Propositional Potentialism

Abstract

A significant part of Kit Fine’s work in metaphysics assumes a very fine-grained individuation of propositions and facts. This article discusses how such fine distinctions lead to inconsistency in ways which are similar to the inconsistency of naive set comprehension. The case of constraints on individuation arising from a relation of metaphysical ground will be considered in particular. Fine has developed a view of sets in response to the inconsistency of naive set comprehension according to which sets are postulated, and so form a merely potential hierarchy. This article explores what an analogous potentialist response might look like in the case of finely distinguished propositions and facts, and applies the resulting view to the case of metaphysical grounding. A potentially problematic feature of the view is highlighted, which is that it is unclear how it might allow one to formulate in suitable generality the claim that grounding is a well-founded relation among possible propositions.

Appendices

Appendix 1: Strengthening Structure

The theory of structured propositions discussed in Fine (1980, p. 192) individuates propositions even more finely than the structure principle (1) considered above. Fine there considers the following principle:

($1^+$):

$\S \varphi \ne \S \psi$    (where $\varphi$ and $\psi$ do not differ merely in their free variables)

As noted above, this strengthens (1) in two ways: First, it counts $\S \varphi$ and $\S \psi$ as distinct even if the formulas $\varphi$ and $\psi$ merely differ in relabeling bound variables. Second, it counts these propositions as distinct if $\varphi$ contains an abstraction term $\S \chi$ where $\psi$ contains a free variable x, even if $\S \chi$ and x co-denote. This appendix considers these two features in turn, and explores their impact on propositional potentialism.

The following instance of (1$^+$) illustrates the first feature, that relabeling bound variables leads to propositional distinctness:

$$ \S \forall x(x=x)\ne \S \forall y(y=y) $$

This is a strikingly strong commitment. As Fine (2007b) argues persuasively, x performs the same role in $\forall x(x=x)$ as y does in $\forall y(y=y)$, so it is hard to see how the two sentences could fail to express the same proposition. Furthermore, this instance of (1$^+$) seems to be in tension with (2): if relabeling bound variables changes the proposition expressed, why should the same not apply to free variables? For example, consider the following consequence of (2):

$$ x=y\rightarrow \S (x=x)=\S (y=y) $$

It’s not clear what conception of variables could support $\S \forall x(x=x)\ne \S \forall y(y=y)$ and $\S (x=x)=\S (y=y)$ at the same time.

The second feature of (1$^+$) may be philosophically more compelling. (1) already commits one to claiming that sentential structure is reflected in propositions to the extent that, e.g., $\S (\varphi \leftrightarrow \psi )$ is distinct from $\S ((\varphi \rightarrow \psi )\wedge (\psi \rightarrow \varphi ))$. Given such a fine-grained view, one may well also hold that the difference in structure between sentences $x=x$ and $\S \varphi =\S \varphi$ is not immaterial, even if x is in fact $\S \varphi$.

However, this second feature of (1$^+$) introduces some logical complications. Assuming also (2), it follows that we can no longer endorse all instances of free universal instantiation:

$$ \forall x\varphi \rightarrow (Et\rightarrow \varphi [t/x]) $$

The problematic cases are those in which t is an abstraction term and x occurs freely inside of the scope of an abstraction term in $\varphi$. By way of example, consider the following instance of free universal instantiation, where t is some abstraction term $\S \psi$, and $\varphi$ is $x=y\rightarrow \S (x=x)=\S (y=y)$:

$$ \begin{aligned}&\forall x(x=y\rightarrow \S (x=x)=\S (y=y))\\ {} &\quad \rightarrow (E\S \psi \rightarrow (\S \psi =y\rightarrow \S (\S \psi =\S \psi )=\S (y=y)))\end{aligned} $$

For any y, the antecedent should be true, since by (2), propositions do not discriminate between alphabetic variation of free variables assigned the same parameter. Assume that y is some existing proposition $\S \psi$. (E.g., we could assume (L) and choose $\psi$ to be a formula without plural parameters.) Then it follows that $\S (\S \psi =\S \psi )=\S (y=y)$. But this is false by (1$^+$), since the formulas $\S \psi =\S \psi$ and $y=y$ cannot be obtained from one another by relabeling free variables, whence $\S (\S \psi =\S \psi )$ and $\S (y=y)$ are distinct. (Both of these propositions can be thought of as a proposition which says of $\S \psi$, i.e., y, that it is identical to itself. But the proposition $\S (\S \psi =\S \psi )$ reflects the syntactic complexity of the expression $\S \psi$, whereas $\S (y=y)$ contains $\S \psi$ as an unstructured individual.)

It therefore follows that universal instantiation must be restricted in cases where the instantiating term is an abstraction term. In cases where the instantiating term is a variable, no corresponding restriction arises: By (2), if $x=y$ then $\S \varphi =\S \varphi [y/x]$. Likewise, if the variable x to be instantiated using a term t does not occur freely in the scope of any abstraction operator in $\varphi$, no obvious problem arises. Thus, the restriction on universal instantiation arising from (1$^+$ and 2) is that we may not have both that the quantified variable occurs freely in the scope of an abstraction term and that it is instantiated using an abstraction term.

Given this restriction on universal instantiation necessitated by (1$^+$ and 2), it is worth investigating whether the inconsistency of (2) can still be derived. The following is a sketch of an argument along the lines of Russell (1903, Appendix B) in sufficient detail to see that the inconsistency does in fact still arise.

By plural comprehension, there are xx such that:

• (R) $\forall x(x\,\varepsilon \,xx\leftrightarrow \exists yy(x=\S \exists y(y\,\varepsilon \,yy)\wedge \lnot x\,\varepsilon \,yy))$

Let $r:=\S \exists y(y\,\varepsilon \,xx)$. Since x does not occur freely in any abstraction term in (R), we can obtain by universal instantiation:

$$ r\,\varepsilon \,xx\leftrightarrow \exists yy(r=\S \exists y(y\,\varepsilon \,yy)\wedge \lnot r\,\varepsilon \,yy)) $$

We first argue that $r\,\varepsilon \,xx$. For assume for reductio that $\lnot r\,\varepsilon \,xx$. Then by duality of quantifiers and truth-functional reasoning, $\forall yy(r=\S \exists y(y\,\varepsilon \,yy)\rightarrow r\,\varepsilon \,yy)$. By an application of universal instantiation in which the instantiating term is a (plural) variable, we obtain $r=\S \exists y(y\,\varepsilon \,xx)\rightarrow r\,\varepsilon \,xx$. The former is true by the reflexivity of identity, so $r\,\varepsilon \,xx$, contradicting the assumption. Thus $r\,\varepsilon \,xx$. So there are yy such that (i) $r=\S \exists y(y\,\varepsilon \,yy)$ and (ii) $\lnot r\,\varepsilon \,yy$. By (ii), $xx\not \equiv yy$. But then (i), which states $\S \exists y(y\,\varepsilon \,xx)=\S \exists y(y\,\varepsilon \,yy)$, contradicts (2).

Thus, even in a classical (as opposed to free) logic in which universal instantiation is restricted to avoid a relatively obvious conflict with (1$^+$ and 2), it turns out that (2) is inconsistent.

Appendix 2: Non-rigid Abstraction Terms

Above, we assumed that abstraction terms are rigid designators: if x is $\S \varphi$, then necessarily so. However, some of Fine’s discussion of procedural postulationism suggests a reading of the modal operators on which abstraction terms should come out as non-rigid. This appendix discusses the reasons for, and some of the details of, such a variant development of propositional potentialism.

In Fine (2006), Fine applies his procedural postulationism to the problem of unrestricted generality. He also makes use of modal operators, which he paraphrases as serving to vary the interpretation of quantifiers in their scope. I will notate these modalities $\Diamond _i$ and $\Box _i$. In the discussion leading up to their use, Fine uses interpretations to index quantifiers, formulating the claim that there is some interpretation I on which $\exists x\forall y(y\in x)$ is true by stating $\exists I\exists _I x\forall _I y(y\in x)$. In general, for any formula $\varphi$ with un-indexed quantifiers, let us write $\varphi ^I$ for the result of indexing all of its quantifiers by I. We can then state $\exists I\varphi ^I$ to say that there is an interpretation I under which $\varphi$ is true. Fine’s modalities function something like quantifiers binding hidden interpretation indices of quantifiers: $\Diamond _i\varphi$, for example, can be read as stating that on some possible interpretation I, $\varphi ^I$ is true.

It is important that in this paraphrase, the existential quantification over interpretations is itself modalized, quantifying over possible interpretations. (Plausibly, the same is meant to apply to postulational modality, which is why I have formulated it above as quantifying over possible postulations.) Indeed, Fine argues (Fine, 2006, p. 34, fn. 13) that no possible world semantics of the interpretational modalities is possible. Nevertheless, as I have suggested, Fine’s discussion indicates the following way of paraphrasing interpretational modalities applied to non-modal statements:

$$\begin{aligned} \begin{array}{lp{2em}l} \Diamond _i\varphi \leftrightarrow \Diamond _i\exists I\varphi ^I &\,& \Box _i\varphi \leftrightarrow \Box _i\forall I\varphi ^I \end{array} \end{aligned}$$

Of course, this does not amount to a reductive account of the modalities, but we can nevertheless derive some non-trivial consequences from them.

One of the consequences of this account of the interpretational modalities is the non-rigidity of abstraction terms under interpretational modal operators. Consider the case of $y=\S \forall x(x=x)$. If abstraction terms are rigid, then $\Box _i(y=\S \forall x(x=x))$, and so $\Box _i\forall I(y=\S \forall _Ix(x=x))$. So necessarily, for any interpretations I and J, $\S \forall _Ix(x=x)=\S \forall _Jx(x=x)$. But presumably, there could be distinct interpretations I and J. It seems highly plausible given the motivations behind the structured theory of propositions encoded in (2) that if I and J are different quantifier-interpretations, then $\forall _I$ and $\forall _J$ are different quantifiers, and so $\S \forall _Ix(x=x)$ and $\S \forall _Jx(x=x)$ are different propositions, contradicting what followed from the assumption of rigidity of abstraction terms.

We cannot immediately conclude that abstraction terms fail to be rigid under postulational modalities, since Fine’s interpretational and postulational readings of the modalities are not interchangeable. E.g., for a given x, there might be an interpretation of the quantifiers on which x does not exist, but once x has been postulated, it cannot be postulated out of existence by making further postulations. But Fine’s discussion in (2006) suggests that the two modalities are closely connected. The most natural connection is that while $\Box _i$ corresponds to truth under all possible interpretations, $\Box$ corresponds to truth under all possible interpretations which extend the current interpretation. (Here, it should not matter whether “possible” is read as $\Diamond$ or $\Diamond _i$.) With this, it is clear that the non-rigidity of abstraction terms also applies to postulational modalities: $\Box (y=\S \forall x(x=x))$ would require that $\S \forall _Ix(x=x)=\S \forall _Jx(x=x)$ for all possible interpretations I and J which extend the current interpretation, which is highly implausible as well. Thus:

$$ \exists y(y=\S \forall x(x=x)\wedge \Diamond (y\ne \S \forall x(x=x))) $$

Admittedly, Fine’s discussion does not unambiguously underwrite this line of reasoning in favor of non-rigid abstraction terms. In (2006, pp. 39–40), Fine suggests that postulational modalities may leave content unaffected, and instead vary a parameter of evaluation he calls the ontology. Because of this, the main text treated the logically simpler case of rigid abstraction terms first. The remainder of this appendix will explore using non-rigid abstraction terms along the lines motivated here.

The failure of rigidity has several important consequences. The first, noted above, is that we need to reformulate the basic potentialist principle (3):

• (3) $\Diamond E\S \varphi$

If abstraction terms of propositions are not rigid, then this formula does not adequately capture the idea that the proposition expressed by any given formula could exist, since it merely expresses that there could be the proposition expressed by $\varphi$, not that the proposition (in fact) expressed by $\varphi$ could exist. The correct formulation of the intended principle poses an interesting problem for the potentialist: If we could get $\S \varphi$ to be assigned to a variable x, we could simply state the possibility of x’s existence as $\Diamond Ex$. But there seems to be no way of assigning $\S \varphi$ to x, since there is in fact no such proposition and we have no way of picking out the proposition after having executed a new postulation, as $\S \varphi$ may then denote a different proposition.

In the case of metaphysical modality, so-called Vlach-operators are standardly appealed to in response to such difficulties, e.g., in Fine (2005 [1977]). In the elegant formulation of Williamson (2010), two operators $\uparrow$ and $\downarrow$ are introduced, $\uparrow$ being interpreted as storing the world of evaluation on a stack, and $\downarrow$ as retrieving it and evaluating its complement clause at it. Using such operators, one might propose to formulate the intended modalized existence principle for propositions as follows:

$$ \uparrow \Diamond \exists x\downarrow (x=\S \varphi ) $$

However, as the explanation of these operators indicates, it is unclear whether they can be regarded as meaningful unless we can make sense of talk of worlds, and adequately reformulate possibility and necessity in terms of truth relative to some or all worlds. And as already noted, Fine (2006, p. 34) explicitly rejects such a reformulation in the case of postulational/interpretational modality, so it is unclear whether we can appeal to Vlach-operators.

I suggest that the potentialist may instead simply solve the problem by introducing as primitive the device they need but are unable to fashion using the resources of the language at hand: Add to the language a symbol $\downarrow$ (unrelated to the Vlach operator used in the previous paragraph) which may be combined with a singular term t, singular variable x, and formula $\varphi$ to produce a formula ${\downarrow ^t_x}\varphi$, which is to be read as “letting x be t, $\varphi$”. With this, it is straightforward to formulate the desired existence principle, since we may now simply use $\downarrow$ to assign $\S \varphi$ to x:

($3^*$):

${\downarrow ^{\S \varphi }_x}\Diamond Ex$

It’s worth noting that with a variable-binder $\lambda$ to form complex predicates, ${\downarrow ^t_x}\varphi$ may be paraphrased as $(\lambda x.\varphi )t$. But $\lambda$ arguably is a somewhat more committing resource than $\downarrow$: it commits one to regard the complex predicate $\lambda x.\varphi$ as meaningful, whereas no such further commitment arises from the use of $\downarrow ^t_x$ as a sentential operator.

Just failing to assert the rigidity of abstraction terms leaves a number of matters open. We can settle some of them by limiting the extend to which the denotation of abstraction terms can vary. First, there is no reason to think that any expressions apart from quantifiers and modal operators are reinterpreted under modal operators, so that $\S \varphi$ should be a rigid term for any formula $\varphi$ free of quantifiers and modal operators:

• (R1) $\downarrow ^{\S \varphi }_x\Box (x=\S \varphi )$      (where $\varphi$ contains no quantifiers or modal operators)

The non-rigidity of propositional abstraction terms also poses the question whether there are cases in which a proposition $\S \varphi$ is such that possibly, it is $\S \psi$ without being $\S \varphi$. This seems counter to the guiding picture: Letting x be $\S \varphi$, if x were not $\S \varphi$, then x would involve quantificational or modal resources on a more narrow interpretation than the one then available, and so should not be expressed by any formula $\S \psi$. Thus x should be $\S \psi$ only if it is still $\S \varphi$:

• (R2) $\downarrow ^{\S \varphi }_x\Box (x=\S \psi \rightarrow x=\S \varphi )$

This also suggests that (1) can be strengthened in a natural way to the following schema:

$$ \downarrow ^{\S \varphi }_x\Box (x\ne \S \psi ) \quad (\text {where}\,\varphi \, \text {and}\,\psi \, \text {as required in (1)}) $$

Another interesting question to consider concerns the original principle (3):

• (3) $\Diamond E\S \varphi$

As discussed, this does not capture the basic potentialist idea if abstraction terms are non-rigid. But we can argue that it is still motivated, at least given (L), since necessarily, there could be more propositions than there are in fact. Necessarily, the number of pluralities of propositions is greater than the number of propositions, so by (2), the number of propositions of the form $\S Exx$, for arbitrary pluralities xx, is greater than the number of propositions. PROP-$\mathscr {L}$ suggests that there could be all these propositions, so it is possible for the number of propositions to be greater than the number of propositions there are now. So it could be that all actual pluralities are limited, which means that according to (L), $\S \varphi$ could exist.

The discussion so far indicates that taking abstraction terms to be non-rigid leads to a substantially different theory. This raises again the question of consistency. As it turns out, the model construction used above can be adapted to prove the consistency of a version of potentialism based on the principles developed in this appendix in relatively standard ways. The main necessary change to the model construction is that A needs to be indexed by worlds. $\downarrow$ can then be given a relatively straightforward clause:

$$ M,w,a\vDash \downarrow ^t_x\varphi \, \text {iff}\, M,w,a[\llbracket t\rrbracket ^{M,w,a}/x]\vDash \varphi $$

It is not too difficult to develop a model theory of non-rigid abstraction terms along these lines which validates the present principles in the sense of validity used above, which requires all sentences obtained by prefixing strings of universal quantifiers and necessity operators to be verified at all worlds. It is important to note a limitation to this, which is that on this notion of validity, generalization by $\downarrow$ need not preserve validity: Since the domains of worlds are increasing under the accessibility relation, making iterated universal necessity claims true only requires truth at a given world relative to existing parameters, so that $\varphi$ can be valid without $\downarrow ^{\S \psi }_x\varphi$ being valid. This is especially noteworthy in the case of (2). Models validating (2) are not too hard to construct, but it is less easy to see how one would make sure that they also validate $\downarrow$-generalizations like $\downarrow ^{\S \varphi }_x\downarrow ^{\S \psi }_y(\S Ex=\S Ey\leftrightarrow x=y)$. It would thus be worth scrutinizing the consistency question more closely, but doing so is beyond the scope of the present paper.

There is one aspect in which non-rigid abstraction terms fit very nicely with the above discussion of grounding: they suggest an attractive principle governing the grounds of potentialist modal statements—something which seems much harder to achieve in the case of metaphysical modality. Recall that postulational necessity can be paraphrased as truth under all possible extended interpretations of the quantifiers. The grounds of universal statements discussed above suggest that the proposition expressing that under any extended interpretation of the quantifiers, $\varphi$ holds, is grounded in just those propositions expressed by $\varphi$ under any possible extended interpretations of the quantifiers. So the proposition that it is necessary that $\varphi$ should be grounded in just those propositions which could be the proposition that $\varphi$:

$$ \forall x(x\prec \S \Box \varphi \leftrightarrow \Diamond x=\S \varphi ) $$

Besides requiring us to adopt different principles of propositional potentialism, non-rigid abstraction terms also require us to weaken natural background logical principles. In fact, the non-rigidity of abstraction terms leads to yet another restriction of universal instantiation, ruling out cases in which an abstraction term is used to instantiate a variable occurring freely in the scope of a modal operator. To illustrate this, consider the following application of (free) universal instantiation violating this restriction:

$$ \forall x(x=y\rightarrow \Box x=y)\rightarrow (E\S \varphi \rightarrow (\S \varphi =y\rightarrow \Box \S \varphi =y)) $$

The antecedent is true by the logic of variables, so this principle fails if y is some existing proposition expressed by a non-rigid term $\S \varphi$.

Thus, there are potentially three independent sources of restriction on universal instantiation. In the case of singular quantification, $\forall x\varphi \rightarrow \varphi [t/x]$ may fail in any of the following three cases:

• (a) If t does not exist.

• (b) If t is an abstraction term and x occurs freely in the scope of a modal operator in $\varphi$.

• (c) If t is an abstraction term and x occurs freely in the scope of an abstraction term in $\varphi$.

(a) arises from the general idea of potentialism, (b) from non-rigidity, and (c) from the strengthened structure principle (1$^+$) discussed in Appendix 1.

Such restrictions on background logical principles have important implications for the assessment of the overall theory. Of course, they immediately lead to a more complex, and so less attractive, theory. But they also affect the assessment in terms of an implicitly modalized theory. Recall that one way of assessing a potentialist theory is to consider the non-modal theory arising from reading quantifiers as implicitly modalized. If abstraction terms are non-rigid, then we cannot even expect the implicitly modalized theory to contain the principles of elementary free quantification theory. The problem arises from the fact that if abstraction terms are non-rigid due to $\Box$ varying the interpretation of quantifiers, then the following should be false, for y being $\S \Box \forall x(x=x)$:

$$ y=\S \Box \forall x(x=x)\rightarrow \Box \forall z(y=\S \Box \forall x(x=x)) $$

This is the modalization of the following formula, which therefore will come out as not included in the implicitly modalized theory:

$$ y=\S \forall x(x=x)\rightarrow \forall z(y=\S \forall x(x=x)) $$

But this is an instance of the law of vacuous quantification, which is included in standard free logic. This is clearly a limitation of the present variant of propositional potentialism. But it is not necessarily a fatal problem: As noted, in the case of propositions, there is not even a standard theory like ZFC the general use of which one would need to vindicate. We might simply insist that theorizing about propositions should be carried out in the explicitly modalized language.

These kinds of complications arising from failures of standard logical principles provide some reasons against the two variations of the theory discussed in the appendices. There are ways in which one might try to avoid them. One option is to do away with term-forming abstraction operators entirely. Instead, one could use an abstraction operator $\int$ which takes an individual term and a sentence as arguments, with $\int (t,\varphi )$ stating that t is the proposition that $\varphi$. Since variables are then the only singular terms, restrictions (b) and (c) immediately become vacuous. And since propositions, once postulated, can’t be postulated out of existence, restriction (a) becomes unnecessary as long as $\Box$ and $\Diamond$ are the only modal operators. But such a language seems problematically limited: Consider the case of a formula $\varphi$ involving a quantifier such that the proposition that $\varphi$ does not exist: $\lnot \exists x\int (x,\varphi )$. As argued above, although this proposition might exist, it might then not be the proposition that $\varphi$. Without a term-forming abstraction operator, there is no obvious way of talking about this proposition; in particular, there is no obvious way of stating that it could exist. The language therefore seems problematically impoverished.