlds frequently. However, a neglected, but important, question concerns whether there are any notions of worlds which are both theoretically adequate and consistent with propositional contingentism. Some notion of a possible world is adequate if the systematic connection between, at least, possibility and truth at some possible world holds. Here, I argue that no adequate notion of a possible world is available to at least those who subscribe to one natural formulation of propositional contingentism. I also show that this result contrasts with a simple and adequate definition of a possible world available to the necessitist—those who hold that necessarily everything necessarily exists.

According to contingentism, there might have been things which might have been nothing. For instance, it strikes many as undeniable that, had my parents never met, I would not have existed—in such a case, I would have been nothing. In fact, some contingentists go further and argue that had I never existed the proposition [I do not exist] would not itself exist. That is to say, in such cases, there just would be no content to the claim that I exist or that I do not exist, see (Adams, 1981), (Fine, 1985), (Fitch, 1996), (Prior, 1967), (Speaks, 2012), and (Stalnaker, 2012). For the sake of simplicity, let propositional contingentism just be the view that some propositions and some non-propositions are contingent, where propositions are contingent because they depend, for their existence, on the existence of other contingent objects.

Propositional contingentists often make use of possible worlds—maximally specific ways the world could have been. For instance, Robert Stalnaker in describing a consequence of his view writes that ‘[t]he singular proposition Socrates does not exist is a proposition that will be true of ... only possible worlds in which that proposition does not exist’ (Stalnaker, 2012, p. 46). Another example is Jason Turner, who formulates one common version of propositional contingentism in terms of possible worlds:

If a proposition p exists and predicates something of an object a, then in any possible world W, if a does not exist in W, then p also does not exist in W. (Turner, 2005, p. 191)

Such talk is ubiquitous, but it prompts a natural although neglected question: to what extent is talk of possible worlds available to the propositional contingentist at all? Of course, some uses of possible worlds are quite innocent, e.g., the nature of ‘worlds’ in a model-theoretic semantics can be somewhat arbitrary. However, worlds are also used in philosophical accounts of modality. Here, I am concerned with this second use. To think that worlds play a significant philosophical role in our account of modality is here understood to at least involve a commitment to the truth of the right sort of biconditional connecting modality and possible worlds, where the latter are entities understood as genuine ways, or specifications of the ways, the world could have been.

I argue that, for the necessitist—one who thinks that necessarily everything, propositions and non-propositions alike, necessarily exists—possible worlds suitable for playing this role can be defined. However, I argue that the contingentist is unable to replicate this success. My strategy throughout the paper is to investigate the question of whether different views can make use of possible worlds by looking at whether we can embed adequate notions of possible worlds into logics for those views. Here, I take a logic for a metaphysical view to be a proof system which has as theorems only truths which hold generally and necessarily according to that view. Developing these logics is not intended to tell us what these modal metaphysical views should think about logic in a grand sense of the word. Rather, such logics are intended to allow us to more rigorously assess what a view is, and can be, committed to.¹ In this framework, a natural measure for whether a certain view can make use of possible worlds is to follow Menzel and Zalta (2014) and say that a notion of a possible world in such a logic is adequate only if there is a theorem in that logic which can be read as stating a biconditional connecting metaphysical possibility and necessity to truth in some, or all, worlds, respectively, i.e., the Leibnizian biconditionals. I assume, here, that regardless of whether some view takes worlds to feature in an analysis, or reduction, of modality, or whether world-facts ground modal-facts, or so on, the Leibnizian biconditionals should be generally and necessarily true, according to that view. A view can make use of possible worlds, if a definition of a world can be given in its logic and such biconditionals are theorems.

Here’s how this paper will proceed. In $\S$1, I start with the simpler case and develop a necessitist logic $\vdash _n$ and show that the systematic connections between possibility as truth at some world and necessity as truth at all worlds, for at least one definition of a possible world, are theorems of $\vdash _n$. Then, in $\S$2, I develop a weaker proof system $\vdash _c$ which captures a natural and promising form of propositional contingentism. I show that the availability of contingentist possible worlds is inconsistent in $\vdash _c$ with some well-motivated claims about possibly indistinguishable entities that the propositional contingentist should accept. Thus, they cannot articulate adequate definitions of possible worlds ($\S$3). I then consider an extension of $\vdash _c$ with an actuality operator $\vdash _c^@$ to investigate whether the contingentist can define worlds for which the connection between possibility and truth at some world holds only actually, if not necessarily ($\S$4). I show that this requirement is equally problematic for the contingentist. Finally, in $\S$6, I prove some formal results underpinning my argument.

Before turning to these arguments, it is worth first saying something in the way of motivating the first-order treatment of propositions in this paper, since this contrasts with recent work on propositional contingentism.² Here, propositions are understood as objects over which first-order variables can range. In contrast to this, propositional contingentism is reasonably well-studied in higher-order settings, notably in the work of (Fritz, 2016, 2017, 2018a, b) and Fritz and Goodman (2016, 2017a). This, in turn, is part of a growing trend of applying the resources of higher-order logic to metaphysical questions, particularly to investigate intensional entities like propositions and properties.³ Now, this paper is not the place to address questions about whether such a trend is on the whole worthwhile. However, it is worth noting that the use of higher-order resources in metaphysics is not widely accepted and taking this approach is far from mandated.⁴ Moreover, a large portion of the recent work on propositional contingentism has been done, assuming a relational type theory, particularly (Fritz & Goodman, 2016) and (Fritz, 2018a, b), and thus it is of a broader theoretical interest to investigate questions about propositional contingentism independently of the specific logical and metaphysical constraints intrinsic to such a setting.⁵ This, in my view, warrants exploring questions about propositional contingentism from a first-order perspective.

1 Necessitism and possible worlds

I begin by outlining the simpler case: a necessitist logic $\vdash _n$ in which we can investigate whether the necessitist can define an adequate notion of a possible world. Outlining the simpler case allows me to discuss principles which carry over to the contingentist logic $\vdash _c$. Moreover, I will show that the necessitist can define an adequate notion of a world, showing that any failure of a contingentist project to make sense of worlds is not the product of a general poverty in the very notion of a possible world.

1.1 The necessitist logic $\vdash _n$

A simple option for a necessitist is to take possible worlds as maximally consistent pluralities of propositions. The necessitist proof system $\vdash _n$ is a plural, two-sorted, first-order modal logic which I will extend to include propositional abstraction and a truth predicate, in a language I call $\mathcal {L}^W$. $\mathcal {L}^W$ is composed of the following lexicon. First, for each natural number n:

• Singular non-propositional variables, $x_n, y_n, z_n$.

• Singular propositional variables, $p_n, q_n, r_n$.

• Plural non-propositional variables, $xx_n, yy_n, zz_n$.

• Plural propositional variables $pp_n, qq_n, rr_n$.

• Denumerably many n-place singular predicates, $R_n^1, R_n^2, ..., R_n^i$.

• Denumerably many n-place plural predicates, $\overline{R}_n^1, \overline{R}_n^2, ..., \overline{R}_n^i$.

We have the following function, logical connectives and logical predicate symbols:

• One function symbol, $\mathtt {\sim }$.

• Singular identity symbol, $=$; an is among symbol, $\prec$; a truth predicate, $\mathrm {T}$.

• $\lnot$, $\vee$, $\wedge$, $\rightarrow$, $\leftrightarrow$, $\forall$, $\exists$, $\lozenge$, $\Box$, (, ) and [, ].⁶

The syntax of $\mathcal {L}^W$ is specified as follows. First, the syntactic rules for complex singular terms in $\mathcal {L}^W$: first, $\mathtt {\sim }t$ is a propositional singular term if t is a propositional singular term and second, $[\phi ]$ is a singular term if $\phi$ is a formula.⁷ The well-formed formulae $\phi \in \mathcal {L}^W$ are defined recursively. That is, first, $Rt_1, ..., t_n$ and $\overline{R}tt_1, ..., tt_n$ are formulae iff R is an n-place singular predicate and $t_1, ..., t_n$ are n singular terms of any sort and $\overline{R}$ is an n-place plural predicate and $tt_1, ..., tt_n$ are n plural terms of any sort, respectively. Second, if $\phi$ is a formula, v and vv are variables of any sort, then $\lnot \phi$, $\Box \phi$ and $\lozenge \phi$, $\forall v \phi$, $\exists v \phi$, $\forall vv \phi$, and $\exists vv \phi$, are also all formulae of $\mathcal {L}^{W}$. Third, if t and tt are singular and plural term of any sort, then $t \prec tt$ is a formula. Finally, if $\phi$ and $\psi$ are formulae, $\phi \vee \psi$, $\phi \wedge \psi$, $\phi \rightarrow \psi$, and $\phi \leftrightarrow \psi$ are formulae.

The necessitist logic $\vdash _n$ is the result of combining the axioms for classical quantification and the axioms for an S5 modal logic, extending this to include plural quantification and propositional abstraction. To define $\vdash _n$, we say, as is standard, that a $\mathcal {L}^W$ substitution instance of a well-formed formula of the language of propositional logic $\Phi$ is any expression which is the result of uniformly replacing every propositional variable in $\Phi$ with a well-formed formula of $\mathcal {L}^W$. Second, I will write $\ulcorner \phi [t/v] \urcorner$ for the result of uniformly replacing all free instances of v in $\phi$ with t—here v and t are either both a singular or plural variable and term of any sort.⁸ Third, we say that a term t is free for variable x just in case no free occurrence of x in $\phi$ lies within the scope of a quantifier $\forall y$ or $\exists y$, where y is a variable which is free in t.

Definition 1

($\vdash _n$) Let $\vdash _n$ be the proof system in $\mathcal {L}^W$ consisting of the following principles. Here $\phi$ and $\psi$ are wff of $\mathcal {L}^W$, v stands for any singular variable of any sort, vv, $vv'$ for any distinct plural variables of any sort, and v for any variable, singular or plural, of any sort. Unless stated otherwise, take v, vv, and $vv'$ to be of the same sort—either all propositional, or all non-propositional. The following are axioms:

• (PC) Any $\mathcal {L}^W$ substitution instance of a tautology.

• ($\forall$1) $\forall {\textbf {v}} \phi \rightarrow \phi [t/{\textbf {v}}]$, provided t is free for v in $\phi$.

• (QE) $\exists {\textbf {v}} \phi \leftrightarrow \lnot \forall {\textbf {v}} \lnot \phi$.

• (Comp) $\exists v \phi (v) \rightarrow \exists vv \forall v (v \prec vv \leftrightarrow \phi (v))$, for $\phi$ free of vv.

• (NE) $\forall vv \exists v (v \prec vv)$.

• (Ext) $\forall vv \forall vv'(\forall v(v \prec vv \leftrightarrow v \prec vv') \rightarrow (\phi (vv) \leftrightarrow \phi (vv')))$.

• (P1) $t \prec tt \rightarrow \Box t \prec tt$, for any terms t, tt.

• (I1) $t = t$, for any term t.

• (I2) $t = t' \rightarrow (\phi [t'/v] \leftrightarrow \phi [t/v])$, for any t, $t'$ free for v in $\phi$.

• (T[]) $\mathrm {T}[\phi ] \leftrightarrow \phi$.

• (PNeg1) $\lnot \mathrm {T}t \leftrightarrow \mathrm {T}\mathtt {\sim }t$, for any propositional term t.

• (PNeg2) $t = [\phi ] \leftrightarrow \mathtt {\sim } t = [\lnot \phi ]$, for any propositional term t.

• (OE) $\Box \phi \leftrightarrow \lnot \lozenge \lnot \phi$.

• (K) $\Box (\phi \rightarrow \psi ) \rightarrow (\Box \phi \rightarrow \Box \psi )$.

• (T) $\Box \phi \rightarrow \phi$.

• (5) $\lozenge \phi \rightarrow \Box \lozenge \phi$.

The following are rules of inference:

• ($\forall$2) $\phi \rightarrow \psi$ / $\phi \rightarrow \forall {\textbf {v}} \psi$, provided v is not free in $\phi$.

• (MP) $\phi$, $\phi \rightarrow \psi$ / $\psi$.

• (N) $\phi$ / $\Box \phi$

Some comments are in order. Pluralities, here, are thought to be nothing over and above the objects which are among them. They are simply the things, considered together in some plural fashion. (Comp), (NE), (Ext) and (P1) govern the behaviour of such pluralities.⁹ Strictly speaking, (P1) is independent of the natural extension of the theory of plural quantification into a theory of modal plural quantification (Hewitt, 2012). However, I take it to be a reasonable modal axiom governing plurals, even if it is not quite wholly uncontentious.¹⁰ The conception of pluralities as nothing over and above the things among them also motivates the idea that pluralities are individuated extensionally and this allows us to define a notion of plural identity.

Definition 2

(PLURAL IDENTITY) Let $\left.{\&\#Xulcorner;}tt = tt' \right.{\&\#Xurcorner;}$ abbreviate $\left.{\&\#Xulcorner;}\forall v (v \prec tt \leftrightarrow v\prec tt') \right.{\&\#Xurcorner;}$, where v, tt and $tt'$ are variables and terms, respectively, each of the same, but any, sort.

Thus, plural parallels for the singular identity axioms are theorems. In other words, for any terms tt, $tt'$: $\vdash _n tt = tt'$ and $\vdash _n tt = tt' \rightarrow (\phi [tt/xx] \leftrightarrow \phi [tt'/xx])$. The latter theorem follows from (2) and (Ext)—the axiom that extensionally equivalent pluralities satisfy the same open-sentences of $\mathcal {L}^{W}$.

Several axioms of $\vdash _n$ govern propositions and, although it is controversial what propositions are, such axioms in $\vdash _n$ characterise propositions minimally. First, (T[ ]) captures the relation between propositions and formulae of $\mathcal {L}^W$—$[\phi ]$ is the proposition expressed by $\phi$, and so, naturally, $[\phi ]$ is true if and only if $\phi$. Second (PNeg1) and (PNeg2) together capture one way in which propositions relate to each other in ways which mirror the relations between sentences indicated by sentential operations. We should at least like to say that, for any proposition p, there is a distinct proposition which is the negation of p. Since propositional quantification in the logic here is nominal, I need propositional analogues to the sentential operators. For my purposes, I need only be explicit about negation–we have $\mathtt {\sim }$ as a function which takes any proposition as argument, and returns that proposition’s negation. Of course, we also need some comprehension principle for propositions, i.e., a principle to govern the conditions under which a proposition exists. In $\vdash _n$, the quantification is classical and so the comprehension principle is derived:

Theorem 3

(COMPREHENSION) $\vdash _n \exists p(p = [\phi ])$, for $\phi \in \mathcal {L}^W$.

It is thus guaranteed that there is a proposition, i.e., $[\phi ]$, for every formula $\phi \in \mathcal {L}^W$. The classical quantification, identity axioms and the rule of necessitation also guarantee the truth of singular and plural necessitism, for both non-propositions and propositions alike in $\vdash _n$. Thus: $\vdash _n \Box \forall x \Box \exists y(y = x)$, $\vdash _n \Box \forall p \Box \exists q (q = p)$, $\vdash _n \Box \forall xx \Box \exists yy (yy=xx)$, and $\vdash _n \Box \forall pp \Box \exists qq(pp = qq)$. One final feature of $\vdash _n$ should be noted, namely that the Barcan Formula is a theorem, i.e. $\vdash _n \lozenge \exists x \phi \rightarrow \exists x \lozenge \phi$. From hereon, I will refer to this as ‘BF’.

1.2 Possible worlds in $\vdash _n$

Possible worlds are supposed to be maximally specific ways the world could have been and there are several ways of implementing this idea. The strategy taken in $\vdash _n$ is to take possible worlds as maximally consistent pluralities of propositions. Of course, maximally consistent pluralities of propositions are not the only way of defining possible worlds. For instance, we have the idea that worlds are maximally consistent sets of propositions (Adams, 1981); maximally inclusive and possible states of affairs (McMichael, 1983; Plantinga, 1976, 1979); maximally consistent individual propositions (Fine, 1977a; Stalnaker, 2012); or certain special properties of the world (Forrest, 1986; Stalnaker, 2012). Pluralities rather than sets are used here in order to avoid worries about cardinality, see (Bringsjord, 1985; Grim, 1986; Menzel, 1986), and (Menzel, 2012).¹¹ But, all in all, very little, if anything, of what follows involves making overly specific assumptions about worlds as pluralities of propositions or making assumptions for which there would not be some analogous assumptions about salient features of alternative conceptions of possible worlds. (I will discuss this point about the generality of the arguments presented in this paper in more detail in $\S$3.)

To start, then, we first define what it means for some propositions to be maximal and some propositions to be consistent:

Definition 4

(MAXIMALITY): Propositions pp are maximal if $\forall p (p \prec pp \vee \mathtt {\sim } p \prec pp)$.

Definition 5

(CONSISTENCY): Propositions pp are consistent if $\lozenge \forall p (p \prec pp \rightarrow \mathrm {T}p)$.

We then define a world:

Definition 6

(WORLD): Propositions pp are a world ($\mathrm {W}pp$) just in case pp are both maximal and consistent.

As possible worlds, such pluralities ought to have certain features. Importantly, some propositions are true, and some propositions are false, relative to a world. Following the notation in Menzel and Zalta (2014), I write $\ulcorner pp \models p \urcorner$ to stand for the claim that p is true relative to pp. In $\vdash _n$ this is defined as follows, where I write $\left.{\&\#Xulcorner;}\overline{\mathrm {T}}pp \right.{\&\#Xurcorner;}$ for $\left.{\&\#Xulcorner;}\forall p (p \prec pp \rightarrow \mathrm {T}p) \right.{\&\#Xurcorner;}$.

Definition 7

(TRUTH IN): Let some proposition p be true in some propositions pp $(\left.{\&\#Xulcorner;}pp \models p \right.{\&\#Xurcorner;})$ just in case $\Box (\overline{\mathrm {T}}pp \rightarrow \mathrm {T}p)$.

The definitions here allow for a simple and attractive view of possible worlds. Simple, since we require no more than (4)–(7) and the uncontroversial characterisation of propositions in $\vdash _n$. Attractive, since the systematic connections between possibility and necessity, on the one hand, and truth relative to some or all worlds, respectively, on the other hold as theorems in $\vdash _n$. Let’s call these systematic connections the Fundamental Theorems of Possibility and Necessity, following (Menzel & Zalta, 2014).

Theorem 8

(FUNDAMENTAL POSSIBILITY): $\vdash _n \lozenge \phi \leftrightarrow \exists pp(\mathrm {W}pp \wedge pp \models [\phi ])$, for any formula $\phi \in \mathcal {L}^W$ with no free ‘pp’.

Theorem 9

(FUNDAMENTAL NECESSITY): $\vdash _n \Box \phi \leftrightarrow \forall pp(\mathrm {W}pp\rightarrow pp \models [\phi ])$, for any formula $\phi \in \mathcal {L}^W$ with no free ‘pp’.

The proof of (8) and (9) to follow is related to the proof in (Menzel & Zalta, 2014, pp. 345–348). My proof requires four preliminary lemmas—one about possibility and possible truth, one about propositions, and two about worlds. Here, t is any singular propositional term.

Lemma 10

For any formula $\phi \in \mathcal {L}^W$: $\vdash _n \lozenge \mathrm {T}[\phi ] \leftrightarrow \lozenge \phi$.

Lemma 11

Let $\left.{\&\#Xulcorner;}\tau (pp) \right.{\&\#Xurcorner;}$ abbreviate $\left.{\&\#Xulcorner;}\forall p (p \prec pp \leftrightarrow \mathrm {T}p) \right.{\&\#Xurcorner;}$. The following hold.

• (i) $\vdash _n \mathrm {T}t \rightarrow \exists pp(\tau (pp) \wedge t \prec pp)$

• (ii) $\vdash _n \tau (pp) \rightarrow \mathrm {W}pp$

Lemma 12

$\vdash _n \mathrm {W}pp \rightarrow \Box \mathrm {W}pp$

Lemma 13

$\vdash _n \mathrm {W}pp \rightarrow (pp \models t \leftrightarrow t \prec pp)$.

Now, here’s a sketch of how we derive (8) and (9). First we use (10) to show that, if there are some propositions which qualify as a world and $[\phi ]$ is true in those, then $\lozenge \phi$ must hold. This establishes the right-to-left direction of (8)—if there exists a world w and $[\phi ]$ is true in w, then $\lozenge \phi$. The left-to-right direction of (8) is more demanding. First, we show that if $\lozenge \phi$ then, by (11)(i), possibly some propositions are such that $\tau (pp)$ and $[\phi ] \prec \tau (pp)$, i.e., $[\phi ]$ is possibly among some propositions which are all and only the true propositions. Second, by (11)(ii), we know that, necessarily, if pp are such that $\tau (pp)$, then pp are a world. By (13), this establishes that if $\lozenge \phi$ holds, then possibly there is a world in which $[\phi ]$ is true. Given the assumption that propositions necessarily exist and metaphysical modality satisfies the principles of S5, it then follows that there is such a world. (9) follows from (8) as a corollary. Formally, the proofs are as follows.

Theorem 8

(FUNDAMENTAL POSSIBILITY): $\vdash _n \lozenge \phi \leftrightarrow \exists pp(\mathrm {W}pp \wedge pp \models [\phi ])$, for any formula $\phi \in \mathcal {L}^W$ with no free ‘pp’.

Proof

First, the right-to-left direction. ¹²

• (1) $\vdash _n \exists pp(\mathrm {W}pp \wedge pp \models [\phi ]) \rightarrow \exists pp(\lozenge \overline{\mathrm {T}} pp \wedge \Box (\overline{\mathrm {T}} pp \rightarrow \mathrm {T}[\phi ]))$ D.4, D.7

• (2) $\vdash _n \exists pp(\lozenge \overline{\mathrm {T}} pp \wedge \Box (\overline{\mathrm {T}} pp \rightarrow \mathrm {T}[\phi ])) \rightarrow \lozenge \mathrm {T}[\phi ]$ QML

• (3) $\vdash _n \exists pp(\mathrm {W}pp \wedge pp \models [\phi ]) \rightarrow \lozenge \phi$ (1), (2), L.10

Second, the left-to-right direction.

• (1) $\vdash _n \lozenge \phi \rightarrow \lozenge \exists pp (\tau (pp) \wedge [\phi ] \prec pp)$ L.10, L.11(i), (N)

• (2) $\vdash _n \lozenge \exists pp (\tau (pp) \wedge [\phi ] \prec pp) \rightarrow \lozenge \exists pp (\mathrm {W}pp \wedge [\phi ] \prec pp)$ L.11(ii), (N)

• (3) $\vdash _n \lozenge \exists pp (\mathrm {W}pp \wedge [\phi ] \prec pp) \rightarrow \lozenge \exists pp (\Box \mathrm {W}pp \wedge \Box [\phi ] \prec pp)$ L.12, (P1), (N)

• (4) $\vdash _n \lozenge \exists pp (\Box \mathrm {W}pp \wedge \Box [\phi ] \prec pp) \rightarrow \exists pp \lozenge (\Box \mathrm {W}pp \wedge \Box [\phi ] \prec pp)$ (BF)

• (5) $\vdash _n \exists pp \lozenge (\Box \mathrm {W}pp \wedge \Box [\phi ] \prec pp) \rightarrow \exists pp (\lozenge \Box \mathrm {W}pp \wedge \lozenge \Box [\phi ] \prec pp)$ QML

• (6) $\vdash _n \exists pp (\lozenge \Box \mathrm {W}pp \wedge \lozenge \Box [\phi ] \prec pp) \rightarrow \exists pp ( \mathrm {W}pp \wedge [\phi ] \prec pp)$ S5

• (7) $\vdash _n \lozenge \phi \rightarrow \exists pp ( \mathrm {W}pp \wedge [\phi ] \prec pp)$ (1)–(6)

• (8) $\vdash _n \lozenge \phi \rightarrow \exists pp ( \mathrm {W}pp \wedge pp \models [\phi ])$ (7), L.13

$\square$

Theorem 9

(FUNDAMENTAL NECESSITY): $\vdash _n \Box \phi \leftrightarrow \forall pp(\mathrm {W}pp\rightarrow pp \models [\phi ])$, for any formula $\phi \in \mathcal {L}^W$ with no free ‘pp’.

Proof

First, the left-to-right direction.

• (1) $\vdash _n \forall pp(\mathrm {W}pp \rightarrow ([\phi ] \prec pp \vee [\lnot \phi ] \prec pp))$ (PNeg2), D.4,D.6

• (2) $\vdash _n \exists pp(\mathrm {W}pp \wedge \lnot (pp \models [\phi ])) \rightarrow \exists pp(\mathrm {W}pp \wedge pp \models [\lnot \phi ])$ (1), L.13

• (3) $\vdash _n \exists pp(\mathrm {W}pp \wedge pp \models [\lnot \phi ]) \rightarrow \lozenge \lnot \phi$ T.8

• (4) $\vdash _n \Box \phi \rightarrow \forall pp(\mathrm {W}pp \rightarrow pp \models [\phi ])$ (2), (3), (QE), (OE)

Second, the right-to-left direction.

• (1) $\vdash _n \lozenge \lnot \phi \rightarrow \exists pp(\mathrm {W}pp \wedge pp \models [\lnot \phi ])$ T.8

• (2) $\vdash _n \exists pp(\mathrm {W}pp \wedge pp \models [\lnot \phi ]) \rightarrow \exists pp(\mathrm {W}pp \wedge \lnot (pp \models [\phi ]))$ D.4, D.6, (PNeg2)

• (3) $\vdash _n \lozenge \lnot \phi \rightarrow \exists pp(\mathrm {W}pp \wedge \lnot (pp \models [\phi ]))$ (1), (2)

• (4) $\vdash _n \forall pp(\mathrm {W}pp \rightarrow pp \models [\phi ]) \rightarrow \Box \phi$ (3), (QE)

$\square$

This completes the exposition of a simple, necessitist theory of possible worlds. The systematic connections between possibility, necessity and truth at some, or all, worlds, respectively, are theorems of $\vdash _n$. As noted in(Menzel & Zalta, 2014, p. 336), few philosophers explicitly argue for the connection between possibility, necessity and truth at a world. Thus, the previous results are interesting. In short, given the conception of pluralities in $\vdash _n$, the necessitist simply needs some propositions from which theoretically adequate possible worlds can be defined.¹³ Thus, the necessitist does not even require a potentially suspect primitive notion of a possible world. Moreover, these results show that, in principle, there is no general poverty in the very notion of a world itself. The question for the rest of the paper then is whether we can do the same with weaker assumptions and formulate adequate worlds consistent with propositional contingentism.

2 Propositional contingentism and logic $\vdash _c$

Before I develop the propositional contingentist logic $\vdash _c$ and investigate whether we can embed theoretically adequate definitions of worlds into such a logic, I should first discuss the nature of propositional contingentism in more detail. Now, it is beyond the scope of this paper to survey all formulations of the view. Instead, I will first sketch one common and compelling view of contingently existing propositions. I will then discuss how we should go about axiomatizing this view to capture the general and necessary truths of propositional contingentism.

The view which interests me finds a clear expression in Prior (1957). His view is that, if some sentence $\phi$ contains a term, then $\phi$ expresses a proposition which exists only in the cases in which the referent of that term exists (Ibid., p. 34). Of course, Prior holds this view for distinctly Priorian reasons: the contingency in what propositions there are arises, since he holds that to say that x exists is just to say that there are facts about x, where facts are, for Prior, true propositions (Ibid.). Nonetheless, this view—that propositions expressed by sentences featuring terms depend ontologically on the referents of those terms—has a wide appeal in the literature, e.g., see Jeff Speaks 2012, p. 529 and Jason Turner 2005, p. 192 for clear endorsements of this view.

Now, how we go about constructing a logic for this view involves making several decisions. To begin with, it is, of course, a contingentist view. Since necessitism results as a combined effect of our quantifier and identity axioms—specifically ($\forall$1) and (I1)—and the necessitation rule (N), we need to amend one or more of those principles. There are two standard approaches. The first is to restrict (N) so that not all theorems are necessary, see (Prior, 1957, pp. 48–49), (Adams, 1981, p. 27), (Deutsch, 1990, p. 99), and (Menzel, 1991, p. 359). The second option is adopt a free logic, allowing for only a restricted rule of existential generalisation. This means that from $\vdash _c x=x$ we can only derive the rather trivial $\vdash _c x = x \rightarrow (\mathrm {E}x \rightarrow \exists y(y=x))$, where ‘$\mathrm {E}$’ is an existence predicate. Thus we can only innocuously conclude that $\vdash _c \forall x\Box (\mathrm {E}x \rightarrow \exists y(y=x))$. (I will discuss the meaning of the existence predicates later in this section.)

In my view, the second option is preferable because we have the following good reasons for retaining (N). First, a minor reason: the few positive results for the contingentist presented in $\S$3 show that (N) is crucial to securing some of the positive results for the propositional contingentist and so restricting (N) only further impoverishes $\vdash _c$. This being said, however, the crucial reason concerns the role of the logic $\vdash _c$. To emphasise again, I am interested in axiomatizing truths which hold generally and necessarily for the propositional contingentist. In other words, the formulae derivable in $\vdash _c$ are only those formulae which remain true for the contingentist when prefixed with necessity operators and universal quantifiers in any sequence. Thus, the set of theorems of $\vdash _c$ should be closed under necessitation—this requires us to adopt (N) as a rule in $\vdash _c$. It is worth stressing here, however, that I do not take accepting (N) in $\vdash _c$ to have any bearing on whether the theorems of some one true modal logic, if there is such a logic, are closed under necessitation. This argument for (N) is particular to the role of $\vdash _c$ as a logic which captures only the general and necessary truths of propositional contingentism.

The logic $\vdash _c$ is, of course, a propositional contingentist logic. Thus, we need a distinctive comprehension principle to capture the contingency of propositions. In contrast to $\vdash _n$, such a principle must be adopted as an explicit axiom in $\vdash _c$. Here’s how we formulate this principle. Recall that the motivating thought was that propositions expressed by sentences featuring free variables or complex terms ontologically depend on the referents of those parameters. Thus, we can read off which propositions are contingent existents by looking at the syntax of the sentences expressing those propositions. Let’s write that a sentence features n terms $t_1, ..., t_n$ by writing $\left.{\&\#Xulcorner;}\phi ^{t_1, ..., t_n} \right.{\&\#Xurcorner;}$. Here, $t_1, ..., t_n$ can contain any sort of term, singular or plural. With this in mind, then, we adopt the following axiom scheme, where $\ulcorner \mathrm {E}/\overline{\mathrm {E}} \urcorner$ stands for the appropriate existence predicate.

• (P) $\vdash _c \exists p (p = [\phi ^{t_1, ..., t_n}]) \leftrightarrow (\mathrm {E}/\overline{\mathrm {E}}t_1 \wedge ... \wedge \mathrm {E}/\overline{\mathrm {E}}t_n)$

Notably, when $n = 0$, we take the right-hand side conjunction to be some tautology, i.e., if $\ulcorner \phi \urcorner$ contains no free variables or complex terms, then $\vdash _c \exists p (p = [\phi ])$.

(P) is clearly not neutral with respect to all choice points in the metaphysics of propositions. The conception of propositions, for instance, is not coarse-grained, i.e., it is not the case that any two propositions with the same truth value in all worlds are identical, e.g., see (Stalnaker, 1976). The contingentist here individuates propositions both by their truth, and existence, conditions. However, note, that (P) alone does not determine how fine such a fine-grained individuation should be. That is, (P) is consistent with any view of individuation which holds that propositions are at least individuated by their truth, and existence, conditions. Now, in order to make the later negative arguments general, no further axioms of $\vdash _c$ concern the individuation of propositions.

I take it that the conception of contingently existing propositions so far outlined does not alone prevent the contingentist from also adopting the compelling principles (T[]), (PNeg1), and (PNeg2) as axioms. That is to say, we should still want every proposition to have a negation and that the truth of a proposition $[\phi ]$ be tied to the truth of the sentence $\phi$. However, it is important to discuss what we should think about serious actualism—the view that in order for an object to exemplify a property, or in this context satisfy a predicate, it must exist. Crucially, (T[]), (P), (N), contingentism and the requirement that a proposition must exist to be true (an instance of serious actualism) are jointly inconsistent.¹⁴ Of course, (P) is an essential element of our axiomatization of propositional contingentism. (N), as I stressed earlier, is also an essential element of $\vdash _c$, given the role $\vdash _c$ plays.¹⁵ Thus, we must choose between (T[]) and serious actualism. In my view, we ought to reject serious actualism, even if some will view this to be a cost. The first point to note is that serious actualism is controversial, it being most notably rejected in (Pollock, 1985, pp. 126–129), (Fine, 1985, pp. 163–171) and (Salmon, 1987, p. 95). In fact, both Fine and Salmon explicitly argue against the weaker idea that a proposition’s truth requires its existence—an idea also rejected in (Mitchell-Yellin & Nelson, 2016, p. 1538). Moreover, it has also been noted how serious actualism in conjunction with propositional contingentism is deeply problematic, see (Fritz & Goodman, 2016, p. 655), and (Jacinto, 2019, pp. 491–496).

Second, (T[]) is an intuitively compelling principle; but it is also a powerful principle when combined with (N), allowing us to derive the necessitated truth schema, i.e., $\Box (\mathrm {T}[\phi ] \leftrightarrow \phi )$. In the present context, what’s important is that we afford the contingentist the strongest possible resources that are within the bounds of their contingentism, investigating whether they can make use of such resources to articulate adequate definitions of worlds. Serious actualism is not entailed by contingentism, nor is the weaker idea that simply the truth of a proposition implies its existence, and so we are free to develop a contingentism which rejects both. Indeed, rejecting it allows us to prove some key positive results for the contingentist, e.g., Theorem 20 below. Thus, in rejecting serious actualism, the contingentist is free to accept principles, like (T[]), which are intuitive and which allow the contingentist to preserve crucial logical resources utilised in developing the simple necessitist theory of worlds in $\S$1.2: rejecting serious actualism puts them in a prima facie stronger position to secure the existence of adequate possible worlds.

Although many contingentists accept serious actualism in some form, it is beyond the scope of this paper to offer anything like a decisive argument against serious actualism.¹⁶ We must make a choice regarding serious actualism, since it is deeply non-obvious that precise discussions of this kind can be done independently of the question of serious actualism. However, whilst rejecting serious actualism may be motivated in the present context, we should be mindful that the arguments which follow thus target only one family of ways of developing propositional contingentism and that the arguments can be blocked by a refusal to deny serious actualism. Of course, this goes without saying that blocking the arguments in this paper by accepting serious actualism is not alone sufficient to show that propositional contingentism is consistent with theoretically adequate worlds, let alone show that propositional contingentism and serious actualism constitute a viable package of views. However, it is also beyond the scope of this paper to do that work here.¹⁷

Now, a few final comments are in order, before I fully outline the contingentist logic. $\vdash _c$ is a plural modal logic and we have to be careful with how we think about pluralities in a contingentist setting. I have, at several points, utilised both a singular and plural existence predicate. Here, I define these as follows.

Definition 14

$\left.{\&\#Xulcorner;}\mathrm {E}t \right.{\&\#Xurcorner;}$ abbreviates $\left.{\&\#Xulcorner;}\exists x (x = t) \right.{\&\#Xurcorner;}$, for distinct terms t and x

Definition 15

$\ulcorner \overline{\mathrm {E}}tt \urcorner$ abbreviates $\left.{\&\#Xulcorner;}\exists xx (xx = tt) \right.{\&\#Xurcorner;}$, for distinct terms tt and xx.

(15) requires that we have an identity sign for pluralities. In $\vdash _c$, the previous definition (2) will not suit.¹⁸ Instead, in $\vdash _c$, we must take plural identity to be defined in terms of necessary coextensiveness: $\ulcorner tt = tt' \urcorner$ abbreviates $\ulcorner \Box \forall y (y \prec tt \leftrightarrow y \prec tt') \urcorner$. Another issue is that we need to more carefully tie down the natural thought that a plurality exists only in cases in which all of the objects among that plurality exist. Of course, a plurality here is nothing over and above those things among it. However, for similar reasons to those discussed with plural identity, we cannot hope to capture this natural idea with a quantificational expression in a free logic. Instead, we partially capture the idea with the following principle, where t and tt are arbitrary terms of any, but the same, sort.

• (PE)$\overline{\mathrm {E}}tt \rightarrow (t \prec tt \rightarrow \mathrm {E}t)$ is an axiom.

Indeed, the same kind of issues arise with how we should think about a plural predicate for the collective truth of some propositions. In $\vdash _c$, the quantifier expression $\ulcorner \forall p(p \prec pp \rightarrow \mathrm {T}p) \urcorner$ only defines joint truth on the assumption that all propositions among some jointly consistent propositions necessarily co-exist and this evidently does not hold.¹⁹ The proposal, then, is to extend $\mathcal {L}^{W}$ to a language $\mathcal {L}^{W}_{\overline{\mathrm {T}}}$ which includes a plural truth predicate $\overline{\mathrm {T}}$ as a primitive.²⁰ The following principle restricts the behaviour of the plural truth predicate and should be included in the contingentist logic, where t, $t'$, and tt are any propositional terms.

($\overline{\mathrm {T}})$:

$\overline{\mathrm {T}}tt \rightarrow (t \prec tt \rightarrow \mathrm {T}t)$ is an axiom.

Of course, without the ability to quantify over non-existent propositions, we are unable to express the sufficient condition for the truth of some pp, i.e., that all p among pp are true. But ($\mathrm {\overline{T}}$) restricts the plural truth predicate enough for what is required here.

We’re now in a position to provide the following formal definition of the propositional contingentist logic $\vdash _c$ within which we can explore the availability of propositional contingentist possible worlds.

Definition 16

($\vdash _c$): Let $\vdash _c$ be the proof system in $\mathcal {L}^{W}_{\overline{\mathrm {T}}}$ consisting of the following principles, where v stands for any variable, plural or singular of any sort, v is an arbitrary singular variable of any sort, vv is an arbitrary plural variable of any sort, t stands for any term of the appropriate sort, and $\phi , \psi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$ are arbitrary formulae.

• ($\forall$IE) $\forall {\textbf {v}} \phi \rightarrow (\mathrm {E}t \rightarrow \phi [t/{\textbf {v}}])$ is an axiom.

• ($\forall$2E) $\phi \leftrightarrow \forall {\textbf {v}} \phi$ is an axiom, provided v is not free in $\phi$

• ($\forall ^{\rightarrow }$) $\forall {\textbf {v}}(\phi \rightarrow \psi ) \rightarrow (\forall {\textbf {v}} \phi \rightarrow \forall {\textbf {v}} \psi )$ is an axiom.

• (UG) $\phi$ / $\forall {\textbf {v}} \phi$.

• (UE) $\forall v \mathrm {E}v$ is an axiom.

• (UEP) $\forall vv \overline{\mathrm {E}}vv$ is an axiom.

As well as (N), (K), (T), (5),²¹ (OE), (PC), (MP) (I1), (I2), (Comp), (NE), (Ext), (P1), (QE), (T[]), (P), (PE), (PNeg1), (PNeg2), and ($\mathrm {\overline{T}}$).

This completes the presentation of the propositional contingentist logic $\vdash _c$.

3 Possible worlds in logic $\vdash _c$

I establish the more general result that no adequate notion of a possible world can be given in $\vdash _c$, by focusing on uncontroversial elements of worlds—that worlds are consistent and that propositions are true relative to them. That is to say, I do not focus on the particularities of this or that definition of a possible world in $\vdash _c$. Instead, I focus on the unproblematic definitions available to the propositional contingentist for consistency and world-relative truth which require little discussion. I then show that any account of worlds in $\vdash _c$ which takes them to have such features will be inadequate. Since all accounts of worlds should incorporate such features, no notion of worlds in $\vdash _c$ is adequate.

I argue for this general result by, first, defining consistency and the relation of world-relative truth, using the notation introduced earlier. Straightforwardly, we say that propositions pp are consistent just in case $\lozenge \mathrm {\overline{T}}pp$, where $\overline{\mathrm {T}}$ is the primitive plural truth predicate outlined earlier. Truth in a world is defined in $\vdash _c$ much in the same way as in $\vdash _n$, i.e., a proposition is true in a world if its truth is necessitated by the joint truth of those propositions which are the world.

Definition 17

($\models$) For any proposition p: $\ulcorner pp \models p \urcorner$ abbreviates $\ulcorner \Box (\mathrm {\overline{T}}pp \rightarrow \mathrm {T}p) \urcorner$.

Second, I let $\mathrm {W}^{\mathcal {C}}$ be a schematic term which stands for any proposed world-hood predicate. Since any viable account of possible worlds in $\vdash _c$ must take them to be at least consistent, contingentists can only endorse some world-hood predicate $\mathrm {W}^{\mathcal {C}}$ for which the following is satisfied.

$$\begin{aligned} (W) \mathrm {W}^{\mathcal {C}}pp \rightarrow \lozenge \mathrm {\overline{T}}pp \end{aligned}$$

To spell this out a little further, suppose the contingentist proposes some world-hood predicate $\mathrm {W}^{\mathcal {C}_1}$ in $\vdash _c$. Needless to say, a discerning factor in whether some pp satisfy $\mathrm {W}^{\mathcal {C}_1}$ is whether pp are consistent. In other words, they should endorse the $\mathrm {W}^{\mathcal {C}_1}$-instance of (W). To be clear, the requirement here is not that the contingentist defines their world-hood predicate in simpler terms—it may be taken as primitive. The claim is simply that for contingentist theory to be adequate, the appropriate instance of (W) must be true. Using (W), we can make a more general argument, by focusing on all and only those potential world-hood predicates which meet this minimal condition.

Crucially, we can show that, although minimal, if the contingentist articulates an account of worlds in $\vdash _c$ which satisfies (W), then one direction of each of the fundamental theorems holds in $\vdash _c$. However, we can also show that no contingentist theory of worlds can guarantee that both directions of each of the fundamental theorems are true, if we assume, as we should, that (W) is satisfied. First, the positive results. Note that, given (T[]), a corresponding version of L.10 holds in $\vdash _c$:

Lemma 18

For any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$: $\vdash _c \lozenge \mathrm {T}[\phi ] \leftrightarrow \lozenge \phi$.

Thus, for any contingentist theory of worlds which satisfies (W), one direction of each of the Fundamental Theorems holds in $\vdash _c$, just as in $\vdash _n$.

Theorem 19

$\vdash _c \exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ]) \rightarrow \lozenge \phi$, for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$ with no free occurrence of ‘pp’, assuming that $\mathrm {W}^{\mathcal {C}}$ satisfies (W).

Proof

By deductions in $\vdash _c$.

• (1) $\vdash _c \exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ]) \rightarrow \exists pp (\lozenge \overline{\mathrm {T}}pp \wedge \Box (\overline{\mathrm {T}}pp \rightarrow \mathrm {T}[\phi ]))$ (W), D.17

• (2) $\vdash _c \exists pp (\lozenge \overline{\mathrm {T}}pp \wedge \Box (\overline{\mathrm {T}}pp \rightarrow \mathrm {T}[\phi ])) \rightarrow \lozenge \mathrm {T}[\phi ]$ QML

• (3) $\vdash _c \exists pp (\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ]) \rightarrow \lozenge \phi$ (1), (2), L.18

$\square$

Theorem 20

$\vdash _c \Box \phi \rightarrow \forall pp(\mathrm {W}^{\mathcal {C }}pp \rightarrow pp \models [\phi ])$, for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$ with no free occurrence of ‘pp’, where $\mathrm {W}^{\mathcal {C}}$ satisfies (W).

Proof

By deductions in $\vdash _c$.

• (1) $\vdash _c \Box \phi \rightarrow \Box (\overline{\mathrm {T}}pp \rightarrow \mathrm {T}[\phi ])$ (T[]), (N), (K)

• (2) $\vdash _c \Box \phi \rightarrow (\mathrm {W}^{\mathcal {C}}pp \rightarrow \Box (\overline{\mathrm {T}}pp \rightarrow \mathrm {T}[\phi ]))$ (1), (PC), (MP)

• (3) $\vdash _c \forall pp (\Box \phi \rightarrow (\mathrm {W}^{\mathcal {C}}pp \rightarrow \Box (\overline{\mathrm {T}}pp \rightarrow \mathrm {T}[\phi ])))$ (2), (UGP)

• (4) $\vdash _c \forall pp \Box \phi \rightarrow \forall pp (\mathrm {W}^{\mathcal {C}}pp \rightarrow \Box (\overline{\mathrm {T}}pp \rightarrow \mathrm {T}[\phi ]))$ (3), ($\forall ^{\rightarrow }$)

• (5) $\vdash _c \Box \phi \rightarrow \forall pp (\mathrm {W}^{\mathcal {C}}pp \rightarrow \Box (\overline{\mathrm {T}}pp \rightarrow \mathrm {T}[\phi ]))$ (4), $\forall$2E

$\square$

The other directions of the theorems are more demanding, however, since they require that the space of possible worlds is, in a sense, complete. In fact, I will now show that the other direction of the fundamental theorem for possibility cannot be taken as a general and necessary truth by the contingentist. This follows from some important results concerning consistent pluralities of propositions and claims about possibly indistinguishable entities in $\vdash _c$.

First, let’s start with the intuitive shape of the problem for the contingentist. Very generally, we say that two objects are indistinguishable in some respect, if there are no propositions available which we can use to distinguish those two objects in that respect. Here, we ‘use propositions’ to distinguish objects by supposing they are true, and seeing what follows. In other words, then, two objects x and y are indistinguishable with respect to, say, $\phi$, if there are no propositions the truth of which necessitate that $\phi (x)$, but not that $\phi (y)$, or vice versa. Of course, it is controversial whether there even possibly are objects which are indistinguishable in every respect, i.e., for any open-sentence $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$. However, for our present purposes, we need only be concerned with the idea of objects being indistinguishable with respect to existence and, for the rest of the paper, I will refer to this simply as ‘indistinguishability’. Formally, this can be defined:

Definition 21

$\ulcorner x \approx _e y \urcorner$ abbreviates $\ulcorner \forall pp(\Box (\mathrm {\overline{T}}pp \rightarrow \mathrm {E}x) \leftrightarrow \Box (\mathrm {\overline{T}}pp \rightarrow \mathrm {E}y)) \urcorner$.

That is to say, x and y are indistinguishable just in case, for any plurality of propositions, pp, necessarily, if pp are true, then x exists if and only if, necessarily, if pp are true, then y exists.

Now, the problem for the contingentist is that the existence of adequate possible worlds is inconsistent with certain well-motivated claims about distinct and yet possibly indistinguishable entities in the above sense. Loosely speaking, if the systematic connection between possibility and truth in some world holds, then all possibility claims are ‘witnessed’, as it were, by some propositions—if some claim is possible, its truth is necessitated by some propositions which qualify as a world. However, as I will argue, there are well-motivated claims for the propositional contingentist about possibilities involving entities which are possibly indistinguishable in the above sense. Thus, I will argue that there are claims which the contingentist must accept as possible and yet the truth of those claims are not necessitated by any propositions—a fortiori those claims are not necessitated by any propositions which qualify as a world. Since the propositional contingentist should accept such claims about genuinely distinct but possibly indistinguishable entities, the failure of the systematic connection between possibility and truth in some world follows. (Later, I forestall worries about this argument and show that there are models of a sound semantics for $\vdash _c$ in which such well-motivated indistinguishability claims are true.)

Here’s how we make this problem precise in $\vdash _c$. First, consider this result in $\vdash _c$.

Lemma 22

$\vdash _c x \approx _e y \rightarrow \lnot \exists pp(\lozenge \mathrm {\overline{T}}pp \wedge (pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y])).$

Proof

By deductions in $\vdash _c$. For convenience, let $\phi ^{x, y} := \mathrm {E}x \wedge \lnot \mathrm {E}y$

• (1) $\exists pp( \lozenge \mathrm {\overline{T}}pp \wedge pp \models [\phi ^{x, y}]) \rightarrow \exists pp(\lozenge \mathrm {\overline{T}}pp \wedge \Box ( \mathrm {\overline{T}}pp \rightarrow \mathrm {T}[\phi ^{x, y}]))$ D.17

• (2) $\Box (\mathrm {T}[\phi ^{x, y}] \rightarrow \mathrm {E}x)$ (T[]), (PC), (N)

• (3) $\Box (\mathrm {T}[\phi ^{x, y}] \rightarrow \lnot \mathrm {E}y)$ (T[]), (PC), (N)

• (4) $\exists pp(\lozenge \mathrm {\overline{T}}pp \wedge pp \models [\phi ^{x, y}]) \rightarrow \exists pp(\Box (\mathrm {\overline{T}}pp \rightarrow \mathrm {E}x) \wedge \Box (\mathrm {\overline{T}}pp \rightarrow \lnot \mathrm {E}y))$ (1), (2), (3)

• (5) $\exists pp(\Box (\mathrm {\overline{T}}pp \rightarrow \mathrm {E}x) \wedge \Box (\mathrm {\overline{T}}pp \rightarrow \lnot \mathrm {E}y)) \rightarrow \lnot x \approx _e y$ D.21, QML

• (6) $\exists pp(\lozenge \mathrm {\overline{T}}pp \wedge pp \models [\phi ^{x, y}]) \rightarrow \lnot x \approx _e y$ (4), (5), (T[]), (N), QML

Contrapose (6) for the result. $\square$

(22) shows that if x and y are indistinguishable in the sense of (21), then there are no possibly true propositions, the joint truth of which necessitate the truth of the proposition that x exists but y does not. As we saw earlier, any adequate notion of a world will satisfy (W), i.e., the propositions which are the world will be at least jointly possibly true. Thus, it is easy to see that (22) shows that if x and y are indistinguishable, then there are no worlds relative to which the proposition that x exists but y does not is true. Note that (22) is not problematic for the propositional contingentist in and of itself. Rather, the problem arises due to the interaction in $\vdash _c$ between (22) and some well-motivated claims for the propositional contingentist about distinct and yet possibly indistinguishable entities. Here, I will focus explicitly on the most plausible and minimal such claim and the argument against contingentist worlds we can develop from this claim.

Consider, then, the following.

• (ID) $\lozenge \exists x \lozenge \exists y (\lozenge x \approx _e y \wedge \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y))$ (To be read: Possibly there is something x, such that possibly there is something y and x and y are possibly indistinguishable and possibly x exists and y does not.)

(ID) does not involve claiming that there are objects which could have been indistinguishable. Rather, it is the claim that it is possible that there is some object such that it is possible for there to be another which is possibly indistinguishable from the first.²² (ID) is a logically weak, and plausible, claim indeed. To see why the propositional contingentist ought to accept it, it’s worth considering what philosophical picture emerges if one rejects (ID) and how implausible such a metaphysics is. According to such a metaphysics, at least the following holds.

• ($\lnot$ID) $\Box \forall x \Box \forall y (\lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y) \rightarrow \Box \lnot x \approx _e y)$ (To be read: Necessarily, for any thing x and necessarily for any thing y, if it is possible that x, but not y, exists, then necessarily x and y are distinguishable.)

That is to say, speaking loosely, for any two individuals x and y across all possible worlds, if it is possible for either one to exist without the other, then in every world w, those x and y are distinguishable in terms of propositions which exist in w. The obvious test case for whether ($\lnot$ID) is acceptable to the propositional contingentist is the case where x and y can both fail to exist. To accept ($\lnot$ID) is to think that, even if both x and y were not to exist, there would be propositional resources available to distinguish x from y. Of course, it’s clear that the necessitist can accept this. In each world, there will be propositions which distinguish x from y and vice versa, e.g., propositions like $[\mathrm {E}x]$ or $[\mathrm {E}y]$. Likewise, a contingentist who only accepts that certain individuals, but not propositions, contingently exist could accept ($\lnot$ID).

However, it is clear that for the propositional contingentist no propositions like $[\mathrm {E}x]$ or $[\mathrm {E}y]$ would exist, if x and y were not to exist. The propositional contingentist can accept ($\lnot$ID) only if they think that, even in such cases as a contingently existing x and y, there are some qualitative propositions which necessarily uniquely specify x and y, i.e., propositions not expressed by sentences featuring ‘x’ or ‘y’. This, however, implies a deeply implausible metaphysical picture. It is one in which, necessarily, for any two entities which satisfy the antecedent of ($\lnot$ID), there is some qualitative feature of either x and y which we can use to lock onto x or y independently of x or y existing. For instance, if x and y were two qualitatively indistinguishable electrons, it is deeply implausible that there necessarily exist propositions which we can use to distinguish x from y.²³

One natural thought is that one may endorse ($\lnot$ID), if one appealed to special kinds of properties like essences or thisnesses. For instance, Plantinga has defended the existence of a special kind of property, known as an essence, which, for each individual, uniquely tracks that individual across all worlds (Plantinga, 1979). An essence for some individual is exemplified only by that individual, in any world that individual exists. Importantly, such essences are qualitative and exist necessarily, regardless of whether the individual itself exists. Likewise, some have argued that there are non-qualitative properties known as thisnesses which similarly uniquely track individuals across worlds and which can exist independently of the existence of those individuals, see (Ingram, 2018b) and (Ingram, 2018a). However, it is implausible that such properties are a well-motivated addition to the propositional contingentist’s ontology. In the case of qualitative essences, it is deeply implausible that, and under-explained how, for every individual, there is a qualitative property which is able to lock on, and uniquely track, an individual across all possible worlds, see (Williamson, 2013, p. 269). In the case of thisnesses, the situation is even less motivated. Given the close parallel between propositions and properties, it is deeply problematic for the propositional contingentist to take certain propositions like [John exists] to ontologically depend on the objects they concern, and yet deny that non-qualitative properties defined in terms of, or built out of, certain individuals ontologically depend on those very individuals. (See (Fine, 1985, p. 149) and (Williamson, 2013, Chap. 6) for a discussion of this, and similar, issues.)

For the propositional contingentist, then, the metaphysical burden of denying (ID) is simply an anathema to their view. Propositional resources which uniquely identify even a contingent individual across all possible worlds are precisely those resources which the propositional contingentist rejects. Thus, the propositional contingentist should accept (ID)—it is possible for there to be an object for which it is possible that there is another and the two are possibly indistinguishable and yet, because they are distinct objects, it is possible for one to exist and yet the other not. The problem, in the present context, is that the following results show that accepting a well-motivated claim like (ID) means that theoretically adequate possible worlds are not available to the propositional contingentist.

Lemma 23

$\vdash _c$ ID $\rightarrow \lozenge \exists x \lozenge \exists y (\lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y) \wedge \lozenge \lnot \exists pp(\lozenge \overline{\mathrm {T}}pp \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y]))$

Proof

The following is derived as a corollary from (22) with (N) and UG:

• (i) $\vdash _c \Box \forall x \Box \forall y (x \approx _e y \rightarrow \lnot \exists pp(\lozenge \mathrm {\overline{T}}pp \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y]))$

From some simple QML and (i), (23) follows. $\square$

Crucially, there is no incoherence in $\vdash _c$ in the supposition that (ID) is true. That is to say, there are models in which all the axioms of $\vdash _c$ are true and the inference rules truth-preserving and in which (ID) holds. In the appendix, I outline such a semantics, showing that $\vdash _c$ is sound with respect to a class of models of that semantics, $\mathbb {M}^{P}$. Here I simply state the relevant result.

Theorem 24

For some $\mathcal {M} \in \mathbb {M}^{P}$: $\mathcal {M}, w, {a} \vDash _c$ ID

(23) and (24) have the following consequence.

Theorem 25

It is not the case that, for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$ with no free pp: $\vDash _c \lozenge \phi \rightarrow \exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ])$, assuming that $\mathrm {W}^{\mathcal {C}}$ satisfies (W)

Proof

Let:

$\xi := \lozenge \exists x \lozenge \exists y (\lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y) \wedge \lozenge \lnot \exists pp(\lozenge \overline{\mathrm {T}}pp \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y]))$$\psi := \lnot \Box \forall x \Box \forall y (\lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y) \rightarrow \Box \exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y]))$

Assume $\mathrm {W}^{\mathcal {C}}$ satisfies (W), then: (i) $\vdash _c \xi \rightarrow \psi$ and thus, given the soundness of the semantics based on $\mathbb {M}^{P}$, (ii) $\vDash _c \xi \rightarrow \psi$. Next, suppose that $\vDash _c \lozenge \phi \rightarrow \exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ])$, for any $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$ with no free pp. Letting $\phi := \mathrm {E}x \wedge \lnot \mathrm {E}y$, it follows, by soundness, the supposition, S5, and applications of (UG) and (N) that $\vDash _c \lnot \psi$. However, by (23), (24), (ii), and soundness, we know that there is an $\mathcal {M} \in \mathbb {M}^{P}$ such that $\mathcal {M}, w, {a} \vDash _c \xi$ and thus $\mathcal {M}, w, {a} \vDash _c \psi$. Thus, it is not the case that $\vDash _c \lozenge \phi \rightarrow \exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ])$, for any $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$ with no free pp, assuming $\mathrm {W}^{\mathcal {C}}$ satisfies (W). $\square$

Theorem 26

$\nvdash _c \lozenge \phi \rightarrow \exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ])$, for any $\phi \in \mathcal {L}^W$ with no free pp, assuming that $\mathrm {W}^{\mathcal {C}}$ is some world-hood predicate which satisfies (W).

Proof

Corollary of (25) and (42) (see Appendix) $\square$

To summarise, (ID) is a particularly logically weak and plausible claim and the propositional contingentist ought to accept it. However, if it holds, then the fundamental theorem for possibility is not provable in $\vdash _c$, for any conception of worlds which takes them to be consistent. Crucially, it is not an option for the propositional contingentist to simply supplement $\vdash _c$ with the fundamental theorems as axioms. The presence of the systematic connection between possibility and truth at some world amongst the general and necessary truths of the propositional contingentist’s view is inconsistent with the claim that there might have existed distinct and yet possibly indistinguishable entities. If the fundamental theorems were stipulated to hold, the propositional contingentist would have to accept ($\lnot$ID) as a theorem—as a general and necessary truth of their view. Such a consequence is, however, deeply unpalatable, as I have already argued.

In the next section, I will outline a further limitative result for the contingentist. But before doing so, it’s worth discussing the generality of the argument presented so far. Explicitly, the arguments presented here concern the prospects of the contingentist securing the existence of theoretically adequate worlds, where worlds are pluralities of propositions. However, in several important respects, the arguments presented are plausibly more general. First, I have not assumed that the contingentist defines worlds as those pluralities of propositions which are at least possibly jointly true. Rather, I plausibly assume that, regardless of the constraints the contingentist imposes on pluralities of propositions, it will be nonetheless true that every such plurality is, as a matter of fact, at least possibly jointly true. Similarly, though the definition of truth in a world is modal, provided a proposition p is true in a world pp only if the truth of p is necessitated by pp, regardless of how this notion is strictly defined, the arguments above apply: the arguments show that the possibility of indistinguishable entities rules out the existence of pluralities pp and propositions p which meet minimal necessary conditions for being a world and for being a proposition true in such a world.

Moreover, the arguments here plausibly generalise to other potential conceptions of contingentist possible worlds, given the close parallels between pluralities of propositions and other entities like individual propositions, sets of propositions, states of affairs, or properties. Here, we define the ‘actuality’, and the consistency or possibility of a world using the notion of singular, and plural, propositional truth. If worlds, however, were, say, properties, these notions wouldn’t apply. However, analogous ones would, e.g., a world w would be actual if instantiated and p would be true in w, if the truth of p were necessitated by the instantiation of w. Now, plausibly, the propositional contingentist is committed to the same patterns of contingency in what properties there are as with propositions (Williamson, 2013, p. 289), and indeed, are plausibly committed to the same patterns of contingency in what sets of propositions, or states of affairs, there are. Thus, the thought goes, the arguments presented here should be of general concern to conceptions of contingentist possible worlds beyond the specific proposal in which such arguments are presented, given that many such alternative conceptions of worlds will involve positing entities exhibiting deeply analogous features, allowing us to reformulate the specific versions of the arguments presented here. It goes without saying, of course, that the arguments here will not generalise to all conceptions of contingentist worlds. For instance, the contingentist could put forward an account of worlds in which the notion of truth in a world is not understood to entail that the relevant proposition’s truth is necessitated by the actuality of the world. However, my point here is only that the arguments can be very plausibly extended to many other approaches to worlds which eschew the specific details involved in formulating them here.

It’s also worth discussing how the results in this section relate to other results in the extant literature, particularly those in Fritz (2016). There, Fritz develops two classes of models to model contingency in what propositions there are.²⁴ Fritz then raises worries for the prospects of understanding world-talk in terms of maximally strong non-trivial propositions (Ibid., pp. 140–141), as suggested in Stalnaker (1976). Now, one may worry that the results presented here are no more than notational variants of results in Fritz (2016). This is not, however, the case. First, we are here concerned with a different conception of propositional contingentism. Distinguish:

• Aboutness View Some propositions p are directly, or singularly about individuals $i_1, ..., i_n$ and because of this relation between p and $i_1, ..., i_n$, if any of $i_1, ..., i_n$ were not to exist, p would not.

• Distinction View A proposition p is in part the proposition that it is because of the distinctions it draws in modal space. Some propositions draw distinctions which essentially involve appealing to individuals $i_1, ..., i_n$. In the absence of $i_1, ..., i_n$, such propositions do not exist.²⁵

Fritz is explicitly interested in developing models for propositional contingentism according to the distinction conception, see (Fritz, 2016, p. 124; 2018aa, p. 408) and see also (Fritz & Goodman, 2016, p. 646; 2017aa, p. 509). The present paper has been concerned with an aboutness view: propositions are contingent entities if they are expressed by sentences which feature terms denoting contingent entities. These two ways of motivating propositional contingentism are fundamentally different and lead to diverging judgements on important cases. For instance, consider a tautologous proposition like $[Fq \vee \lnot Fq]$. Such a proposition, according to the aboutness view is a contingent entity, if q is contingent. However, according to the distinction view, $[Fq \vee \lnot Fq]$ is a necessarily existing proposition: the trivial distinction it draws in modal space does not essentially involve appealing to the individual q (Stalnaker, 2012). One cannot assume that results about one conception of propositional contingentism simply carry over to the other.

More importantly, the results presented here are independent to those in Fritz (2016) and, in a sense, more flexible. Fritz is concerned with one plausible way the propositional contingentist could understand world-talk. As I’ve stressed, the results presented here make few assumptions about worlds and generalise to many conceptions of worlds which make the same, or analogous, assumptions. The arguments here also show that such minimal assumptions entail that the contingentist should reject even very weak claims like (ID). This contrasts with the discussion in Fritz (2016). His primary concern there is how one particular strategy for understanding world-talk interacts with generalised quantifier phrases like ‘there are uncountably many worlds such that...’, especially given the limitative results about such quantifiers in higher-order modal languages obtained in Fritz (2018b). Thus, even if it were legitimate to generalise results in Fritz (2016) about the distinction view to the aboutness view, my results show that, for the contingentist, it is not enough to respond to Fritz’s worries, since there are independent and stubborn worries about contingentist possible worlds.

4 Actualised fundamental theorem for possibility

I want to finally discuss one further limitative result since it shines a light on a tempting, but mistaken, response which the propositional contingentist may wish to take. One may naturally worry that what has driven the arguments here is the requirement that the systematic connections between possibility, necessity, and truth at possible worlds hold of unqualified necessity. Instead, one might think that what is important is that such connections hold here, in the actual world. After all, the results presented so far simply show that the fundamental theorems possibly fail to be true, if some claims about distinct but merely possible individuals are true. Instead, one might think that the machinery of possible worlds can help elucidate modality even if it just gives us purchase on those modal claims which are actually true. The objection would then run: what needs to be shown, and crucially what hasn’t been shown, is that the fundamental theorems fail to actually hold and without this result my argument is in jeopardy. For the rest of the paper, then, I will argue for just that—even if we only require the fundamental theorem to actually hold, we can outline a similar limitative argument against the contingentist.

To better understand this response, we should first spend some time talking about how an actuality operator should work in some suitable extension of $\vdash _c$. If we extend $\mathcal {L}^{W}_{\overline{\mathrm {T}}}$ to include an operator @ for actuality, we will need to supplement (16) to include some axioms or rules to govern it as well as extending the notion of a formula of $\mathcal {L}^{W}_{\overline{\mathrm {T}}}$. Both tasks are straightforward. First, we say that if formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}@}$, then formula $@\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}@}$, where $\mathcal {L}^{W}_{\overline{\mathrm {T}}@}$ is $\mathcal {L}^{W}_{\overline{\mathrm {T}}}$ extended to include @. Second, we supplement $\vdash _c$ with the following principles for @ to get a proof system $\vdash _c^@$:

• (@1) $\phi / @\phi$.

• (@2) $@(\phi \rightarrow \psi ) / @\phi \rightarrow @\psi$

• (@3) $\vdash _c^@ \lnot @ \phi \leftrightarrow @ \lnot \phi$

• (@4) $\vdash _c^@ @\phi \rightarrow \Box @ \phi$

Note that these principles are not intended to be a full axiomatization of @. (@1)–(@4) are the least which hold of ‘@’. Now, weakening the requirement to only the actualised fundamental theorem means that the propositional contingentist takes the following to be a validation of the adequacy of some notion of worlds $\mathrm {W}^{\mathcal {C}}$, for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}_@}$:

• $\vdash _c^@ @ (\lozenge \phi \leftrightarrow \exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ]))$

This becomes the requirement that, for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}_@}$:

• (@) $\vdash _c^@ \lozenge \phi \leftrightarrow @\exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ])$

This follows straightforwardly from (@1)–(@4) and, given that the underlying modal logic is S5, the fact that $\vdash _c^@ \lozenge \phi \leftrightarrow @\lozenge \phi$.

Thus the question of whether the propositional contingentist can endorse the mere actuality of the fundamental theorems becomes the question of whether the propositional contingentist can guarantee that (@) holds.²⁶ The following argument shows that they cannot meet this requirement, since the same sort of problem reemerges in this setting. First, consider the following.

• (ID@) $\lozenge \exists x \exists y (@ x \approx _e y \wedge \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y))$ (To be read: Possibly there exist two things x and y and x and y are actually indistinguishable and it is possible that x exists and y does not.)

Of course, insofar as we cannot distinguish x and y in terms of actually existing propositions, x and y are entities which do not actually exist. However, crucially, just as before, in endorsing (ID@) we do not talk about two merely possible and indistinguishable individuals in particular but only two merely possible and indistinguishable individuals whichever two things in particular they would turn out to be. The problem for the propositional contingentist is that (ID@) is just as problematic as (ID).

Lemma 27

$\vdash _c^@$ ID@ $\rightarrow \lozenge \exists x \exists y (\lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y) \wedge \lnot @ \exists pp(\lozenge \overline{\mathrm {T}}(pp) \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y]))$

Proof

Again, from (22), we know that

$$\begin{aligned} \vdash _c^@ x \approx _e y \rightarrow \lnot \exists pp(\lozenge \mathrm {\overline{T}}pp \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y]) \end{aligned}$$

From this we derive the result, using (@1)-(@3) and (UG), (N), and (W) $\square$

Again, we appeal to some model-theoretic results to show that there is no incoherence in the supposition that there might have been some individuals which are actually indistinguishable and which might have existed without the other existing. Only this time, we appeal to a different class of models, $\mathbb {M}^@$. I show that $\vdash _c^@$ is sound with respect to $\mathbb {M}^@$ in the appendix. Here is the relevant result, where $\vDash _c^@$ is the notion of truth in a $\mathbb {M}^@$ model.

Theorem 28

For some $\mathcal {M} \in \mathbb {M}^@$: $\mathcal {M}, w, {a} \vDash _c^@$ ID@

(27) and (28) have much the same effect as the previous results. If (@) holds, then so too does the following.

• (@*) $\vdash _c^@ \Box \forall x \forall y (\lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y) \rightarrow @\exists pp(\lozenge \overline{\mathrm {T}}pp \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y])$

However, by (28) we know that there is some model $\mathcal {M} \in \mathbb {M}^@$ in which ID@ is true at some world, under an assignment. Thus, given (27) and the fact that the semantics defined over $\mathbb {M}^@$ is sound for $\vdash _c^@$, it follows that:

Theorem 29

It is not the case that $\vDash _c^@ \lozenge \phi \rightarrow @\exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ])$, for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}@}$ with no free pp, assuming that $\mathrm {W}^\mathcal {C}$ satisfies (W).

Proof

Proof. Corollary of (27)–(28) and (42) (see Appendix). $\square$

Theorem 30

It is not the case that $\vdash _c^@ \lozenge \phi \rightarrow @\exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ])$, for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}@}$ with no free pp, assuming satisfies (W).

Proof

Proof. Corollary of (29) and (42) (see Appendix). $\square$

Thus, even if the contingentist only demands that possible worlds are adequate if the fundamental theorem for possibility actually holds, they are unable to define theoretically adequate worlds. The requirement that the fundamental theorem of possibility holds of unqualified necessity and not merely actually is not an essential element of the argument I have presented here.

5 Concluding remarks

I have presented in detail the two most promising limitative results which show that the propositional contingentist is unable to embed theoretically adequate notions of possible worlds into a natural and promising logic which characterises their view. First, I presented an argument which focused on a compelling claim about distinct and yet indistinguishable entities, showing that there is an inconsistency between such a claim and the availability of adequate possible worlds in $\vdash _c$. I also appealed to the existence of a model for such a claim in a sound semantics for the logics which characterise their view—the existence of such a model is proven in the Appendix. Consequently, the fundamental theorem for possibility cannot be a theorem of $\vdash _c$. Given the role of $\vdash _c$, this shows that the systematic connection between possibility and truth at some world is not one of the general and necessary truths distinctive of their view. As a second argument, I considered the question of whether the systematic connection between possibility and truth at some world could be taken to be a general, but only actual, truth of the propositional contingentist view. I showed that a parallel argument can be given against the availability of contingentist possible worlds, even if the requirement is weakened in this way. Each of these results contrasts with the situation for the necessitist.

At the very least then, the propositional contingentist cannot utilise the notion of a possible world unreflectively. There are three options for such contingentists. They may wish to theorize about modality without recourse to possible worlds; they may wish to abandon their propositional contingentism in favour of propositional necessitism, a view which can unproblematically theorise with worlds; or they may attempt to circumvent my argument and articulate a theoretically useful notion of a possible world. There are options available to the contingentist who wishes to take the final option. My argument here has involved several choices in formulating contingentism which one could attempt to challenge—perhaps most controversially, I rejected serious actualism. However, what is clear is that the contingentist must first do substantial theoretical work to establish that a useful notion of a possible world is available—the results presented here show that even assuming that this latter option is an available one is problematic for the propositional contingentist, since it very plausibly at least requires rejecting claims about possibly indistinguishable entities—claims they ought to accept.

Technical appendix

Here, I present both a sound semantics for $\vdash _c$ and $\vdash _c^@$, I prove that both $\vdash _c$ and $\vdash _c^@$ are consistent, and prove (24) and (28). The semantics presented involves extending the standard model-theoretic semantics for first-order modal logic with variable domains to include machinery for handling the plural fragment of $\mathcal {L}^{W}_{\overline{\mathrm {T}}}$, propositions and the term-forming brackets.

First, we define a frame.

Definition 31

(FRAME): Let a frame be a triple $\langle W, D_i, D_{p} \rangle$ where W is a non-empty set, $D_i$ is some function which maps each $w \in W$ to a set $D_i(w)$, and $D_{p}$ is some function which maps each $w \in W$ to a set $D_{p}(w) \subseteq \mathcal {P}(W) \times \mathcal {P}(W)$.

Intuitively, W is a non-empty set of possible worlds, $D_i(w)$ is the set of non-propositional individuals which exist at w and $D_{p}(w)$ is the set of propositions which exist at w. Following (Fine, 1980), we identify propositions with pairs of sets of worlds, i.e. $\langle \alpha , \beta \rangle \in \mathcal {P}(W) \times \mathcal {P}(W)$. Each pair $\langle \alpha , \beta \rangle$ represents a proposition which exists in each world in $\beta$ and is true in each world in $\alpha$. Earlier ($\S$2), I noted that (P) is consistent with conceptions of propositions which individuate propositions at least by their truth, and existence, conditions. Here, we only want to develop a sound semantics and, to keep things simple, we select the simplest conception of propositions to build into our model theory in order to validate (P): propositions are individuated by their truth, and existence, conditions and nothing else. Moreover, there should always be a background worry that more fine-grained views are incoherent, via Russell-Myhill style arguments, e.g., (Russell, 1937) and (Myhill, 1958).

A model based on frame is defined as follows, where $D^A_{p}$ is $\mathcal {P}(W) \times \mathcal {P}(W)$ and $D^A_i$ is $\bigcup \limits _{x \in W}D_{i}(x)$.²⁷

Definition 32

(MODEL): Let a model be a quadruple $\langle W, D_i, D_{p}, v \rangle$, where W, $D_i$, and $D_{p}$ constitute a frame and v is the valuation function:

• (i) v maps each $w \in W$ and non-logical n-place predicate of $\mathcal {L}^{W}_{\overline{\mathrm {T}}}$ to some set of n-tuples of $d \in D^A_i \cup D^A_{p}$.

• (ii) v maps each $w \in W$ and non-logical n-place plural predicate of $\mathcal {L}^{W}_{\overline{\mathrm {T}}}$ to some set of n-tuples of subsets of $D^A_i \cup D^A_{p}$.

We evaluate formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$ in models relative to assignments. Truth in a model $\vDash _c$ and denotation relative to an assignment $\delta _{ {a}}$ are defined in tandem.

Definition 33

(DENOTATION AND $\vDash _c$): Let an assignment be a function which maps each individual singular variable to some $d \in D^A_i$, each propositional singular variable to some $d \in D^A_{p}$, each individual plural variable to some non-empty $d_s \subseteq D^A_i$ and each propositional plural variable to some non-empty $d_s \subseteq D^A_{p}$. Let the variant assignment ${a}[v/d]$ be the assignment which assigns the same values to the same terms as ${a}$, except assigns d to v. We specify the truth-set of $\phi$ relative to ${a}$ as:

$$\begin{aligned} {ts}_{ {a}}(\phi ) = \{ w \in W : \mathcal {M}, w, {a} \models \phi \} \end{aligned}$$

We specify the existence-set of $\phi$ relative to ${a}$:

$$\begin{aligned} {es}_{ {a}}(\phi ^{t_1, ..., t_n}) = \{w \in W: \mathcal {M}, w, {a} \models \mathrm {E}/\overline{\mathrm {E}}t_1 \wedge ... \wedge \mathrm {E}/\overline{\mathrm {E}}t_n \}. \end{aligned}$$

Finally let $v(R)_w$ be the extension of predicate R at w. With these in mind:

• (i) The denotation function $\delta _{ {a}}$, relative to ${a}$, is given:

  • (a) $\delta _{ {a}}(t) = {a}(t)$, where t is a variable.

  • (b) $\delta _{ {a}}([\phi ]) = \langle ts_{ {a}}(\phi ), es_{ {a}}(\phi ) \rangle$.

  • (c) $\delta _{ {a}}(\mathtt {\sim }p) = \langle (W - \alpha ), \beta \rangle$, if $\delta _{ {a}}(p) = \langle \alpha , \beta \rangle$

• (ii) The relation of $\vDash _c$, relative to ${a}$, is given:

  • (a) $\mathcal {M}, w, {a} \vDash _c Rt_1, ..., t_n$ iff $\langle \delta _{ {a}}(t_1), ..., \delta _{ {a}}(t_n) \rangle \in v(R)_w$

  • (b) $\mathcal {M}, w, {a} \vDash _c \overline{R}tt_1, ..., tt_n$ iff $\langle \delta _{ {a}}(tt_1), ..., \delta _{ {a}}(tt_n) \rangle \in v(\overline{R})_w$

  • (c) $\mathcal {M}, w, {a} \vDash _c \lnot \phi$ iff $\mathcal {M}, w, {a} \nvDash \phi$²⁸

  • (d) $\mathcal {M}, w, {a} \vDash _c \forall x \phi$ iff for every $d \in D_i(w)$, $\mathcal {M}, w, {a}[x/d] \vDash _{c} \phi$

  • (e) $\mathcal {M}, w, {a} \vDash _c \forall p \phi$ iff for every $d \in D_{p}(w)$, $\mathcal {M}, w, {a}[p/d] \vDash _{c} \phi$

  • (f) $\mathcal {M}, w, {a} \vDash _c \forall xx \phi$ iff for every non-empty $d_s \subseteq D_i(w)$, $\mathcal {M}, w, {a}[xx/d_s] \vDash _{c} \phi$

  • (g) $\mathcal {M}, w, {a} \vDash _c \forall pp \phi$ iff for every non-empty $d_s \subseteq D_{p}(w)$, $\mathcal {M}, w, {a}[pp/d_s] \vDash _{c} \phi$²⁹

  • (h) $\mathcal {M}, w, {a} \vDash _c \Box \phi$ iff for every $w \in W$ $\mathcal {M}, w, {a} \vDash _{c} \phi$

  • (i) $\mathcal {M}, w, {a} \vDash _c \lozenge \phi$ iff for some $w \in W$ $\mathcal {M}, w, {a} \vDash _{c} \phi$

  • (j) $\mathcal {M}, w, {a} \vDash _c t_1 = t_2$ iff $\delta _{ {a}}(t_1) = \delta _{ {a}}(t_2)$

  • (k) $\mathcal {M}, w, {a} \vDash _c t \prec tt$ iff $\delta _{ {a}}(t) \in \delta _{ {a}}(tt)$

Let $\ulcorner w \rhd \delta _{ {a}}(t) \urcorner$ be $\ulcorner \delta _{ {a}}(t) =$ some $\langle \alpha , \beta \rangle$ such that $\alpha , \beta \in \mathcal {P}(W)$ and $w \in \alpha \urcorner$ and $\ulcorner w \rhd \delta _{ {a}}(tt) \urcorner$ be $\ulcorner \delta _{ {a}}(tt) =$ some $d_s \subseteq \mathcal {P}(W) \times \mathcal {P}(W)$ and, for every $\langle \alpha , \beta \rangle \in d_s$, $w \in \alpha \urcorner$:

• (l) $\mathcal {M}, w, {a} \vDash _c \mathrm {T}t$ iff $w \rhd \delta _{ {a}}(t)$

• (m) $\mathcal {M}, w, {a} \vDash _c \overline{\mathrm {T}}tt$ iff $w \rhd \delta _{ {a}}(tt)$

A formula is true simpliciter at a world, if it is true under every assignment at that world. It is valid in a model, if it is true at any world in that model. It is valid relative to a frame, if it is true in any model on that frame. It is valid simpliciter, if it is true in any model based on any frame. For soundness, we first show that a truncated proof system $\vdash ^{-P}_c$ is sound with respect to the semantics defined by the more general class of models.

Definition 34

($\vdash ^{-P}_c$): Let $\vdash ^{-P}_c$ be $\vdash _c$ without (P) as an axiom.

Theorem 35

($\vdash ^{-P}_c$ SOUNDNESS): If $\vdash ^{-P}_c \phi$, then $\vDash _c \phi$, for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$.

Proof

We show that each axiom except (P) is valid and the inference rules preserve truth in a model. It is immediate that the modal axioms, (PC), the axioms for both singular and plural quantification are valid, and the rules for quantification and (MP) preserve truth. What remains are the axioms for identity, pluralities, and propositions. In the case where the axiom applies to both propositional and non-propositional sorts, I explicitly prove only the non-propositional version, leaving the second to be established by symmetry with the first.

IDENTITY. (I1) is valid since trivially, for any $t \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$, $\delta _{ {a}}(t) = \delta _{ {a}}(t)$ and (I1) is valid just in case, for arbitrary model $\mathcal {M}$, $w \in W$ and ${a}$:

$$\begin{aligned} \mathcal {M}, w, {a} \vDash _c t = t \text { iff } \delta _{{a}}(t) = \delta _{{a}}(t) \end{aligned}$$

(I2) is valid iff, for arbitrary model $\mathcal {M}$, world w and assignment ${a}$:

$$\begin{aligned} \mathcal {M}, w, {a} \vDash _c t_1 = t_2 \rightarrow (\phi [t_1/x] \leftrightarrow \phi [t_2/x]) \end{aligned}$$

If $\mathcal {M}, w, {a} \vDash _c t_1 = t_2$, then $\delta _{ {a}}(t_1) = \delta _{ {a}}(t_2)$. $\mathcal {M}, w, {a} \vDash _c \phi [t_1/x] \leftrightarrow \phi [t_2/x]$ follows by routine induction on the complexity of $\phi$.

PLURALITIES. (NE) is valid on the semantics, since $\delta _{ {a}}(xx)$, for any xx is a non-empty $d_s \subseteq D_i(w)$. (P1) is valid since $\mathcal {M}, w, {a} \vDash _c t \prec tt$ iff $\delta _{ {a}}(t) \in \delta _{ {a}}(tt)$ which holds, if at all, independently of the world parameter. (Ext) is valid, since, if $\mathcal {M}, w, {a}\vDash _c \forall x (x \prec xx \leftrightarrow x \prec yy)$, for arbitrary xx and yy, then $\delta _{ {a}}(xx) = \delta _{ {a}}(yy)$. We then show that $\mathcal {M}, w, {a} \vDash _c \phi (xx) \leftrightarrow \phi (yy)$, assuming $\delta _{ {a}}(xx) = \delta _{ {a}}(yy)$, by induction on the complexity of $\phi$. (Comp) is valid iff, for arbitrary model $\mathcal {M}$, world w and assignment ${a}$:

$$\begin{aligned} If \mathcal {M}, w, {a} \vDash _{c} \exists x \phi (x) \, then\, \mathcal {M}, w, {a} \vDash _c \exists xx \forall y (y \prec xx \leftrightarrow \phi (y)) \end{aligned}$$

$\mathcal {M}, w, {a} \vDash _c \exists xx \forall y (y \prec xx \leftrightarrow \phi (y))$ iff, for some non-empty $d_s \subseteq D_i(w)$:

$$\begin{aligned} (i) \, For \, any \, d \in D_i(w): d \in d_s \leftrightarrow d \in v(\phi )_w. \end{aligned}$$

Let $v(\phi )_w$ be the set $d_s \subseteq D_i(w)$ such that each $y \in d_s$ iff $\mathcal {M}, w, {a}[x/y] \vDash _c \phi (x)$. If $\mathcal {M}, w, {a} \vDash _c \exists x \phi (x)$, then $v(\phi )_w$ is non-empty. Letting $d_s = v(\phi )_w$, we satisfy (i). Finally, (PE) is valid, since, if $\delta _{ {a}}(xx) \in D_i(w)$, for any ${a}$ and w, then if $\delta _{ {a}}(y) \in \delta _{ {a}}(xx)$, it follows that $\delta _{ {a}}(y) \in D_i(w)$.

PROPOSITIONS. (T[]) is valid, since, by (33), for arbitrary $\mathcal {M}$, w, ${a}$:

$$\begin{aligned} \begin{aligned} \mathcal {M}, w, {a} \vDash _c \mathrm {T}[\phi ]&\, {\text {iff}} \, w \rhd \delta _{ {a}}([\phi ])\\&\, {\text {iff}}\, w \in ts_{ {a}}(\phi ) \\&\, {\text {iff}} \, \mathcal {M}, w, {a} \vDash \phi \end{aligned} \end{aligned}$$

($\mathrm {\overline{T}}$) is valid, since $\mathcal {M}, w, {a} \vDash _c \overline{\mathrm {T}}pp$ iff each $\langle \alpha , \beta \rangle \in \delta _{ {a}}(pp)$ is such that $w \rhd \langle \alpha , \beta \rangle$ and, if $\delta _{ {a}}(p) \in \delta _{ {a}}(pp)$, (i.e., if $\mathcal {M}, w, {a} \vDash _c p \prec pp)$, then $w \rhd \delta _{ {a}}(p)$ (i.e., then $\mathcal {M}, w, {a} \vDash _c \mathrm {T}p$). (PNeg1) is valid, since for arbitrary $\mathcal {M}$, w, and ${a}$:

$$\begin{aligned} \begin{aligned} \mathcal {M}, w, {a} \vDash _c \mathrm {T}\mathtt {\sim }p&\, {\text {iff}} \, w \rhd \delta _{ {a}}(\mathtt {\sim }p)\\&\, {\text {iff}}\, w \in \alpha : \langle \alpha , \beta \rangle = \delta _{ {a}}(\mathtt {\sim }p) \\&\, {\text {iff}} \, w \notin (W - \alpha ) \\&\, {\text {iff}} \, \lnot (w \rhd \delta _{ {a}}(p)) \\&\, {\text {iff}}\, \mathcal {M}, w, {a} \vDash _c \lnot \mathrm {T} p \end{aligned} \end{aligned}$$

(PNeg2) is valid, since for arbitrary $\mathcal {M}$, w, and ${a}$:

$$\begin{aligned} \begin{aligned} \mathcal {M}, w, {a} \vDash _c p = [\phi ]&\, {\text {iff}} \, \delta _{ {a}}(p) = \langle ts_{ {a}}(\phi ), es_{ {a}}(\phi ) \rangle \\&\, {\text {iff}}\, \delta _{ {a}}(\mathtt {\sim }p) = \langle W - ts_{ {a}}(\phi ), es_{ {a}}(\phi ) \rangle \\&\, \text {iff}\, \delta _{ {a}}(\mathtt {\sim }p) = \delta _{ {a}}([\lnot \phi ]) \\&\, \text {iff} \, \mathcal {M}, w, {a} \vDash _c \mathtt {\sim }p = [\lnot \phi ] \end{aligned} \end{aligned}$$

$\square$

Next we define a specified class of models $\mathbb {M}^{P}$ and $\vDash _c^{P}$—the notion of truth in a $\mathbb {M}^{P}$–model. The intention is for $\mathbb {M}^{P}$ to be the class of models of $\mathbb {M}$ in which (P) is valid. So we define $\mathbb {M}^{P}$ as such, as well as specifying a restriction on the existence sets of the ordered pairs representing propositions.³⁰ We then define another class of models $\mathbb {M}^{@}$ and $\vDash _c^{@}$—the notion of truth in a $\mathbb {M}^{@}$ model. The semantics defined on $\mathbb {M}^{@}$ is proven to be sound for the proof system $\vdash _c^@$.

Definition 36

($\mathbb {M}^{P}$) Let $\mathbb {M}^{P}$ be the subclass of $\mathbb {M}$ such that, for $\mathcal {M} \in \mathbb {M}^{P}$: (i) $\langle \alpha , \beta \rangle \in D_{p}(w) \rightarrow w \in \beta$; (ii) $\mathcal {M} \vDash _c \exists p (p = [\phi ^{t_1, ..., t_n}]) \leftrightarrow \mathrm {E}/\overline{\mathrm {E}}t_1 \wedge ... \wedge \mathrm {E}/\overline{\mathrm {E}}t_n$.

Definition 37

($\vDash _c^{P}$) We say that $\vDash _c^{P} \phi$ iff $\mathcal {M} \vDash _c \phi$, for any $\mathcal {M} \in \mathbb {M}^{P}$.

Theorem 38

(SOUNDNESS): If $\vdash _c \phi$, then $\vDash _c^{P} \phi$, for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$.

Proof

Immediate, given (35)–(37). $\square$

Definition 39

($\vdash _c^@$) Let $\vdash _c^@$ be the proof system in $\mathcal {L}^{W}_{\overline{\mathrm {T}}@}$ identical to $\vdash _c$ supplemented with the following rules of inference and axioms.

• (@1) $\phi$ / $@\phi$ (@2) $@(\phi \rightarrow \psi )$ / $@\phi \rightarrow @\psi$

• (@3) $\vdash _c^@ @\lnot \phi \leftrightarrow \lnot @ \phi$ (@4) $\vdash _c^@ @\phi \leftrightarrow \Box @ \phi$

Definition 40

($\mathbb {M}^@$) Let $\mathbb {M}^@$ be a class of models $\mathcal {M} = \langle W, D_i, D_{p}, v, w* \rangle$, where $W, D_i, D_{p}$, and v are defined by (31)–(33) and (33) (i)–(ii) $w* \in W$ and (36)(i)–(ii).

Definition 41

($\vDash _c^@$) Let $\vDash _c^@$ be an extension of $\vDash _c$ with the following principle, where $\mathcal {M} \in \mathbb {M}^@$, for arbitrary $w \in W$, and assignment ${a}$.

• (i) $\mathcal {M}, w, {a} \vDash _c^@ @ \phi$ iff $\mathcal {M}, w*, {a} \vDash _c^@ \phi$

Theorem 42

(SOUNDNESS $\vdash _c^@$) If $\vdash _c^@ \phi$, then $\vDash _c^@ \phi$, for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}@}$.

Proof

Given (31)–(39), it suffices to show that $\vdash _c^@$ is sound for the semantics $\vDash _c^@$ by showing that (@1)–(@4) are either logical truths of $\vDash _c^@$ or preserve truth. In the following $\mathcal {M}, w \in W$ and ${a}$ are arbitrary.

$$\begin{aligned} \vDash _c^@ \phi&\, {\text {then}},\quad \mathcal {M}, w, {a} \vDash _c^@ \phi \\&{\text {then},} \, \mathcal {M}, w*, {a} \vDash _c^@ \phi \\&{\text {then},} \vDash _c^@ @ \phi \end{aligned}$$

(@1)

$$\begin{aligned} \vDash _c^@ @(\phi \rightarrow \psi ) \, \text {and} \vDash _c^@ @\phi&\,{\text {then}} \,\mathcal {M}, w*, {a} \vDash _c^@ \phi \rightarrow \psi \text {and} \mathcal {M}, w*, {a} \vDash _c^@ \phi \\&\, {\text {then}} \, \mathcal {M}, w*, {a} \vDash _c^@ \psi \\&\, {\text {then}} \,\mathcal {M}, w, {a} \vDash _c^@ @ \psi \\&\, {\text {then}}\, \mathcal {M}, w, {a} \vDash _c^@ @\phi \rightarrow @ \psi \end{aligned}$$

(@2)

$$\begin{aligned} \vDash _c^@ @ \lnot \phi&{\text {iff}} \mathcal {M}, w*, {a} \vDash _c^@ \lnot \phi \\&{\text {iff}} \mathcal {M}, w*, {a} \nvDash _c^@ \phi \\&{\text {iff}} \mathcal {M}, w, {a} \nvDash _c^@ @ \phi \\&{\text {iff}} \vDash _c^@ \lnot @ \phi \end{aligned}$$

(@3)

For (@4), note that if $\mathcal {M}, w, {a} \vDash _c^@ @\phi$ holds, it holds independently of the world parameter. The converse follows from the validity of (T). $\square$

Next we show that $\mathbb {M}^{P}$ and $\mathbb {M}^@$ are non-empty classes of models in our semantics and thus $\vdash _c$ and $\vdash _c^@$ are consistent.

Theorem 43

$\vdash _c$ is consistent.

Proof

Let W and X be two distinct non-empty sets. Let $\mathcal {M} = \langle W, D_i, D_{p}, v \rangle$,where $D_i(w)= X$ and $D_{p}(w) = \{\langle z, W \rangle : z \subseteq W \}$, for any $w\in W$; and v is any valuation function satisfying (32)(i)–(ii). (33)(i) is satisfied by construction of $\mathcal {M}$. (P) is vacuously true in $\mathcal {M}$. Thus 33(ii) is satisfied. $\square$

Theorem 44

$\vdash _c^@$is consistent.

Proof

Consider the model $\langle W, D_i, D_{p}, v, w* \rangle$ which is an extension of $\mathcal {M} = \langle W, D_i, D_{p}, v \rangle$ in the proof of (43), where $w* \in W$. (33)(i) and (P) are satisfied, given the construction. (@1)–(@4) are satisfied, given (42). $\square$

Finally, we prove (24) and (28). Before we establish these, we fix some symbolism and introduce the notion of an automorphism.³¹

Definition 45

An automorphism on $\mathcal {M} = \langle W, D_i, D_{p}, v \rangle$ is a pair $\langle f, g \rangle$ of structure preserving permutations. Here, f is a permutation of W, i.e., a bijection from W to itself, and g is a permutation on $D_i^A$. Let ${a}^{\langle f, g \rangle }$ be the pointwise application of $\langle f, g \rangle$ to assignment ${a}$. This is defined as follows. Let $\mathrm {aut}^g(x) = \{ g(y): y \in x \}$ and $\mathrm {aut}^f(x) = \{ f(y): y \in x \}$, where x is any set. For each sort of term in $\mathcal {L}^{W}_{\overline{\mathrm {T}}}$, we have:

• (i) (t): ${a}^{\langle f, g \rangle }(t) = g( {a}(t))$

• (ii) (tt): ${a}^{\langle f, g \rangle }(tt) = \mathrm {aut}^g( {a}(tt))$

• (iii) (p): ${a}^{\langle f, g \rangle }(p) = \langle \mathrm {aut}^f(\alpha ), \mathrm {aut}^f(\beta ) \rangle$, where ${a}(p) = \langle \alpha , \beta \rangle$

• (iv) (pp): ${a}^{\langle f, g \rangle }(pp) = \{\langle \mathrm {aut}^f(\alpha ), \mathrm {aut}^f(\beta ) \rangle : \langle \alpha , \beta \rangle \in {a}(pp) \}$

An automorphism $\langle f, g \rangle$ is structure-preserving iff the following holds, where x is a term of any sort, singular or plural.

• (a) ${a}(x) \in v(F)_w$ iff ${a}^{\langle f, g \rangle }(x) \in v(F)_{f(w)}$

• (b) ${a}(x) \in D_i(w)$ iff ${a}^{\langle f, g \rangle }(x) \in D_i(f(w))$

• (c) ${a}(x) \in D_{p}(w)$ iff ${a}^{\langle f, g \rangle }(x) \in D_p(f(w))$

Lemma 46

If $\langle f, g \rangle$ is an automorphism on $\mathcal {M} = \langle W, D_i, D_{p}, v \rangle$, then for any formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$, $w \in W$ and ${a}$:

$$\begin{aligned} \mathcal {M}, w, {a} \vDash _c \phi iff \mathcal {M}, f(w), {a}^{\langle f, g \rangle } \vDash _c \phi \end{aligned}$$

Proof

By induction on the complexity of $\phi$. $\square$

To prove each of (24) and (28), we just need to define two models, $\mathcal {M}_1 \in \mathbb {M}^{\mathrm {P}}$ and $\mathcal {M}_@ \in \mathbb {M}^@$.

Definition 47

($\mathcal {M}_1$) For convenience, in the following, we let $\ulcorner \langle \textit{n}_1\textit{n}_2..., \textit{m}_1\textit{m}_2 ...\rangle \urcorner$ stand for $\ulcorner \langle \{n_1, n_2, ...\}, \{m_1, m_2, ...\} \rangle \urcorner$. Let $\mathcal {M}_1 = \langle W, D_i, D_{p}, v \rangle$, where $W = \{1, 2, 3, 4 \}$, $D_i(1) = \varnothing$, $D_i(2) = \{5, 6 \}$, $D_i(3) = \{5, 7 \}$, and $D_i(4) = \{6, 7 \}$. Let v be the valuation function such that $v(F)_w = D_i(w)$, for any non-logical predicate $F \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$. $D_{p}$ is specified:

• $D_{p}(1) = \{\langle 1, W \rangle , \langle 234, W \rangle , \langle W, W \rangle , \langle \varnothing , W \rangle \}$

• $D_{p}(2) = D_{p}(1) \cup \{\langle x, 2 \rangle | x \subseteq W \} \cup \{\langle x, 23 \rangle | x \subseteq W \} \cup \{ \langle x, 24 \rangle | x \subseteq W \}$

• $D_{p}(3) = D_{p}(1) \cup \{\langle x, 3 \rangle | x \subseteq W \} \cup \{\langle x, 23 \rangle | x \subseteq W \} \cup \{ \langle x, 34 \rangle | x \subseteq W \}$

• $D_{p}(4) = D_{p}(1) \cup \{\langle x, 4 \rangle | x \subseteq W \} \cup \{\langle x, 24 \rangle | x \subseteq W \} \cup \{ \langle x, 34 \rangle | x \subseteq W \}$

Definition 48

($\mathcal {M}_@$) Let $\mathcal {M}_@ = \langle W, D_i, D_{p}, v, w* \rangle$, where $W, D_i, D_{p}, v$ are as defined in (47) and let $w* = 1$

Theorem 24

For some $\mathcal {M} \in \mathbb {M}^{P}$: $\mathcal {M}, w, {a} \vDash _c$ ID

Proof

We show (a) $\mathcal {M}_1 \in \mathbb {M}^{P}$ and that (b) for some $w \in W$ and assignment ${a}$: $\mathcal {M}_1, w, {a} \vDash _c \lozenge \exists x \lozenge \exists y (\lozenge x \approx _e y \wedge \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y))$. First, (a) holds iff

• (i) $\langle \alpha , \beta \rangle \in D_{p}(w) \rightarrow w \in \beta$

• (ii) $\mathcal {M}_1 \vDash _c \exists p (p = [\phi ^{t_1, ..., t_n}]) \leftrightarrow \mathrm {E}/\overline{\mathrm {E}}t_1 \wedge ... \wedge \mathrm {E}/\overline{\mathrm {E}}t_n$, for arbitrary $\phi ^{t_1, ..., t_n}$.

By construction of the model, we can see that (i) holds. For convenience, let $\ulcorner \delta _{ {a}}(t) \gg D(w) \urcorner$ hold just in case either $\delta _{ {a}}(t) \in D_i(w)$, $\delta _{ {a}}(t) \in D_{p}(w)$, $\delta _{ {a}}(t) \subseteq D_i(w)$, or $\delta _{ {a}}(t) \subseteq D_{p}(w)$. Thus (ii) holds iff

$$\begin{aligned} (ii*) {\delta _{ {a}}([\phi ^{t_1, ..., t_n}]) \in D_{p}(w) iff each t_i in t_1, ..., t_n is such that \delta _{ {a}}(t_i) \gg D(w) } \end{aligned}$$

The left-to-right direction of (ii*) follows from (i). The right-to-left direction of (ii*) is established by going through two cases, where $w = 1, 2$, establishing $w = 3, 4$, by symmetry with $w=2$.

CASE I (w = 1): Suppose, for some ${a}$ and $t_1, ..., t_n$ such that each $t_i$ in $t_1, ..., t_n$ is such that $\delta _{ {a}}(t_i) \gg D(1)$. By construction of $\mathcal {M}_1$: if $\delta _{ {a}}(t_i) \gg D(1)$, then, for any $w \in W$, $\delta _{ {a}}(t_i) \gg D(w)$. Thus $\delta _{ {a}}([\phi ^{t_1, ..., t_n}])$ is some $\langle \alpha , W \rangle$, for $\alpha \subseteq W$. We define two automorphisms on $\mathcal {M}_1$, $\langle f, g \rangle$ and $\langle f', g' \rangle$:

$\langle f, g \rangle$::

$f(1) = 1; f(2) = 4; f(3) = 3$; and $f(4) = 2$; $g(5) = 7$; $g(6) = 6$; and $g(7) = 5$.

$\langle f', g' \rangle$:

$f'(1) = 1; f'(2) = 2; f'(3) = 4$; and $f'(4) = 3$; $g'(5) = 6$; $g'(6) = 5$; and $g'(7) = 7$.

Given the specification of $\langle f, g \rangle$ and $\langle f', g' \rangle$, it follows that, for any $t_i$ such that $\delta _{ {a}}(t_i) \gg D(1)$: ${a}^{\langle f, g \rangle }(t_i) = {a}(t_i)$ and ${a}^{\langle f', g' \rangle }(t_i) = {a}(t_i)$. Thus, given (46), and the fact that $f(2) = 4$ and $f'(3) = 4$ it follows that, for any $\phi ^{t_1, ..., t_n} \in \mathcal {L}^W_{\overline{\mathrm {T}}}$ where all $t_i$ in $t_1, .., t_n$ are such that $\delta _{ {a}}(t_i) \gg D(1)$:

• (1) $\mathcal {M}_1, 2, {a} \vDash _c^@ \phi ^{t_1, ..., t_n}$ iff $\mathcal {M}_1, 4, {a} \vDash _c^@ \phi ^{t_1, ..., t_n}$

• (2) $\mathcal {M}_1, 3, {a} \vDash _c^@ \phi ^{t_1, ..., t_n}$ iff $\mathcal {M}_1, 4, {a} \vDash _c^@ \phi ^{t_1, ..., t_n}$

From (1) and (2), it follows that all $\langle \alpha , \beta \rangle \in D_{p}(1)$ are such that $2 \in \alpha$ iff $3 \in \alpha$ iff $4 \in \alpha$. Thus, if all $t_i$ in $t_1, .., t_n$ are such that $\delta _{ {a}}(t_i) \gg D(1)$, then:

• (3) $\delta _{ {a}}([\phi ^{t_1, ..., t_n}]) \in D_{p}(1)$ iff $\Big (2 \rhd \delta _{ {a}}([\phi ^{t_1, ..., t_n}])$ iff $3 \rhd \delta _{ {a}}([\phi ^{t_1, ..., t_n}])$ iff $4 \rhd \delta _{ {a}}([\phi ^{t_1, ..., t_n}])\Big )$

The left-to-right direction of (3) is immediate. The right-to-left direction of (3) follows from the fact that if all $t_i$ in $t_1, .., t_n$ are such that $\delta _{ {a}}(t_i) \gg D(1)$ and $2 \rhd \delta _{ {a}}([\phi ^{t_1, ..., t_n}])$ iff $3 \rhd \delta _{ {a}}([\phi ^{t_1, ..., t_n}])$ iff $4 \rhd \delta _{ {a}}([\phi ^{t_1, ..., t_n}])$, then $\delta _{ {a}}([\phi ^{t_1, ..., t_n}]) = \langle \alpha , W \rangle$, where either $2, 3, 4 \in \alpha$ or $2, 3, 4 \notin \alpha$. The set of $\langle \alpha , W \rangle$ such that either $2, 3, 4 \in \alpha$ or $2, 3, 4 \notin \alpha$ is

$$\begin{aligned} D* = \{\langle 1, W \rangle , \langle 234, W \rangle , \langle W, W \rangle , \langle \varnothing , W \rangle \} \end{aligned}$$

By inspection, $D_p(1) = D*$ and thus $\delta _{ {a}}([\phi ^{t_1, ..., t_n}]) \in D_{p}(1)$, for arbitrary $\phi ^{t_1, ..., t_n}$ and ${a}$.

CASE II (w = 2): We establish (ii) when $w = 2$, for arbitrary $\phi ^{t_1, ..., t_n} \in \mathcal {L}^{W}_{\overline{\mathrm {T}}}$ and ${a}$ using the following facts.

• 1. By (i) each $t_i$ in $t_1, ..., t_n$ is such that $\delta _{ {a}}(t_i) \gg D(2)$.

• 2. By the construction of $\mathcal {M}_1$, it follows that

  • a. If $\delta _{ {a}}([\phi ^{t_1, ..., t_n}]) \in D_{p}(1)$, then, all $w \in W$: $\delta _{ {a}}([\phi ^{t_1, ..., t_n}]) \in D_{p}(w)$.

  • b. If $\delta _{ {a}}([\phi ^{t_1, ..., t_n}])$ is some $\langle \alpha , \beta \rangle$ such that $2, 3, 4 \in \beta$, then $1 \in \beta$.

It follows from (1.) and (2.) that if $\delta _{ {a}}([\phi ^{t_1, ..., t_n}]) \in D_{p}(2)$, then $\delta _{ {a}}([\phi ^{t_1, ..., t_n}]) = \langle \alpha , \beta \rangle$, where $\alpha \subseteq W$ and either $\beta = W$, $\beta = \{2\}$, $\beta = \{2, 3\}$, $\beta = \{2, 4\}$. We can see that, by construction of $\mathcal {M}_1$, for any $\alpha \subseteq W$: $\langle \alpha , 2 \rangle \in D_{p}(2)$, $\langle \alpha , 23 \rangle \in D_{p}(2)$, and $\langle \alpha , 24 \rangle \in D_{p}(2)$. Moreover, given CASE I, for any $\langle \alpha , W \rangle$ such that $2 \in \alpha$ iff $3 \in \alpha$, $\langle \alpha , W \rangle \in D_{p}(2)$. Thus $\delta _{ {a}}([\phi ^{t_1, ..., t_n}]) \in D_{p}(2)$, for arbitrary $\phi ^{t_1, ..., t_n}$.

Cases of $w=3, 4$ are established by symmetry with CASE II.

Next, (b), i.e. for some $w \in W$ and ${a}$: $\mathcal {M}_1, w, {a} \vDash _c \lozenge \exists x \lozenge \exists y (\lozenge x \approx _e y \wedge \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y))$. For this we show that, for some$d_s \in D_i(2)$, some $w' \in W$, and $d_s* \in D_i(w')$:

• (i) $\mathcal {M}_1, w', {a}[x/d_s, y/d_s*] \vDash _c \lozenge x \approx _e y$

• (ii) $\mathcal {M}_1, w', {a}[x/d_s, y/d_s*] \vDash _c \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y)$.

Let $w' = 2$, $d_s = 5$, and $d_s* = 6$. (i) holds iff for some $w'' \in W$:

$$\begin{aligned} \mathcal {M}_1, w'', {a}[x/d_s, y/d_s*] \vDash _c \forall pp(\Box (\overline{\mathrm {T}}pp \rightarrow \mathrm {E}x) \leftrightarrow \Box (\mathrm {\overline{T}}\mathrm {pp} \rightarrow \mathrm {E}y)) \end{aligned}$$

Let $w'' = 1$, $d_s = 5$, and $d_s* = 6$. Thus, (i) holds if for any $d_s' \subseteq D_{p}(1)$:

• (iii) $\mathcal {M}_1, 1, {a}[x/5, y/6, pp/d_s'] \vDash _c \Box (\overline{\mathrm {T}}pp \rightarrow \mathrm {E}x) \leftrightarrow \Box (\mathrm {\overline{T}}\mathrm {pp} \rightarrow \mathrm {E}y)$.

There are fifteen non-empty $d_s' \subseteq D_{p}(1)$. We immediately note that eight such $d_s' \subseteq D_{p}(1)$ are such that $\langle \varnothing , W \rangle \in d_s'$ and so trivially satisfy (iii). Two further subsets of $D_{p}(1)$, namely $\{ \langle 1, W \rangle , \langle 234, W \rangle \}$ and $\{\langle 1, W \rangle , \langle 234, W \rangle , \langle W, W \rangle \}$, also trivially satisfy (iii). Grouping common cases, we have the following two:

CASE I: (Either $d_s' = \{ \langle 1, W \rangle \}$, $d_s' = \{ \langle W, W \rangle \}$ or $d_s' = \{ \langle 1, W \rangle , \langle W, W \rangle \}$.) In each case, $\mathcal {M}_1, 1, {a}[x/5, y/6, pp/d_s'] \vDash _c (\mathrm {\overline{T}}pp \wedge \lnot \mathrm {E}x) \wedge (\mathrm {\overline{T}}pp \wedge \lnot \mathrm {E}y)$. Thus, it follows:$\mathcal {M}_1, 1, {a}[x/5, y/6, pp/d_s'] \vDash _c \lozenge (\mathrm {\overline{T}}pp \wedge \lnot \mathrm {E}x) \wedge \lozenge (\mathrm {\overline{T}}pp \wedge \lnot \mathrm {E}y)$ and thus (iii).

CASE II: (Either $d_s' = \{ \langle 234, W \rangle \}$ or $d_s' = \{ \langle 234, W \rangle , \langle W, W \rangle \}$.) In both cases, it follows that (i) $\mathcal {M}_1, 4, {a}[x/5, y/6, pp/d_s'] \vDash _c \mathrm {\overline{T}}pp \wedge \lnot \mathrm {E}x$ and (ii) $\mathcal {M}_1, 3, {a}[x/5, y/6, pp/d_s'] \vDash _c \mathrm {\overline{T}}pp \wedge \lnot \mathrm {E}y$. From (i) and (ii) it follows that $\mathcal {M}_1, 1, {a}[x/5, x/6, pp/d_s'] \vDash _c \lozenge (\mathrm {\overline{T}}pp \wedge \lnot \mathrm {E}x) \wedge \lozenge (\mathrm {\overline{T}}pp \wedge \lnot \mathrm {E}y)$. Thus (iii) holds.

Thus (iii) and thus (i). (ii) holds given the following: $\mathcal {M}_1, 3, {a}[x/5, x/6, pp/d_s'] \vDash _c \mathrm {E}x \wedge \lnot \mathrm {E}y$. Therefore: $\mathcal {M}_1, 2, {a}[x/5, x/6, pp/d_s'] \vDash _c \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y)$. $\square$

Theorem 28

For some $\mathcal {M} \in \mathbb {M}^@$: $\mathcal {M}, w, {a} \vDash _c^@$ ID@

Proof

We use $\mathcal {M}_@$. $\mathcal {M}_@, 1, {a} \vDash _c^@ \lozenge \exists x \exists y(@x \approx _e y \wedge \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y))$ holds iff, for some $w' \in W$, assignment ${a}$ and some $d_s, d_s* \in D_i(w')$:

• (1) $\mathcal {M}_@, w', {a}[x/d_s, y/d_s*] \vDash _c^@ @ x \approx _e y$

• (2) $\mathcal {M}_@, w', {a}[x/d_s, y/d_s*] \vDash _c^@ \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y)$

Let $w' = 2$, $d_s = 5$, and $d_s* = 6$. Thus (1) and (2) hold if, respectively:

(1$'$):

$\mathcal {M}_@, 2, {a}[x/5, x/6] \vDash _c^@ @x \approx _e y$

(2$'$):

$\mathcal {M}_@, 2, {a}[x/5, x/6] \vDash _c^@ \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y)$

Given that $w* = 1$ it follows from $\mathcal {M}_@, 1, {a}[x/5, x/6] \vDash _c^@ x \approx _e y$ that (1$'$) and thus (1). $\mathcal {M}_@, 3, {a}[x/5, x/6] \vDash _c^@ \mathrm {E}x \wedge \lnot \mathrm {E}y$ holds. Thus (2$'$) and thus (2). $\square$

Finally I address the point in fn. 26. Namely, that the unactualised fundamental theorem for possibility, i.e., $\phi \leftrightarrow \exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ])$, for any $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}@}$ with no free $'pp'$, assuming that $\mathrm {W}^{\mathcal {C}}$ satisfies (W), is not real world valid.

Definition 49

(REAL WORLD VALIDITY) Formula $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}@}$ is real world valid just in case for any $\mathcal {M} \in \mathbb {M}^{@}$ and assignment ${a}$: $\mathcal {M}, w*, {a} \vDash _c^@ \phi$.

Theorem 50

It is not the case that, for any $\phi \in \mathcal {L}^{W}_{\overline{\mathrm {T}}@}$ with no free $'pp'$: $\phi \leftrightarrow \exists pp(\mathrm {W}^{\mathcal {C}}pp \wedge pp \models [\phi ])$ is real world valid assuming that $\mathrm {W}^{\mathcal {C}}$ satisfies (W)

Proof

Given (27) and the proof of (28) above: $\mathcal {M}_@, 1, {a} \vDash _c^@ \lnot @ \exists pp(\mathrm {W}(pp \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y]))$. Thus: $\mathcal {M}_@, 1, {a} \vDash _c^@ @ \lnot \exists pp(\mathrm {W}(pp \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y]))$ and $\mathcal {M}_@, 1, {a} \vDash _c^@ \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y)$. Since $w* = 1$ in $\mathcal {M}_@$, it follows that $\mathcal {M}_@, w*, {a} \nvDash _c^@ \exists pp(\mathrm {W}(pp \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y]))$ and $\mathcal {M}_@, w*, {a} \vDash _c^@ \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y)$. Therefore: $\mathcal {M}_@, w*, {a} \nvDash _c^@ \lozenge (\mathrm {E}x \wedge \lnot \mathrm {E}y) \leftrightarrow \exists pp(\mathrm {W}(pp \wedge pp \models [\mathrm {E}x \wedge \lnot \mathrm {E}y]))$. $\square$