## Abstract

In a recent paper, Pruss (Can J Philos 43:430–437, 2013) proves the validity of the rule beta-2 relative to Lewis’s semantics for counterfactuals, which is a significant step forward in the debate about the consequence argument. Yet, we believe there remain intuitive counter-examples to beta-2 formulated with the actuality operator and rigidified descriptions. We offer a novel and two-dimensional formulation of the Lewisian semantics for counterfactuals and prove the validity of a new transfer rule according to which a new version of the consequence argument can be formulated. This new transfer rule is immune to the counter-examples involving the actuality operator and rigidified descriptions. However, we show that counter-examples to this new rule can also be generated, demanding that the Lewisian semantics be generalized for higher dimensions where counter-examples can always be generated.

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## Notes

A statement here is simply a disambiguated declarative sentence.

Although, as we shall see, Pruss’s proof is syntactic in nature; but the rules used in the proof can all be shown to be sound according to a (modal) Lewisian semantics for counterfactuals.

A small observation: throughout his paper, Pruss uses the letters

*p*,*q*,*r*, etc., to stand in place of arbitrary statements, whereas we use letters from the end of the Greek alphabet.What we make, precisely, of the notion of closeness of worlds is orthogonal to the main issue here, so one may simply take Lewis’s (1973: 48) minimal conditions for comparative similarity, for example.

Pruss uses “taut con” (see the justification for lines 7 and 17) as a (convenient) rule for tautological consequences. We follow him in using this rule in the Fitch-style proof found in Sect. 4.1.

We should note that Kearns (2011) has independently argued that the actuality operator creates trouble for the incompatibilist in the context of the direct argument. Of course, we agree with him, and the present arguments can be seen as a generalization (and endorsement) of his case against the direct argument.

Note, additionally, that our cases are compatible with—although not dependent upon—the

*necessitation of grounding*, which is widely held (see Witmer et al. 2005; Rosen 2010; Trogdon 2013; Dasgupta 2014, and many others). This is the thesis that if the fact \([\varphi ]\) grounds a fact \([\psi ]\), then \([\psi ]\) is necessitated by \([\varphi ]\).See Yalcin (2015). In particular, Yalcin rejects the claim that there are two semantic roles performed by “actually”, namely, a logical and a rhetorical one.

Pretend that in this alternate world Paul was having troubles singing and playing the bass simultaneously.

We should note that our cases also provide compelling reasons for rejecting the consequence argument formulated with Pruss’s

*M*operator, which in turns corresponds to a more formal rendering of Huemer’s (2000) \(N_S\) operator:This is supposed to capture the idea that \(\varphi \) is true no matter what. Yet, even though the statements that Nina actually raises her hand and that the (actual) lead singer of The Beatles is John Lennon are true no matter what in the sense above, they were up to Nina and John. So, again, even though Pruss’s proof of the validity of gamma-2, that is,

$$\begin{aligned} {\textsc {gamma-2}},\, M\varphi ,\square (\varphi \rightarrow \psi )\vdash M\psi \end{aligned}$$still holds if the formal language is enriched with an actuality operator—or rigidified descriptions—we think the rule is intuitively invalid, for it validates inferences that we would otherwise reject.

A similar idea is defended by Weatherson (2001) and Wehmeier (2013) for indicative conditionals, where the change of actual world reflected an epistemic feature of indicatives, instead of the metaphysical feature of subjunctives. Yet, these semantics do not assume that there is a relation of similarity for indicatives, in which case an indicative conditional really quantifies over every possible world (in a model) taken as actual.

Another feature of this semantics is that it provides an account of why—regardless of their agential powers—agents cannot falsify some contingent a priori truths, i.e. contingent truths that are true at every possible world

*w*from the perspective of*w*as the actual world (in other words, contingent truths that hold at every diagonal point). We should note that Fusco (2019) explores this topic in her account of some of the puzzles of deontic logic by making use of a two-dimensional semantics involving a diagonal consequence relation, which relates to the present project given the strong connection between free will and moral responsibility.More generally, we should speak of the semantics for subjunctive conditionals developed by Stalnaker (1968), Lewis (1973), and Kratzer (1979). Even though there are differences amongst these, the general idea is the same. But since Pruss proves the validity of beta-2, in the Lewisian semantics, this is the one we consider here.

It is more proper to use “counterfactuals” for subjunctive conditionals with false antecedents, but we pass over this distinction here since it is not important in what follows.

Our use of the term “counteractual” is similar—though not quite the same, we think—to Yablo’s (2002) original use. Rather than considering a world

*w*as counterfactual, Yablo talks about considering a world*w*as counteractual, i.e. as a “hypothesis about what*this*world is like.” (2002: 449). This is done as a proposal to understand the notion of conceptual necessity and its differences with respect to metaphysical necessity: the former would consider worlds as counteractuals, whereas the latter as counterfactuals. Yet, we are not trying to draw a distinction between conceptual and metaphysical necessity here, for this would bring obvious questions about how we should think of the reference of names in counteractual worlds, and the use of language more broadly, all of which is beyond the scope of this paper. We should also note that Stefánsson (2018) develops a multidimensional semantics for counterfactuals to avoid what has been called*counterfactual skepticism*, i.e. (roughly) the view that all counterfactuals are false. In his framework, however, the word “counteractual” stands for something else: a possibly counteractual world is a truth-maker for a counterfactual statement (cf. Stefánsson (2018: 885)).One might object that the semantics for has an epistemic feature, since it quantifies over worlds taken as actual. But we do not think this conditional ought to be taken as epistemic. The notion of necessity formalized by \(\mathcal {F}@\), for example, even though it has epistemic applications, is not epistemic, as Davies and Humberstone are clear about the fact that they are not offering an epistemic logic. Moreover, we should point out that the fixedly operator itself appeared earlier in Crossley and Humberstone (1977), with motivations that are similar to ours. When Crossley and Humberstone axiomatized the modal logic of actuality they noted that some—although they were not committed with this claim—might think the axiom schema \(@\varphi \rightarrow \square @ \varphi \) counter-intuitive for the reason that it might seem contingent in some sense which world turned out to be actual. Because \(@\varphi \rightarrow \mathcal {F}@\varphi \) is not valid in the modal logic of fixedly actually, the necessity formalized by \(\mathcal {F}@\) might then be taken as the necessity one has in mind in denying that actual facts are all necessarily actual. The application to a priori knowledge, however, was suggested only later in Davies and Humberstone (1980) in connection to Evans’s work.

The language here is propositional and so it does not contain rigidified descriptions. The main philosophical points we will be making here carry over to the case of rigidified descriptions since any non-rigid description can be rigidified by using the actuality operator.

We drop the outer parentheses of formulas when convenient.

That is, \({\mathbf {F}}=\mathbf {CAL}+\square +@\). The reason why we use \({\mathbf {F}}\) is because there are models which we define below with a set of accessibility relations, one such relation for each formula, again, but over a different language. So \({\mathbf {F}}\) makes the notation of these models simpler.

For instance, see Davies and Humberstone (1980: 4–5).

See Lewis (1973: 14) for minimal conditions on systems of spheres.

This and the previous constraint are added to the (non-vacuist) semantics for subjunctive conditionals in Berto et al. (2018: 697).

That the

**T**axiom schema is not generally valid in the logic of fixedly actually is pointed out in Davies and Humberstone (1980: 6). The culprit here is the actuality operator, for without it**T**is generally valid in the semantics defined here.The idea that the a priori involves some sort of diagonal necessity is defended by many others. See Chalmers (2006) for an overview of a number of distinct formulations of this and other core notions of two-dimensional semantics.

This corresponds to axiom schema \((\mathcal {F}6)\) from Davies and Humberstone (1980: 4).

Or quadruples, if one adds an accessibility relation.

We have in mind here Kaplan’s (1989) distinction between context of utterance and context of evaluation. The former includes things like the speaker, time, place, and world in which a certain sentence is uttered. The latter provides the circumstances against which that sentence receives a truth-value.

Corresponding to a generalization of what Crossley and Humberstone (1977) call

*real-world validity*.By transporting this notion of logical truth to the two-dimensional framework, that is, truth at \(w*\) with respect to \(w*\) taken as actual in every model, then \(({\mathsf {A}}\varphi \leftrightarrow \varphi )\) turns out to be a logical truth that is also counteractually contingent—or deeply contingent, in Davies and Humberstone’s sense. Furthermore, if one takes \(\blacksquare \) as an operator for a priori knowledge, this means that there are logical truths that are not a priori knowable in the sense of being true along the diagonal. That is, besides being able to distinguish the concept of necessity from what is a priori knowable by means of Kripke’s (1980) examples, and necessity from logical truth by means of Zalta’s (1988) examples, the case at hand shows that it is possible to distinguish logical truth from what is a priori knowable, too. In Lampert (2018), this is taken as a possible motivation to call into question the usual interpretation of \(\blacksquare \) as an a priori operator. But, of course, one might simply accept that some logical truths are either a posteriori or unknowable, or even take this as a reductio against this notion of logical truth.

Proof: suppose that \(\mathfrak {M},w,v\vDash {\mathsf {A}}\varphi \), in which case \(\mathfrak {M},w*,w*\vDash \varphi \), by the truth conditions for \({\mathsf {A}}\)-formulas. Now assume that \(\mathfrak {M},w,v\nvDash \blacksquare {\mathsf {A}} \varphi \), to derive a contradiction. Then \(\mathfrak {M},w,v\vDash \lnot \blacksquare {\mathsf {A}}\varphi \), and so there is a possible world

*v*in \(\mathfrak {M}\) such that \(\mathfrak {M},v,v\vDash \lnot {\mathsf {A}}\varphi \). But then \(\mathfrak {M},v,v\nvDash {\mathsf {A}}\varphi \), whence \(\mathfrak {M},w*,w*\nvDash \varphi \), by the truth conditions for \({\mathsf {A}}\)-formulas, which in turn implies that \(\mathfrak {M},w*,w*\vDash \lnot \varphi \), contradicting the assumption. Note: because general implies diagonal and real-world validity, this argument establishes that \({\mathsf {A}}\varphi \rightarrow \blacksquare {\mathsf {A}}\varphi \) is valid according to all three notions of validity mentioned here.As an anonymous reviewer suggested, one could also add a set of worlds, say,

*D*, to the class of models, and an operator \([3]_D\) that diagonalizes only over the points in*D*. Then we can have sentences that are necessary in the usual sense even though they are*choice-contingent*, i.e. contingent relative to the points in*D*. In this case, there is a sense in which*D*could be said to represent the agent’s available (free) options even if they include necessary truths. Although we do not have the space needed to develop this suggestion further, see, for example, the (act-conditional) semantics for ‘can’ offered in Mandelkern et al. (2017), which involves some form of quantification over a contextual set of actions/options. Also, given a contextually supplied set of options for an agent, see Hedden (2012) for a theory of options underlying the subjective*ought*.

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## Acknowledgements

We would like to thank two anonymous referees for many helpful comments and suggestions that improved this paper. We are also grateful to Kai Wehmeier for valuable advice. Pedro Merlussi’s research is funded by São Paulo Research Foundation - FAPESP - (Grant No. 2017/20532-8), with additional support from the LATAM Free Will, Agency and Responsibility project (Grant No. 61255) funded by the John Templeton Foundation.

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Lampert, F., Merlussi, P. Counterfactuals, counteractuals, and free choice.
*Philos Stud* **178**, 445–469 (2021). https://doi.org/10.1007/s11098-020-01440-z

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DOI: https://doi.org/10.1007/s11098-020-01440-z

### Keywords

- Incompatibilism
- Choices
- Free will
- Consequence Argument
- Counterfactuals
- Actuality
- Conditionals