2. Closure and Four Challenges to the Existence of Free Will

Challenges to the existence of free will have been a recurring theme in Western philosophy since classical times. Rather than surveying their historical development, we will focus instead on four simple arguments. As we will see, each of these arguments shares a common structure, relying on a familiar inference rule we call Closure. To introduce the arguments, let us begin with a simple example, an everyday action that just about anyone could presumably freely perform. Suppose that you are about to raise your arm at some future time, t. Will you raise your arm freely at t? The following are four informal arguments for answering in the negative:Footnote ¹

These are simple versions of widely discussed arguments. Using more or less standard terminology, (I) is a reconstruction of the argument for logical fatalism (Taylor, Reference Taylor1962; Haack, Reference Haack1974), (II) is a reconstruction of the argument for theological fatalism (Pike, Reference Pike1965; Fischer, Reference Fischer2016), (III) is a version of the Consequence Argument for the incompatibility of free will and determinism (Huemer, Reference Huemer2000; Van Inwagen, Reference Van Inwagen1983), and (IV) is a version of the Mind Argument for the incompatibility of free will and indeterminism (Van Inwagen, Reference Van Inwagen1983; Nelkin, Reference Nelkin2001).Footnote ²

Those who are not free-will skeptics will be interested in these arguments mainly as furnishing what one may describe as conditional challenges to the existence of free will. Received in this spirit, the first argument shows that you have no choice about raising your arm on the supposition of a non-open future; the second does the same on the supposition of divine omniscience, etc.Footnote ³

All four arguments above are schematic: in each case, the conclusion could involve any true proposition concerning a human action, and the premises stand in for the corresponding assumptions about future truths, divine knowledge, and so on. Assuming the arguments are valid, the upshot is that if their respective premises are true, then so are their conclusions; thus, no human ever acts freely.

In each argument, the first premise is typically supported by some version of the principle that, if p is a truth about the past, then it is not up to anyone whether p (at least not anymore). This claim, that the past is fixed, has been challenged in various ways (see Dorr, Reference Dorr2016; Lampert, Reference Lampert2022, Merricks, Reference Merricks2009, Reference Merricks2011; Perry, Reference Perry2008; Saunders, Reference Saunders1966; among others). Still, it remains a cornerstone of the case for the claim that free will is precluded by determinism, divine foreknowledge, and so on (see, e.g., Fischer & Todd, Reference Fischer and Todd2013).Footnote ⁴

The second premise in each argument is relatively uncontroversial. In (I), it can be motivated by the general idea that prior truths about later times settle how things must be at those later times. In (II), it reflects the widely accepted principle of divine infallibility, according to which God could not hold false beliefs. In (III), the premise follows from determinism: for any truth p, the conjunction of L and H entails p. In (IV), the second premise corresponds to a theorem of standard modal logic: $ \square \left(\left(p\wedge \left(p\to q\right)\right)\to q\right) $ .

Furthermore, the arguments in question all share the same “shape.” Where p is any truth:

1. (1) S has no choice about: p.

2. (2) Necessarily, if p then q.

   —————

3. (3) S has no choice about: q.

More schematically, let ‘Np’ abbreviate ‘p and S does not have, never had, and never will have a choice about whether p’—or, simply, ‘p and S has no choice about whether p.’ Then, arguments (I)–(IV) each instantiate an inference rule that aims to derive (3) from premises (1) and (2), namely:

If Closure is valid, then each of the four arguments above is likewise valid, and the central question becomes whether their first premises are true. If, on the other hand, Closure is invalid, then the arguments are unsound. It is therefore crucial to assess the plausibility of Closure itself.Footnote ⁵

3. Whence Closure?

To better understand Closure in the context of the free will literature, it is helpful to briefly summarize some relevant recent history. Closure was first formulated in connection with van Inwagen’s Consequence Argument for the incompatibility of free will and determinism. In his original presentation, van Inwagen introduced the ‘no choice’ operator N, where ‘Np’ abbreviated ‘p, and no one has, or ever had, any choice about whether p.’ To explicate this operator, van Inwagen invoked the notion of rendering a proposition false: one has no choice about a truth if and only if one is not able to render it false.Footnote ⁶ In this sense, if I ordered coffee after lunch, though I could have ordered tea or nothing at all, then I had a choice about the fact that I ordered coffee, since there was something I could have done such that, had I done it, the proposition that I ordered coffee would have been false.

Van Inwagen then proposed the following inference rules involving the operator N:

According to Alpha, no one has a choice about necessary truths. According to Beta, if no one has a choice about a truth p and no one has a choice about the conditional $ p\to q $ , then no one has a choice about q. In other words, choicelessness distributes across material conditionals that no one has a choice about.

We can now present van Inwagen’s original Consequence Argument. Let p be any truth about a human action—for instance, the fact that you will (or did) raise your arm. It is a definitional consequence of determinism that the conjunction of the laws of nature L and a complete description of the remote past H together necessitate every truth (see premise (1) of argument (III)). Thus:

$$ {\displaystyle \begin{array}{lll}1.& \square \left(\left(L\wedge H\right)\to p\right)& \mathrm{Premise},\mathrm{from}\ \mathrm{Determinism}\\ {}2.& NL& \mathrm{Premise},\mathrm{Fixity}\ \mathrm{of}\ \mathrm{the}\ \mathrm{Laws}\ \mathrm{of}\ \mathrm{Nature}\\ {}3.& NH& \mathrm{Premise},\mathrm{Fixity}\ \mathrm{of}\ \mathrm{the}\ \mathrm{Past}\\ {}4.& \square \left(L\to \left(H\to p\right)\right)& 1,\ \mathrm{by}\hskip0.45em \mathrm{Modal}\ \mathrm{Exportation}\\ {}5.& N\left(L\to \left(H\to p\right)\right)& 4,\ \mathrm{by}\ \mathrm{A}\mathrm{LPHA}\\ {}6.& N\left(H\to p\right)& 2\hskip0.45em \mathrm{and}\hskip0.45em 5,\ \mathrm{by}\ \mathrm{B}\mathrm{ETA}\\ {}7.& Np& 3\hskip0.45em \mathrm{and}\hskip0.45em 6,\ \mathrm{by}\ \mathrm{B}\mathrm{ETA}\end{array}} $$

Since p is arbitrary, given determinism, it only matters whether p is true. But, p is also about human action. Since the same line of reasoning applies to any truth whatsoever, a fortiori it applies to any truth concerning human actions. Thus, if determinism is true, no one has a choice about any human action. Therefore, no one has free will.

The derivation itself relies on two extralogical inference rules, namely Alpha and Beta. But are these rules valid? What can we say in their favor?

There are no standard arguments in defense of Alpha. Its validity is typically taken to be self-evident and has only rarely been questioned (see, for instance, Section 6).Footnote ⁷ Van Inwagen claims that no one could or would dispute the validity of Alpha, and asserts that it is obviously valid (see Van Inwagen, Reference Van Inwagen1989: 405, Reference Van Inwagen1990: 283, Reference Van Inwagen2000: 2). Vihvelin (Reference Vihvelin1988: 230) remarks that Alpha “seems uncontroversial,” while Finch & Warfield (Reference Finch and Warfield1998: 517) describe it as “surely unobjectionable.” McKay & Johnson (Reference McKay and Johnson1996: 115) go so far as to call it “impeccable,” adding that it “needs no defense.”

So much for Alpha; what about Beta? Van Inwagen (Reference Van Inwagen1983: 96) himself concedes that it is “the most difficult of the premisses (…) to defend.” As it turns out, Beta is now widely regarded as invalid. The most familiar counterexamples to Beta are also counterexamples to another inference rule, namely Agglomeration:

Agglomeration is derivable from Alpha and Beta.Footnote ⁸ Therefore, a counterexample to Agglomeration would be a counterexample to Alpha and Beta taken together, thereby undermining van Inwagen’s formulation of the Consequence Argument. Proposed counterexamples to Agglomeration, canonically formulated by McKay & Johnson (Reference McKay and Johnson1996), are widely known. Pruss (Reference Pruss2013: 431) considers their counterexamples to be “conclusive,” and van Inwagen himself (Van Inwagen, Reference Van Inwagen2000) claims that they undermine his original rule Beta. Since Alpha is widely accepted and Agglomeration is derivable from Alpha and Beta, there is an entrenched consensus that Beta is therefore invalid.Footnote ⁹

In response, some defenders of the Consequence Argument have proposed replacing Beta with a different principle, namely, what we above called Closure.Footnote ¹⁰ Replacing Beta with Closure allows for a streamlined and more elegant formulation of the Consequence Argument:

$$ {\displaystyle \begin{array}{lll}1.& N\left(L\wedge H\right)& \mathrm{Premise},\mathrm{Fixity}\ \mathrm{of}\ \mathrm{the}\ \mathrm{Laws}\ \mathrm{of}\ \mathrm{Nature}\ \mathrm{and}\ \mathrm{the}\ \mathrm{Past}\\ {}2.& \square \left(\left(L\wedge H\right)\to p\right)& \mathrm{Premise},\mathrm{from}\ \mathrm{Determinism}\\ {}3.& Np& 1\hskip0.42em \mathrm{and}\hskip0.44em 2,\mathrm{by}\hskip0.42em \mathrm{C}\mathrm{LOSURE}\end{array}} $$

The validity of this argument clearly hinges on the validity of Closure alone, as the argument is simply an instance of it. This argument may also be regarded as a regimentation of (III) above.Footnote ¹¹

Closure is now a standard feature of arguments challenging the existence of free will. For instance, variants of arguments (I) and (II) appealing to Closure have been presented and discussed by Finch & Warfield (Reference Finch and Warfield1999), Rea (Reference Rea2006), Finch & Rea (Reference Finch and Rea2008), Fischer & Todd (Reference Fischer and Todd2015), Merricks (Reference Merricks2009, Reference Merricks2011), Todd (Reference Todd2023), Lampert (Reference Lampert2025), among others. Nelkin (2001) develops a prominent version of (IV) employing Closure as well. More generally, some of the most prominent contemporary arguments for various forms of incompatibilism all depend straightforwardly on Closure.

But the significance of Closure is hardly limited to those arguments alone. For example, the validity of van Inwagen’s original Consequence Argument also requires the validity of Closure. Recall that the original argument invokes the rules Alpha and Beta. From these, however, one can straightforwardly derive Closure:

$$ {\displaystyle \begin{array}{lll}1.& Np& \mathrm{Premise}\\ {}2.& \square \left(p\to q\right)& \mathrm{Premise}\\ {}3.& N\left(p\to q\right)& 2,\ \mathrm{by}\;\mathrm{A}\mathrm{LPHA}\\ {}4.& Nq& 1\hskip0.42em \mathrm{and}\ 3,\ \mathrm{by}\ \mathrm{B}\mathrm{ETA}\end{array}} $$

Thus, defenders of the original Consequence Argument presuppose, in a sense, the validity of Closure, for any counterexample to Closure would undermine Alpha and Beta, and with it van Inwagen’s argument.Footnote ¹²

So, what can be said in favor of Closure?

With a few exceptions (which we will discuss in due course), it is only the apparent obviousness of Closure that has counted in its favor—and that almost conclusively. For this reason, partisans of Closure rarely offer arguments in its defense. Finch & Warfield (Reference Finch and Warfield1998: 522) remark:

Elsewhere, they maintain in the same spirit that Closure is “clearly valid” (Finch & Warfield, Reference Finch and Warfield1998: 525). Similarly, Kane (Reference Kane, Fischer, Kane, Pereboom and Vargas2007: 11), in the absence of a compelling defense of Closure, appeals to intuitive plausibility and illustrative examples. On similar grounds, he claims in an earlier work that Closure is “as plausible” as Alpha (Kane, Reference Kane2005: 26). It is also worth noting that principles closely resembling Closure appear in related contexts, particularly in debates about the compatibility of determinism and moral responsibility. Like Closure, these principles are rarely defended through rigorous argumentation, yet they are frequently regarded as intuitively compelling.Footnote ¹³

There is, then, widespread acceptance of Closure (and closely related principles) among philosophers, which is motivated largely by appeals to intuition rather than by rigorous (let alone formal) argument. (In the following section we note a few exceptions.) Our main focus, however, is a simple and powerful argument for Closure. Reflecting on this argument leads us to our central contention: that the key issue in the debate over Closure is whether the N operator is intensional or hyperintensional.

4. The Intensional Argument

Scattered attempts to vindicate Closure and related principles are available, though these are almost exclusively confined to the literature on the Consequence Argument. Notably, informal arguments in support of Closure can be found in Carlson (Reference Carlson2000) and Huemer (Reference Huemer2000), while formal derivations are presented in Pruss (Reference Pruss2013). Despite their clear merits, existing strategies for defending Closure are explicitly narrow in scope: they proceed by first identifying plausible, but nevertheless highly specific, interpretations of the N operator, and then arguing that those interpretations validate Closure.

To clarify, note that we do not expect the sentence ‘p and no one can render p false’ to wear its meaning on its sleeve in ordinary discourse. The modal verb ‘can’ serves a wide variety of roles, and even within the class of so-called “agentive” or “ability” modals, its truth-conditions in a given context are highly context-sensitive. That is to say nothing of the expression ‘render false,’ which, in this context, is surely a philosopher’s invention.

Now, one cannot sensibly argue that Closure is a truth-preserving inference rule without addressing the truth-conditional interpretation of the sole non-logical constant occurring therein, namely the operator N. Accordingly, defenders of Closure typically proceed by identifying a single interpretation of N, and then arguing that Closure, so interpreted, is valid. One may attempt to isolate a suitable reading either by appealing to an “intended interpretation” of the operator N, or by stipulating a particular technical reading. As a result, the free will literature has witnessed a proliferation of proposed interpretations of N. Footnote ¹⁴

But this narrow strategy is hardly the only way to defend Closure, since one can profitably address the truth-conditional interpretation of N without committing to any highly specific reading. For obvious reasons, more general strategies for vindicating Closure are likely to be more promising, if feasible. We believe that one such broadly applicable strategy is indeed feasible.

Very roughly, instead of focusing on the meaning of ‘no choice,’ or on any single reading of the N operator, we propose investigating the behavior of N within the framework of a background normal modal logic, under any interpretation that satisfies minimal formal constraints. This amounts to a somewhat superficial investigation of the bimodal logic of the operators $ \square $ and N. One advantage of this approach is that it allows us to set aside narrower questions about how N should be understood—questions that are difficult at best and merely verbal at worst. Any results of this investigation will thus be extremely general.Footnote ¹⁵

Even on the most cursory consideration of the issues, two minimal requirements on the logic of the N operator readily suggest themselves. First, N is factive:

Second, N is closed under conjunction elimination:

Factivity is straightforwardly borne out by every proposed reading of the N operator to date, and for this reason we treat it as part of the common ground. After all, N is definitionally only applied to truths that no one has any choice about.

Conjunction is almost as compelling as Factivity. Nevertheless, since we cannot plausibly claim that Conjunction holds by definition alone, it is worth offering some straightforward, if nontrivial, remarks in its support.

First, under any intuitive characterization of the N operator, rejecting Conjunction leads to paradoxical results. Suppose there is a counterexample to Conjunction. Then: (a) someone has no choice about the conjunction $ p\wedge q $ ; (b) by Factivity, $ p\wedge q $ is true; and yet (c) someone does have a choice about one of the conjuncts, say, p. But, following van Inwagen (Reference Van Inwagen1983), having a choice about a truth is very close to simply being able to “render it false” (however this is understood). Taken together, we then get the paradoxical conclusion that someone could render p false without having any choice about whether $ p\wedge q $ is true—that is, without being able to render $ p\wedge q $ false.

There are nevertheless stronger reasons for favoring Conjunction. If other principles proposed by incompatibilists for governing the N operator hold, then so does Conjunction. The principles in question are the standard inference rules appealed to in different versions of the Consequence Argument. In particular, Conjunction can be proved from Alpha and Beta, regardless of how the N operator is understood.Footnote ¹⁶ If we drop the pairing of Alpha and Beta in favor of Closure alone, we obtain the same conclusion.Footnote ¹⁷

Finally, a more modest argument for Conjunction assumes only that N is Boolean. That is, suppose N is closed under the consequence relation of the classical propositional calculus: given $ Np $ and some classical consequence q of p, we infer $ Nq $ . Since any conjunction has each of its conjuncts as tautological consequences, Conjunction follows immediately.

In this way, Conjunction can be seen as a logical common core underlying the principles previously recommended for governing the N operator. However, we are not arguing for Conjunction on the assumption that the more substantive rules governing N are truth-preserving; doing so would be dialectically backwards. Rather, we proceed on the assumption that defenders of Closure have some understanding of what ‘no choice’ means in their mouths. Thus even if their endorsement of principles like Closure does not guarantee that such rules are truth-preserving, it still provides evidence that violations of principles like Conjunction involve a change of subject matter.

Those are our two proposed minimal requirements on the logic of N. How might we more adventurously move beyond them? What further, general considerations about the N operator can be brought to bear on the question of whether Closure is valid? As it turns out, venturing just a little farther, we can say quite definitively what it takes for Closure to be truth-preserving. There is a simple, straightforward proof of Closure that follows from our minimal logical assumptions together with one further condition, namely, that N is an intensional operator.

To clarify, contemporary philosophers use the word ‘intensional’ and its cognates in diverse ways. Adhering to a somewhat dated usage, in some quarters ‘intensional’ is used interchangeably with ‘non-extensional.’ According to this older usage, any sentential context in which material equivalents cannot be substituted truth-preservingly for one another counts as an intensional context. This, however, is not the sense of ‘intensional’ we have in mind here.

Whereas the older usage emphasizes the contrast between the extensional and the non-extensional, more recent usage emphasizes the contrast between the hyperintensional and the non-hyperintensional. Roughly speaking, a hyperintensional sentential context is one in which necessarily equivalent expressions are not guaranteed to be truth-preservingly substitutable, or substitutable salva veritate. A paradigmatic hyperintensional context is an open attitude ascription. Consider the sentential context ‘Peter believes that ….’ Philosophical lore holds that from the sentence ‘Peter believes that Superman flies,’ one cannot truth-preservingly deduce ‘Peter believes that Clark Kent flies,’ even though the two embedded complement sentences are necessarily equivalent—that is, the propositions expressed by the two sentences are true in exactly the same possible worlds. Such sentential contexts are regarded as paradigmatically hyperintensional. Thus, rather than taking intensional as contrasting primarily with extensional, more recent usage contrasts intensional with hyperintensional. This is the usage we adopt in what follows.

In contrast to hyperintensional sentential operators, intensional operators thus permit the substitution of necessary equivalents into operand position salva veritate. A paradigmatic intensional operator is the necessity operator. Necessity is intensional, for if p and q are necessarily equivalent, $ \square p $ and $ \square q $ have the same truth value. We can emphasize the intensionality of $ \square $ by noting that, whereas philosophical lore does not want the following conditional to come out true:

it does take the following to be true:

Thus, the lore dictates that belief contexts are hyperintensional, whereas the necessity operator is intensional.

So, how might one regiment the thesis we wish to consider, that is, the thesis that N is an intensional operator? For our purposes, it will suffice to explicate this thesis by means of the following inference rule:Footnote ¹⁸

Hereafter, when we say that N is an intensional operator, we stipulatively mean that Intensionality holds.

As advertised, we are now in a position to state what it takes for Closure to be truth-preserving. Given our minimal assumptions, along with the additional assumption that N is an intensional operator, a short proof of Closure is readily available:

$$ {\displaystyle \begin{array}{lll}1.& Np& \mathrm{Premise}\\ {}2.& \square \left(p\to q\right)& \mathrm{Premise}\\ {}3.& \square \left(p\leftrightarrow \left(p\wedge q\right)\right)& 2,\ \mathrm{by}\ \mathrm{Modal}\ \mathrm{Logic}\\ {}4.& N\left(p\wedge q\right)& 1\ \mathrm{and}\ 3,\ \mathrm{by}\ \mathrm{I}\mathrm{NTENSIONALITY}\\ {}5.& Nq& 4,\ \mathrm{by}\ \mathrm{C}\mathrm{ONJUNCTION}\end{array}} $$

Call this argument for Closure the Intensional Argument.

At the risk of repetition, we note again that the Intensional Argument proceeds independently of any particular reading of N, thereby distinguishing it from extant defenses of Closure. In this sense, the argument is highly general: however N is defined, the Intensional Argument shows that Closure is valid so long as N is intensional. Note also that if Closure is valid, then so is Intensionality; if N is closed under entailment, it is a fortiori closed under necessary equivalence. Thus, against our background assumptions, the validity of Intensionality is both necessary and sufficient for the validity of Closure. This, then, is what it takes for Closure to be truth-preserving: N must be an intensional operator.Footnote ¹⁹

Thus, we arrive at the central contention of this paper: the status of Closure—and with it, all four of the arguments identified earlier—turns on whether N is an intensional operator. If we reframe disputes about these arguments in terms of the narrower and more precise question of whether N is intensional, there is real hope for more progress on some of the grand old questions surrounding free will. After all, debates about intensionality and hyperintensionality are well-trodden territory for contemporary logicians, philosophers of language, and formally inclined metaphysicians. In the next two sections, we undertake a preliminary exploration of these questions.

5. N as an Intensional Operator

As the previous section argues, Closure is valid if N is an intensional operator. But if Closure is valid, so are the four arguments discussed above that challenge the existence of free will. Thus, if we can show that N is an intensional operator, we will thereby have an answer to the open question whether the four arguments are valid—namely, that they are valid. So, what can be said in favor of the thesis that N is intensional?

Thus far, the literature on free will has largely overlooked this question. However, some observations drawing on the existing literature do suggest that N should be treated as an intensional operator. For one thing, an argument for the intensionality of N was already available to van Inwagen when he originally formulated the Consequence Argument. In that setting, Intensionality is implicit in two of his operational assumptions. First, to have a choice about p is to stand in some special relation R to the proposition that p. Second, propositions are identified with sets of possible worlds: that is, the proposition that p is the set of worlds in which p is true (see van Inwagen, Reference Van Inwagen1983: 58–59).

Given these assumptions, it follows that N is intensional, and, by the argument of the previous section, that Closure is valid. For suppose that $ Np $ is true and that $ \square \left(p\leftrightarrow q\right) $ holds, so that p and q are true in exactly the same possible worlds. Given the possible-worlds theory of propositions, the proposition that p just is the proposition that q. Given that $ Np $ , no one has a choice about whether p, and so no one stands in relation R to the proposition that p. But then, given the aforementioned identity, it follows that no one stands in relation R to the proposition that q; hence, $ Nq $ is true.

Note that this argument does not really require endorsing a possible-worlds theory of propositions—something van Inwagen himself was unwilling to do. In the passage referred to above, van Inwagen suggests that such a theory is not serviceable as a general account of propositions. Nevertheless, he maintains that it is serviceable for clarifying the key technical notions at issue in the Consequence Argument. Sensibly enough, then, we need only say—more modestly—that, for van Inwagen, the most distinctive and bold feature of such a theory, namely, that it individuates propositions by necessary equivalence, generates the right predictions about the interaction of the standard modal operators on the one hand and the N operator on the other. This modest retreat leaves the above argument essentially unaltered.

A further remark by van Inwagen supports treating the N operator as intensional. At one point in An Essay on Free Will, van Inwagen suggests the possibility of giving a basic possible-worlds interpretation of the N operator:

We could generalize this proposal by positing compositional possible-worlds truth-conditions for sentences formed with the operator N. This is a highly plausible proposal, since it effectively treats ‘no one has a choice about whether p’ in a manner analogous to standard treatments of agentive modals in formal semantics, which draw heavily on the influential work of Kratzer (Reference Kratzer1977).

As it happens, this proposal also straightforwardly validates Intensionality. To see why, suppose that N is defined in the way van Inwagen describes. Then there exists some set S of possible worlds such that $ Np $ is true if and only if p is true in all worlds in S. Now suppose that $ Np $ is true. Then p is true throughout S. If $ \square \left(p\leftrightarrow q\right) $ is also true, then p and q are true in exactly the same possible worlds. It follows that q must also be true in all worlds in S, and hence $ Nq $ is true. Thus, N satisfies Intensionality.Footnote ²⁰

One further thing these considerations bring out is that any characterization of N—or informal understanding of ‘no choice’—that appeals solely to intensional notions will itself support Intensionality. Paradigmatic intensional notions of this sort include those involving counterfactual dependence or causal relations. This observation connects our present discussion to the existing literature in two important respects. First, in the literature on the Consequence Argument several readings of N have in fact been proposed that rely exclusively on intensional resources. Given our minimal formal assumptions, the Intensional Argument shows that such interpretations will validate Closure, and thereby all four arguments discussed above. Second, the Intensional Argument offers a unifying framework that generalizes certain arguments already present in the literature for the validity of Closure. Indeed, the assumptions behind the Intensional Argument are sufficiently modest that plausibly the best existing defenses of Closure are special cases of the Intensional Argument.

This very brief discussion highlights underexplored routes to the conclusion that the four arguments challenging free will from Section 2 are all valid, namely, through the metaphysics of propositions and the semantics of ability modals. Moreover, there are further avenues for exploration that we have yet to consider: seasoned metaphysicians, for instance, may find promising approaches by appealing to well-known general theses about supervenience, intrinsicality, or grounding, all of which can be connected to necessity and possibility in various ways. If it can be argued convincingly that N is insensitive to putative differences between necessary equivalents, the implications for the free will literature would be far-reaching.

That said, the Intensional Argument also motivates a distinct line of inquiry for those who wish to argue that one or more of the arguments challenging free will we focus on are invalid. It is to this investigation that we now turn.

6. N as a Hyperintensional Operator

What the Intensional Argument clarifies is that if N is an intensional operator, then Closure, and with it the four arguments discussed in Section 2, are valid. Accordingly, any counterexample to Closure must also be a counterexample to Intensionality; that is, it must involve N being a hyperintensional operator. Parallel to the last section, this insight motivates us to propose some general considerations against the validity of Closure drawing on discussions beyond the free will literature.

Let us outline one such consideration here. The following three theses enjoy some popularity in contemporary philosophy:

1. (A) All intentional action is acting on know-how (Anscombe, Reference Anscombe1957).

2. (B) All know-how is propositional knowledge (Williamson & Stanley, Reference Williamson and Stanley2001).

3. (C) Propositional knowledge is hyperintensional.

Together, these three theses make it plausible that N is hyperintensional. Given (B) and (C), it is plausible that someone might, on occasion, know how to render p false despite not knowing how to render q false, even when p and q are necessarily equivalent. Consequently, given (A), it is also plausible that one might be able to perform the intentional action of rendering p false without fulfilling a necessary requirement on being able to render q false. Thus, if the intended sense of N is limited to the ability to perform intentional actions, theses (A)–(C) suggest that N is hyperintensional. If, moreover, such possible cases show that N is hyperintensional, they serve as counterexamples to Intensionality and, therefore, make it plausible that Closure is invalid.

It is therefore not only defenders of Closure who can advance new arguments in light of the Intensional Argument, as the latter also provides systematic guidance to detractors of standard incompatibilist arguments. To emphasize this point, we will next highlight a family of possible counterexamples to Closure and show how they conform to the pattern suggested by the Intensional Argument—namely, that these are cases where one would be inclined to regard N as hyperintensional.

To begin, suppose someone has a choice about a necessary truth. It is not hard to see why such cases would challenge Closure. Treated abstractly, suppose p is a truth that no one has a choice about, and that q is a necessary truth that someone does have a choice about. Since q is necessary, $ \square \left(p\to q\right) $ is true. So, both $ Np $ and $ \square \left(p\to q\right) $ are true. Nonetheless, because someone has a choice about q, $ Nq $ is false. We thus have a counterexample to Closure.

Now, for our purposes, all we require are cases where someone has a choice about some necessary truth. Moreover, it does not matter for our purposes whether these cases are rare or marginal. With that said, let us begin by considering the following principle:

Spencer (Reference Spencer2017) presents several tempting counterexamples to (POSS), and these can easily be extended to counterexamples to Closure. Consider, once more, the conjunction of L and H, as specified in (III). Suppose that an agent, G, is able to know $ L\wedge H $ but does not in fact come to know it—for instance, perhaps G was absent the day $ L\wedge H $ was taught in school. Since knowledge is factive, if G were to know $ L\wedge H $ , then $ L\wedge H $ would be true. Given the deterministic nature of L, any possible world in which $ L\wedge H $ holds must be a duplicate of the actual world. But, by assumption, G does not know $ L\wedge H $ in the actual world. It follows that G does not know $ L\wedge H $ in any world where $ L\wedge H $ holds. Therefore, it is impossible for G to know $ L\wedge H $ . Nonetheless, by hypothesis, G is able to know $ L\wedge H $ . Thus, G is able to do the impossible.Footnote ²¹ If Spencer is right, then G has a choice, in the relevant sense, about whether she does not know $ L\wedge H $ , even though it is necessary that she does not know $ L\wedge H $ . This presents a case in which someone has a choice about a necessary truth; if a case like this is possible, there are counterexamples to Closure.

Now, even if cases like this are bona fide counterexamples to Closure, we can only appreciate them by assuming from the outset that an agent inhabiting a deterministic world might possess unexercised—indeed, necessarily unexercised—abilities. But, this is precisely what the standard incompatibilist denies: according to the incompatibilist about determinism and free will, agents in a deterministic world are only able to do what they in fact do. For this reason, such cases are dialectically inert when evaluating incompatibilist arguments.Footnote ²²

A less recherché variation on this theme comes from Lampert & Merlussi (Reference Lampert and Merlussi2021a):

The case as described employs the standard rigidifying ‘actually’ operator, @.Footnote ²⁴ Though facts about how contingent matters stand are contingent, facts about how contingent matters actually stand are not. This idea makes intuitive sense if we reason our way through it in terms of possible worlds. Let ‘ $ \alpha $ ’ designate the actual world. Truths about what things are like at $ \alpha $ are noncontingent: facts about what is true may depend on which world is actual, but facts about what is true at a particular possible world do not depend on which world is actual. So, suppose $ @p $ is true. Then p is true at $ \alpha $ . But at any other world (accessible from $ \alpha $ ) it will be true that p is true at $ \alpha $ , and so it will be true that $ @p $ ; therefore, $ @p $ is necessarily true. Thus Nina actually raises her hand only if she necessarily actually raises her hand. Now suppose Nina has a choice about whether she raises her hand. Then, plausibly enough, she has a choice about whether she actually raises her hand. But then Nina has a choice about some necessary truth. So, once again, we have a counterexample to Closure.Footnote ²⁵

Lampert and Merlussi also produce further tempting counterexamples to Closure that do not rely on an ‘actually’ operator but instead employ rigid definite descriptions and other theoretical tools familiar to contemporary metaphysicians.Footnote ²⁶ The authors moreover suggest that having a choice about a proposition can be understood, broadly, with ground-theoretic notions: one has a choice about a proposition when its truth obtains in virtue of one’s actions, and the truth of a necessary proposition such as that Nina actually raises her hand could be said to obtain in virtue of Nina’s actions (see Lampert & Merlussi, Reference Lampert and Merlussi2021a: 452).Footnote ²⁷

These may serve as putative counterexamples to Closure. And, in line with the anti-incompatibilist insight of the Intensional Argument, they require understanding N in a hyperintensional manner.Footnote ²⁸

Now, we began by showing that any case where one has a choice about some necessary truth is a counterexample to Closure, and the cases so far discussed have all been cases of this sort. Lest the reader conclude that all potential counterexamples worth their salt are like this, we wish to note that similar cases arise concerning contingent a priori truths about which no one has a choice. These, in turn, entail contingent truths about which one does have a choice, thereby once again violating Closure.

Consider the following case.Footnote ²⁹ Take any actual truth p about which someone has a choice. If you like, let p be the truth that Nina raises her hand. Consider now the logical truth $ @p\to p $ .Footnote ³⁰ Does anyone have a choice about this? Though Nina has a choice about whether or not she raises her hand, plausibly no one, not even Nina, has a choice about whether Nina’s raising her hand is materially implied by Nina’s actually raising her hand; the latter is an apparently trivial, a priori truth. But it is also a contingent truth: it is true in the actual world and false in any world where p is false.Footnote ³¹ We therefore have two truths, $ @p\to p $ and p; nobody has a choice about the first, whereas someone does have a choice about the second. Since the first entails the second (assuming p is actually true), this is a counterexample to Closure.Footnote ³² Since this phenomenon is very general, we therefore have a whole family of further, candidate counterexamples to Closure.

The previous argument invokes a schema from standard modal logic that we call ACT:

$$ @p\to \square \left(\left(@p\to p\right)\to p\right) $$

In the presence of some plausible assumptions, ACT gives us a proper reductio of Closure, which by our lights further bolsters the case against Closure. Suppose first that there is some actual truth nobody has a choice about; let it be p. Suppose next that nobody has a choice about the logical truth $ @p\to p $ . Thus, we have $ @p $ , Np and $ N\left(@p\to p\right) $ . In the presence of all three, ACT and Closure generate a contradiction.Footnote ³³ Since ACT is provable in standard modal logic, one could therefore claim that the blame must fall squarely on Closure.

Now, incompatibilists may well object to $ N\left(@p\to p\right) $ on the grounds that one could have rendered p, and therefore $ @p\to p $ , false. After all, if p is actually true, then $ @p\to p $ must be false in any circumstance where p is false, and so one could falsify $ @p\to p $ in any circumstance where one could falsify p. But notwithstanding these considerations, denying $ N\left(@p\to p\right) $ remains somewhat puzzling. As we see it, this imposes a certain argumentative burden on anyone committed to the validity of Closure (and so also to N’s intensionality), for it shows that Closure is inconsistent with the plausible claim that no one has a choice about whether $ @p $ materially implies p, which once again seems plausible even where p is a truth someone has (or once had) a choice about.

The issues here are subtle and far-reaching. Earlier, we saw that an important motivation for Intensionality, and thus for Closure, is the view that propositions are individuated by necessary equivalence, which is entailed by the possible-worlds theory of propositions. But now consider the following argument, showing a strengthening of ACT:

$$ {\displaystyle \begin{array}{lll}1.& @p\to \square @p& \mathrm{R}\mathrm{IGIDITY}\\ {}2.& \square \left(@p\to \left(\left(@p\leftrightarrow p\right)\leftrightarrow p\right)\right)& \mathrm{Necessitated}\ \mathrm{Tautology}\\ {}3.& \square @p\to \square \left(\left(@p\leftrightarrow p\right)\leftrightarrow p\right)& 2,\ \mathrm{by}\ \mathrm{K}\hskip0.4em \mathrm{Axiom}\ \mathrm{and}\ \mathrm{Modus}\ \mathrm{Ponens}\\ {}4.& @p\to \square \left(\left(@p\leftrightarrow p\right)\leftrightarrow p\right)& 1\hskip0.45em \mathrm{and}\hskip0.45em 3,\hskip0.45em \mathrm{by}\hskip0.45em \mathrm{Hypothetical}\ \mathrm{Syllogism}\\ {}& & \end{array}} $$

Thus, assuming the possible-worlds theory of propositions, given the truth of $ @p $ it follows that $ @p\leftrightarrow p $ and p are the same proposition. So, for any actually true p, the assumption that no one has a choice about $ @p\leftrightarrow p $ leads directly to the claim that no one has a choice about p. In other words, given the general (and, to us, prima facie plausible) claim that nobody has a choice about logical truths, the possible-worlds theory of propositions overgenerates, yielding the fatalistic conclusion that nobody has a choice about anything. To avoid this conclusion, incompatibilist advocates of the possible-worlds theory of propositions can be expected, once again, to say that we do have a choice about some logical truths. On the other hand, those who insist that we do not have a choice about logical truths can be expected to reject the possible-worlds theory of propositions—or some other coarse-grained theory identifying necessarily equivalent propositions.

All of this brings out that the opposing verdicts here—endorsing or rejecting $ N\left(@p\leftrightarrow p\right) $ —cannot simply be opposed intuitive verdicts, but they have to interface with a theorist’s larger, more global commitments about our abilities vis-a-vis modality and logical truth. This general point also applies, mutatis mutandis, to the sort of modal semantics for N discussed in the previous section.Footnote ³⁴

Taking stock, it is important to make a final point about the foregoing material. What we have seen is a handful of putative counterexamples to Closure. One might be tempted to discount one or another of these examples on the grounds that they involve rarefied pieces of logical machinery or abstruse phenomena such as the contingent a priori. And these cases may simply appear bizzare, which could lead some theorists—with their theory-building leashed to common sense—to ignore them altogether. Both impulses should be resisted. The curious resources these cases appeal to (the contingent a priori, actuality operators, ridigified descriptions, etc.) are well-understood; in constructing these cases, nothing is assumed that cannot be motivated by “business as usual” in contemporary semantics and modal metaphysics. Moreover, these counterexamples and the associated problems for Closure are systematic: what they bring out are not simply parochial tensions between Closure and particular cases, but they instead illustrate issues with Closure that are more general.

This, finally, is exactly the sort of upshot one could have expected as a takeaway from the Intensional Argument, with its lesson that any counterexamples to Closure must involve hyperintensionality. If N is an intensional operator, the lesson of the Intensional Argument is that Closure is valid, and so are the four arguments discussed in Section 2. This, we think, really does give those arguments a new lease on life: defenders of such arguments can fruitfully turn their attention to arguing that N is an intensional operator. But there is a corresponding upshot as well for those of us who are skeptical of the arguments: to undercut these arguments, one can investigate systematic pathways to the conclusion that N is a hyperintensional operator. As the foregoing discussion suggests, some recent contributions to the literature have already been pointing in this direction. There are well-understood pathways for motivating hyperintensionality in metaphysics, and what we have seen is that these appear no less fruitful in the metaphysics of free will. Accordingly, it is not difficult to imagine what a rigorous case against Closure might look like.