Abstract
Peter van Inwagen’s original formulation of the Consequence Argument employed an inference rule (rule beta) that was shown to be invalid given van Inwagen’s interpretation of the modal operators in the Consequence Argument (McKay and Johnson in Philos Top 24:113–122, 1996). In response, van Inwagen (Metaphysics. The big questions, Blackwell, Oxford, 2008a, Harv Rev Philos 22:16–30, 2015) recently suggested a revised interpretation of his modal operators. Following up on a debate between Blum (Dialectica 57:423–429, 2003) and Schnieder (Synthese 162:101–115, 2008), I analyze van Inwagen’s revised interpretation in terms of explanatory notions and I argue that van Inwagen faces a dilemma: he either has to admit that beta entails fatalism, or he has to admit that a new counterexample invalidates beta. Either way, it seems reasonable to reject beta and to conclude that the Consequence Argument fails. Further, I argue that Widerker’s (Analysis 47:37–41, 1987) well-known substitute for rule beta is faced with a similar dilemma and, therefore, is bound to fail as well. I conclude that, if the modal operators are interpreted in terms of explanatory notions, neither van Inwagen’s nor Widerker’s rule of inference turns out to be valid.
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Notes
Speak (2011, p. 128).
In what follows I often use ‘it is unavoidable that p’ instead of ‘N p’.
Van Inwagen (1983, p. 93).
It is not difficult to imagine, how a similar argument for the incompatibility of free will and divine foreknowledge might look like. For let ‘p’ be a correct and complete description of an arbitrary state of the world and let ‘\(\hbox {p}_{0}\)’ be a correct and complete description of God’s foreknowledge. Since a correct and complete description of God’s foreknowledge (at least given certain extra assumptions) obviously entails any correct and complete description of an arbitrary state of the world, a parallel argument would show that free will and divine foreknowledge are incompatible. As it turns out, van Inwagen defends the incompatibility of free will and divine foreknowledge but he presents his argument in a slightly different manner (2008b, pp. 217–218).
Accordingly, I reject one of the best arguments for the incompatibility of human freedom and divine foreknowledge (see fn. 4 above and fn. 33 below), since its validity obviously depends on the validity of either van Inwagen’s or Widerker’s rule of inference.
Van Inwagen (2015, p. 18).
McKay and Johnson (1996, p. 115).
It has proven useful to analyze van Inwagen’s argument in terms of counterfactual conditionals. See Lewis (1981), McKay and Johnson (1996), Kapitan (1996), Carlson (2000), Schnieder (2004). Schnieder (2004, pp. 412–417) has shown, however, that standard explications of van Inwagen’s argument lead to counterintuitive consequences. Accordingly, he suggests to analyze van Inwagen’s modal operators in terms of explanatory notions (2004, pp. 418–421, 2008, pp. 105–107). Following up on this, I take my explication of ‘R p’ to be equivalent to ‘there is someone who is (or has been) able to do something such that, if he did it, it would be a reason why p’. See the argument below for further distinctions.
Van Inwagen (2015, p. 19).
Van Inwagen (2008a, p. 452).
I follow Lewis (1973, p. 2) in using ‘\(\lozenge \rightarrow \)’ for the ‘might’ counterfactual, in using ‘\(\square \rightarrow \)’ for the ‘would’ counterfactual, and in assuming the following interdefinability: ‘(p \(\lozenge \rightarrow \) q) \(\leftrightarrow \)\(\sim \) (p \(\square \rightarrow \)\(\sim \) q)’.
Given this interdefinability, it is easy to see that the involved inference rule is valid. For suppose ‘p & q’ is true. If ‘p \(\lozenge \rightarrow \) q’ would be false, ‘p \(\square \rightarrow \)\(\sim \) q’ would be true. But then ‘q & \(\sim \) q’ would be true (because ‘p’ is true). Hence, the involved inference rule is valid.
My argument does not presuppose any particular account of this restrictive reading of ‘consequence’. One might, for instance, connect this restrictive reading of ‘consequence’ with Fine’s notion of exact verification assuming the following bridge principle: necessary, x is a full reason why p if and only if x contains an exact verifier for p (see Fine 2016, 2017). As it turns out, the notion of a full reason, thus connected, coincides with Fine’s notion of inexact verification (see Fine 2017, pp. 558, 565). Given this bridge principle, a couple of important premises of my argument would gain additional support from Fine’s truthmaker semantics and his logic for exact entailment (Fine 2016, 2017). See fn. 18, 23, 38, 40 and 41 below for further clarification.
Again, my argument does not presuppose any particular account of this permissive reading of ‘consequence’. One might connect this permissive reading of ‘consequence’ with Schnieder’s account of truthmaking assuming the following bridge principle (I slightly modify Schnieder’s proposal for the purposes of my argument): necessary, x is a partial reason why p if and only if x is a truthmaker of p, that is, if and only if p because x exists (see Schnieder 2006, pp. 30–31). As will become evident, given this bridge principle, a couple of premises and presuppositions of my argument would gain additional support from Schnieder’s logic for ‘because’ (Schnieder 2011). See fn. 19, 21, 29 and 44 below for further clarification. Of course, Schnieder himself establishes a connection between the debate on the Consequence Argument and the debate on truthmaking and grounding (2004), (2006), and (2008).
It turns out, in fact, that this principle gains additional support from Kit Fine’s truthmaker semantics (see Fine 2016, pp. 204–206, 2017, pp. 559–563) provided we adopt the account of a full reason that has been sketched above (see fn. 16). For suppose that it is realizable that p & q, that is, suppose that there is someone who is able to do something such that, if he did it, it might be a full reason why p & q. If so, he is able to do something such that, if he did it, it might contain an exact verifier for p & q. It is necessary, though, that every exact verifier for p & q contains an exact verifier for p and an exact verifier for q (see Fine 2016, pp. 204–205). It follows, given the transitivity of containment (see Fine 2016, p. 204), that he is able to do something such that, if he did it, it might contain an exact verifier for p and an exact verifier for q. It follows that it is realizable that p and it is realizable that q.
One might argue for this claim given the account of a partial reason that has been suggested above (see fn. 17) and given that ‘because’ does not distribute over conjunctions (see Schnieder 2011, pp. 454–455).
Schnieder (2008, pp. 111–112) employs a similar counterexample.
An anonymous referee pointed out that one might reject this counterexample invoking a distinction between partial reasons and reasons for a part. Given this distinction, one might argue that Ralph’s action is not a partial reason why p & q, but only a reason for a part of ‘p & q’. And clearly, if Ralph’s action is not a partial reason why p & q, my counterexample fails. I concede that, at least given some notions of a partial reason, Ralph’s action might not be a partial reason why p & q [even though Ralph’s action definitely is a partial reason why p & q, given the account of a partial reason that has been suggested above (see fn. 17)]. It is important to note, however, that a failure of my counterexample would by no means threaten the argument of my paper. Quite the contrary: if the principle of conjunction would not only be valid given 2A but also given 2B, then Blum’s argument would suffice to show that van Inwagen’s rule of inference is invalid (see the argument below for further clarification).
Blum (2003, p. 426). In his argument against the validity of \(\upbeta \) Blum presupposes an explication of van Inwagen’s old interpretation of ‘N’.
This additional premise seems plausible in its own right, but given the account of a full reason that has been suggested above (see fn. 16), one might argue for this additional premise as follows (I confine myself to a rough sketch of the argument): according to Kit Fine’s axioms and rules of his logic for exact entailment ‘\(\sim \) (\(\sim \) p \(\vee \) q)’ exactly entails ‘p & \(\sim \) q’ (see Fine 2016, pp. 200–202). Hence, if the material conditional is treated like a disjunction such that ‘\(\sim \) (\(\sim \) p \(\vee \) q)’ and ‘\(\sim \) (p \(\rightarrow \) q)’ are taken to be equivalent, it follows that ‘\(\sim \) (p \(\rightarrow \) q)’ exactly entails ‘p & \(\sim \) q’. It follows (from Fine’s semantics for the logic of exact entailment and some reasonable assumptions) that, necessary, every exact verifier for \(\sim \) (p \(\rightarrow \) q) is an exact verifier for p & \(\sim \) q (see Fine 2016, p. 211). If so, it is necessary that everything that contains an exact verifier for \(\sim \) (p \(\rightarrow \) q) also contains an exact verifier for p & \(\sim \) q. It follows (given the notion of a full reason that has been supposed), that, necessary, everything that is a full reason why \(\sim \) (p \(\rightarrow \) q) also is a full reason why p & \(\sim \) q. Alternatively, one might derive this premise from the clauses of Kit Fine’s truthmaker semantics and some reasonable extra assumptions (see Fine 2016, pp. 204–206, 2017, pp. 559–563). I thank an anonymous referee for pressing me on this issue.
Lewis’ logic for counterfactual conditionals (1973, pp. 21–22, 132) provides some justification for this inference, since, according to Lewis’ deductive system, the following inference rule is valid: ‘p \(\lozenge \rightarrow \) q, \(\square \) (q \(\rightarrow \) r) \(\vdash \) p \(\lozenge \rightarrow \) r’.
Blum (2003, pp. 425–426).
Schnieder (2008, pp. 110–112). Schnieder presupposes an explication of van Inwagen’s old interpretation of ‘N’. This doesn’t affect the argument of this paper.
Note that it nonetheless remains false that Ralph’s declaration might be a reason why the atom decays this evening.
The involved inference rule ‘p \(\lozenge \rightarrow \) q \(\vdash \) p \(\lozenge \rightarrow \) (p & q)’ is intuitively valid and is derivable from the axioms, rules and definitions of Lewis’ (1973, pp. 21–22, 132) logic for counterfactual conditionals.
This premise seems reasonable enough. But if one takes ‘p’ to be ‘it is not the case that the declaration exists’ (instead of ‘it is not the case that Ralph does the declaration’) and if one assumes the account of a partial reason that has been suggested above (see fn. 17), then one is able to derive this premise from the rules of Schnieder’s logic for ‘because’ (see Schnieder 2011, pp. 448–451). For suppose Ralph does the declaration and the atom decays this evening. It follows, first, that the declaration exists. If so, it is not the case that p \(\vee \) q because the declaration exists [this follows from Schnieder’s introduction rules for negative disjunctions and double negations together with Schnieder’s rule of transitivity (2011, pp. 448–451)]. Hence (given the notion of a partial reason that has been supposed), it is necessary that the declaration is a partial reason why \(\sim \) (p \(\vee \) q), if Ralph does the declaration and the atom decays this evening. Nonetheless, I prefer a different formulation of the argument, for I do not want to take a stand on the problem of negative existential statements (at least not in this paper).
The involved inference rule ‘p \(\lozenge \rightarrow \) q, \(\square \) (q \(\rightarrow \) r) \(\vdash \) p \(\lozenge \rightarrow \) r’ is intuitively valid and is derivable from the axioms, rules and definitions of Lewis’ logic for counterfactual conditionals (1973, pp. 21–22, 132).
To be sure: it has been argued that successful counterexamples to \(\upbeta \) must not presuppose indeterminism, since otherwise principle \(\upbeta \) could easily be substituted by principle \(\updelta \), whereby \(\updelta \) allows the inference from ‘N p’ and ‘N (p \(\rightarrow \) q)’ to ‘N q’ only if determinism is true (see McKay and Johnson 1996, p. 118; Crisp and Warfield 2000, pp. 179–180). I admit that such a substitution would preserve the validity of the Consequence Argument, but I strongly doubt that such a substitution would preserve the plausibility of the Consequence Argument as well. In any case, I do not think that my counterexample does presuppose indeterminism but since my only aim here is to argue that \(\upbeta \) is a modal fallacy, my counterexample would clearly do the job even if it presupposed indeterminism. Further, I regard it as a virtue of my counterexample that it clearly does not presuppose compatibilism.
See Widerker (1987, p. 41).
Needless to say, that the argument for the incompatibility of free will and divine foreknowledge could easily be modified as well (see Plantinga (1986, pp. 237–239), and Zagzebski (1991, pp. 7–8), for a similar formulation of the argument). Again, I contend that the argument is not valid, provided that the operators are interpreted in terms of explanatory notions.
I have not been able to derive agglomeration from Widerker’s rule \(\upbeta \)’. Blum (2000) provides a different derivation of agglomeration, but he needs some extra assumptions that are clearly false given an interpretation of the operators in terms of explanatory notions. Therefore, I have to come up with an argument against the validity of \(\upbeta \)’ that does not rely on the McKay-Johnson-counterexample.
Widerker himself seems to adopt van Inwagen’s old interpretation of ‘N’ (1987, pp. 37–38).
Blum (2003, p. 428).
For ‘p’ would be realizable, if ‘p & \(\sim \) q’ would be realizable (given that the principle of conjunction holds for 1A and 2A).
Note that this is true, even if there is someone who is able to do something such that, if he did it, it would or might be a full reason why \(\sim \) q (in fact, Ralph is able to refrain from proposing marriage to Heidi).
This assumption is derivable from the principle of distributivity (see fn. 38) together with Lewis’ definitions, axioms and rules of his logic for counterfactual conditionals (see Lewis 1973, pp. 21–22, 132). The derivation of this assumption becomes even more straightforward, once Stalnaker’s instead of Lewis’ axioms and rules are assumed (see Stalnaker 1986, pp. 105–106). For the purposes of my argument, I nonetheless prefer to stick to Lewis’ deductive system.
This additional premise seems plausible in its own right, but one might employ a similar strategy as above in order to argue for this additional premise (see fn. 23).
Lewis’ logic for counterfactual conditionals provides some justification for this inference (1973, pp. 21–22, 132). See fn. 24 for a similar case.
The involved inference rule ‘p \(\square \rightarrow \) q \(\vdash \) p \(\square \rightarrow \) (p & q)’ is intuitively valid and is derivable from the axioms, rules and definitions of Lewis’ logic for counterfactual conditionals (1973, pp. 21–22, 132).
This additional premise seems reasonable enough, but if one takes ‘p’ to be ‘it is not the case that the acceptance exists’ (instead of ‘it is not the case that Heidi does the acceptance’), then one might employ a similar strategy as above and argue for this additional premise (see fn. 29).
The involved inference rule ‘p \(\square \rightarrow \) q, \(\square \) (q \(\rightarrow \) r) \(\vdash \) p \(\square \rightarrow \) r’ is intuitively valid and is derivable from the axioms, rules and definitions of Lewis’ logic for counterfactual conditionals (1973, pp. 21–22, 132).
Of course, other substitutes for rule \(\upbeta \) (e. g. O’Connor (1993)), other interpretations of the modal operators (e. g. Vihvelin (1988); McKay and Johnson (1996); Kapitan (1996); Carlson (2000)) and other versions of the arguments have been suggested (e. g. Pike (1965); Van Inwagen (1983); Fischer (1994)). I have to leave it open here whether these alternative proposals are successful or whether—under close scrutiny—they prove fallacious, too.
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Acknowledgements
I would like to thank Thomas Buchheim, Christopher Erhard, Benjamin Schnieder and two anonymous referees of this journal for helpful comments on an earlier version of this paper and I would like to thank Amit Kravitz for helpful discussion.
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Hausmann, M. The consequence argument ungrounded. Synthese 195, 4931–4950 (2018). https://doi.org/10.1007/s11229-017-1436-6
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DOI: https://doi.org/10.1007/s11229-017-1436-6