How Not to Argue for Incompatibilism

MICHAEL KREMER HOW NOT TO ARGUE FOR INCOMPATIBILISM ABSTRACT. Ted A. Warfield has recently employed modal logic to argue that compatibilism in the free-will/determinism debate entails the rejection of intuitively valid inferences. I show that Warfield’s argument fails. A parallel argument leads to the false conclusion that the mere possibility of determinism, together with the necessary existence of any contingent propositions, entails the rejection of intuitively valid inferences. The error in both arguments involves a crucial equivocation, which can be revealed by replacing modal operators with explicit quantifiers over possible worlds. I conclude that the modal-logical apparatus used by Warfield obscures rather than clarifies, and distracts from the real philosophical issues involved in the metaphysical debate. These issues cannot be settled by logic alone. Modal logic and the associated apparatus of possible worlds semantics have become part of the stock-in-trade of metaphysicians in the past forty years. However, as often as not, one has the feeling that such use of heavyduty technology produces more obscurity than it does clarity. In this paper I examine a case in point: An argument for incompatibilism in the freewill/determinism debate put forward by Ted A. Warfield, using the devices of modal logic. (Warfield 2000).1 I show that the modal-logical apparatus used by Warfield leads to equivocations which render his argument fallacious. Moreover, Warfield’s symbol-chopping distracts from the real philosophical issues involved in the metaphysical debate.2 Logic alone cannot settle such issues; hard metaphysical work is what is needed here. I begin in Section 1 with a brief introduction to the contemporary freewill debate. I then summarize Warfield’s argument in Section 2. In Section 3, I show that Warfield’s style of reasoning can be extended analogically to yield clearly false conclusions. In Section 4, I isolate the error in the argument of Section 3, and in Section 5, I point out the analogous equivocation in Warfield’s argument. I conclude in Section 6 by drawing some morals concerning the use of logic in metaphysics. Finally, in an appendix, I discuss a recent, less far-reaching, critique of Warfield’s argument. 1. INTRODUCTION : THE FREE - WILL DEBATE3 The contemporary debate over free will (like much older forms of the debate) focuses primarily on the question of the relationship between two Erkenntnis 60: 1–26, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. 2 MICHAEL KREMER propositions. The first (F) is the claim that human beings possess free will, in a sense that is commonly held to be necessary for attributions of moral responsibility. The second (D) is the thesis of determinism, that all events (at any rate all events that have a past) are necessitated by antecedent “determining conditions”. Since each determining condition is itself necessitated by further antecedent determining conditions, determinism is commonly taken to be equivalent to the claim that all present and future events are fixed by a complete description of the universe at any past time, in conjunction with the laws of nature. Of course, given that modern physics appears to be indeterministic at the quantum level, there might seem to be no interesting philosophical issue left about determinism. However, philosophers have continued to debate the question of free will and determinism, on the grounds that the indeterminism of quantum mechanics does not carry over to the macroscopic level at which we can speak of human beings and their behavior (Kane 2002, p. 5). Three positions have classically been taken with regard to our two propositions. Libertarians assert (F) but deny (D) while hard determinists assert (D) but deny (F). Both libertarians and hard determinists agree that free will requires that the agent choose between genuine alternative possibilities, and both see this freedom as incompatible with determinism, since according to determinism all events, including human acts of choice, are necessitated in a way that rules out genuine alternative possibilities. Thus both libertarians and hard determinists are incompatibilists concerning freedom and determinism. In contrast, compatibilists assert both (F) and (D). Compatibilists try to respond to the claim that (F) and (D) are inconsistent by providing an analysis of freedom of the will which does not depend on the existence of genuine alternative possibilities. The simplest forms of compatibilism, which go back at least as far as Hobbes, employ a conditional analysis of freedom of action along something like the following lines: S is free to do A iff (if S were to will to do A, then S would do A). Such an analysis simply leaves to one side the question of whether S’s willing to do A, or not to do A, is itself determined. However, compatibilists hold that all events, including the event of S’s willing to do A, are determined; and this can seem to imply that the conditional analysis of freedom of action does not make room for genuine freedom of the will, as might be supposed to be necessary for responsibility. Classical compatibilists such as Hobbes accepted this conclusion, arguing that the very idea of “freedom of the will” is confused, since the question of whether HOW NOT TO ARGUE FOR INCOMPATIBILISM 3 an event is a free act turns on the comparison of what actually happens with what is willed to happen – freedom presupposes the will, and so cannot be properly applied to it. However, contemporary compatibilists, influenced by (Frankfurt 1971), have instead tried to argue that we can call acts of the will “free”, even if they are determined, so long as they are determined in the right way. In one form of this style of argument (fairly close to Frankfurt’s original essay), the above conditional analysis is simply directly applied to acts of the will: S is free to will to do A iff (if S were to will to will to do A, then S would will to do A). Here, freedom of the will is seen to turn on the relation of higher-order to lower-order acts of the will. In another form of this view, developed by Susan Wolf, questions of freedom of the will are made to depend further on the sanity of the agent (Wolf 1987, 1990). Recent discussions of compatibilism have focused on an argument for incompatibilism due to Peter Van Inwagen, dubbed the “consequence argument” in the literature. (Van Inwagen 1983, pp. 55–105). Van Inwagen states the argument informally as follows (Van Inwagen 1983, p. 56): If determinism is true, then our acts are the consequences of the laws of nature and events in the remote past. But it is not up to us what went on before we were born; and neither is it up to us what the laws of nature are. Therefore the consequences of these things (including our own acts) are not up to us. In order to present a formal version of this argument (Van Inwagen 1983, pp. 93–95), Van Inwagen adopts the notation “Np” to stand for “p, and no one has, or ever had, any choice about whether p”. He states two principles governing this operator: (α)
p ⊢ Np (β) N(p ⊃ q), Np ⊢ Nq. Using these principles he then argues that “if determinism is true, then no one ever has any choice about anything”, which he takes to be tantamount to incompatibilism. In his argument, he uses “P0 ” to abbreviate some sentence about the total state of the world at some moment in the distant past (say, before there were any human agents), he uses “L” as an abbreviation for the conjunction of all the laws of nature, and he uses “P” as an abbreviation for any true sentence whatsoever. His argument proceeds as follows: (1.)
((P0 & L) ⊃ P) (consequence of determinism) 4 MICHAEL KREMER (2.)
(P0 ⊃ (L ⊃ P)) ((1), elementary modal logic) (3.) N(P0 ⊃ (L ⊃ P)) ((2), (α)) (4.) NP0 (5.) N(L ⊃ P) (6.) NL (since L is the conjunction of the laws of nature) (7.) NP ((5), (6), (β)) (since P0 is about the distant past) ((3), (4), (β)) There have been many responses to this argument, including criticisms of the crucial “transfer principle” (β), and arguments that the conclusion (7), even when generalized, need not entail incompatibilism.4 Warfield, however, introduces a novel objection: He claims that the argument does not establish what it is supposed to, namely the incompatibility of free will and determinism. (Warfield 2000, p. 171). His point is that the argument makes use of contingently true premises, in particular “NL” and “NP0 ”. Because of this, Warfield claims, the argument at best establishes the material conditional: (8.) D ⊃∼F whereas to establish incompatibilism Van Inwagen would need to show the necessitation of this: (9.)
(D ⊃∼F). Warfield’s aim is to provide an argument for incompatibilism that is not open to this objection.5 2. WARFIELD ’ S ARGUMENT6 Warfield argues that compatibilism entails the rejection of the following intuitive principle of reasoning: (∗ ) P is true and there’s nothing anyone is free to do in the circumstances that even might result in ∼P; therefore, P is true and there’s nothing anyone is free to do in the circumstances that would definitely result in ∼P. HOW NOT TO ARGUE FOR INCOMPATIBILISM 5 He argues in two stages. In the first stage, he argues that compatibilists are committed to denying the following principle:7 (10.)
∀x∀s(Fsx ⊃ ♦(H & x)) where “Fsx” abbreviates “subject s is free to make it the case that x”, and “H” abbreviates “the conjunction of the complete state of the world in the distant past with the laws of nature”.8 He tries to show that compatibilists are committed to denying (10) by arguing that (10) entails the incompatibilist CONCLUSION:
∀x∀s((D & x) ⊃∼Fs∼x) or equivalently
(D ⊃ ∀x(x ⊃ ∀s∼Fs∼x)).9 Here “D” represents the thesis of determinism, and the CONCLUSION asserts that determinism strictly implies that if a proposition is true, no one is free to make it not true. Warfield’s argument that (10) implies the CONCLUSION makes use of the following principle, which Warfield thinks is an analytic consequence of the concept of determinism: (11.)
∀x(D ⊃ (x ⊃
(H ⊃ x))). From (10) and (11) Warfield derives the CONCLUSION in a few short, apparently simple steps of modal reasoning (though we will return to this below). The second stage of Warfield’s reasoning is to argue that the incompatibilist’s denial of (10) brings with it the denial of the validity of inference (∗ ) above. In order to argue for this, Warfield introduces the following analyses of the premise and conclusion of (∗ ): (12.) P is true and there’s nothing anyone is free to do in the circumstances that even might result in ∼P is to be analyzed as (13.) P & ∀s∀x(Fsx ⊃
((x & H) ⊃ P)) and (14.) P is true and there’s nothing anyone is free to do in the circumstances that would definitely result in ∼P 6 MICHAEL KREMER is to be analyzed as (15.) P & ∼ ∃s∃x(Fsx &
((x & H) ⊃∼P)) or equivalently (16.) P & ∀s∀x∼(Fsx &
((x & H) ⊃∼P)).10 Warfield now argues that if (10) is false, there will be possible worlds in which an instance of (13) is true and the corresponding instance of (16) is false, so that (∗ ) is invalid. He assumes that (10) is false. Then there is a world w, a subject b, and a proposition a such that Fba & ∼ ♦(H & a) is true in w. Instantiating with ∼a for P in (13) and (16), we get (17.) ∼a & ∀s∀x(Fsx ⊃
((x & H) ⊃∼a) and (18.) ∼a & ∀s∀x∼(Fsx &
((x & H) ⊃∼∼a). Now Warfield argues that (17) is true in w and (18) is false in w. For (17), Warfield observes “because a is incompatible with the past and laws of w, it follows that ∼a is a truth in w. Indeed it follows that ∼a is a strict consequence of the past and laws of w”. (Warfield 2000, p. 175).11 Hence the first conjunct, ∼a, of (17) is true in w, and further
(H ⊃∼a) is true in w, so the second conjunct of (17) is true in w as well. On the other hand, Fba is true in w, and of course
((a & H) ⊃∼∼a) is true in w, so (Fba &
((a & H) ⊃∼∼a)) is true in w, so the instance ∼(Fba &
((a & H) ⊃∼∼a)) of (18) is false in w. Hence (18) is false in w. Thus Warfield argues, anyone who accepts (10) – as the compatibilist must to avoid the CONCLUSION – is committed to denying the validity of (∗ ). Warfield concludes: “believe it if you can and if you are a compatibilist you have to believe it. . . I for one choose incompatibilism” (Warfield 2000, p. 177). 3. A GAUNILO - STYLE OBJECTION I will now mount an objection to Warfield’s argument like the well-known objection presented to Anselm’s ontological argument by Gaunilo: If this form of argument worked, it would lead to ridiculous conclusions.12 In Gaunilo’s case, the absurd conclusion was that there exists a most perfect HOW NOT TO ARGUE FOR INCOMPATIBILISM 7 K, for any kind K, for example a most perfect island. In the case of my objection to Warfield’s argument, the absurd conclusion will be that the mere possibility that determinism is true, together with the necessary existence of contingent propositions (propositions which are neither necessary nor impossible), implies the invalidity of the patently valid form of inference (∗∗ ): (∗∗ ) P is true and there’s nothing which is logically possible that even might result in ∼P; therefore, P is true and there’s nothing which is logically possible that would definitely result in ∼P. Therefore, if Warfield’s style of reasoning were correct, we could conclude that either determinism is necessarily false or it is possible that no proposition is contingent (necessity collapses into truth). In other words, the mere possibility of determinism implies the possibility of the collapse of modality. This conclusion is too strong, however. The possibility of determinism cannot be equated with the possibility that no proposition is contingent. Determinism is entirely compatible with the entire sequence of events in the universe being contingent – many sequences of events, each determined by preceding ones, may conform to the same natural laws. Even the necessary truth of determinism is compatible with the necessary existence of contingent propositions (each set of natural laws might correspond to a plurality of worlds with different histories conforming to those laws). Hence, there is something wrong with Warfield’s style of reasoning.13 In Sections 4 and 5, I locate the error in his reasoning, showing that it involves a non-standard use of propositional letters within the scope of modal operators, a use which allows a crucial ambiguity to creep into his argument. Use of explicit quantifiers over possible worlds then permits us to distinguish two readings of the argument. On one reading Warfield’s argument is valid but, compatibilists will claim, its conclusion is not equivalent to the invalidity of inference (∗ ). On the other reading, although compatibilists may concede that its conclusion is equivalent to the invalidity of inference (∗ ), Warfield’s argument is itself invalid. To construct a parallel to Warfield’s argument, I will first observe by (11) that anyone who accepts determinism as even possibly true is committed to accepting that (19.) ∀x(x ⊃
(H ⊃ x)) is true in some world w. Following Warfield’s lead, let’s represent the premise (20.) P is true and there’s nothing which is logically possible that even might result in ∼P 8 MICHAEL KREMER of (∗∗ ) by (21.) P & ∀x(♦x ⊃
((x & H) ⊃ P)) and the conclusion (22.) P is true and there’s nothing which is logically possible that would definitely result in ∼P of (∗∗ ) by (23.) P & ∼ ∃x(♦x &
((x & H) ⊃∼P)) or equivalently (24.) P & ∀x∼(♦x &
((x & H) ⊃∼P)). Now assume that determinism is possibly true, so that in some world w, (19) is true; and assume that necessarily, some propositions are contingent. Let P be any proposition which is true in w, but not necessarily true in w.14 By (19),
(H ⊃ P) is true in w. Letting Q be any proposition whatsoever, we then have that
((Q & H) ⊃ P), and so ♦Q ⊃
((Q & H) ⊃ P). As Q was arbitrary, (21) is true. On the other hand,
((∼P & H) ⊃∼P) is clearly true in w. Also ♦ ∼P is true in w, since P is not necessarily true in w, so ∼(♦ ∼P &
((∼P & H) ⊃∼P)) is false in w, so ∀x∼(♦x &
((x & H) ⊃∼P)) is false in w, so (24) is false in w. Hence anyone who asserts that determinism is even possibly true, and admits that necessarily, some propositions are contingent, is committed to denying the obviously correct inference from (20/21) to (22/24). Therefore, either it is possible that no proposition is contingent, or determinism is necessarily false. Yet surely something has gone wrong here, and seeing what has gone wrong will enable us to see what has gone wrong in the more complex instance of Warfield’s argument as well. 4. ANALYSIS OF THE ERROR IN SECTION 3 Where has the argument in Section 3 gone wrong? We should look first at principle (19), and principle (11) from which it derived: (11.)
∀x(D ⊃ (x ⊃
(H ⊃ x))) (19.) ∀x(x ⊃
(H ⊃ x)). HOW NOT TO ARGUE FOR INCOMPATIBILISM 9 Warfield explains (11) as follows: “Determinism is the thesis that the conjunction of the past and laws implies all truths”. (Warfield 2000, p. 173). Now (11) plausibly captures this only if “H” does not univocally represent one proposition, “the conjunction of the past and laws” of the actual world. Rather, “H” must represent “the conjunction of the past and laws” of whatever world we are considering under the first necessity operator “
” of (11). We would not want to claim that in any deterministic world, every true proposition is strictly implied by the laws and past history of the actual world. What we want to claim is that in any deterministic world, every true proposition is strictly implied by the laws and past history of that world. This distinction cannot be made explicit, however, within the modal-logical apparatus used by Warfield. In fact, Warfield must implicitly contravene the usual conventions governing this apparatus, according to which propositional letters such as “H” univocally represent single propositions both within the scope of modal operators, and outside that scope. The result is that (11) as used by Warfield is ambiguous; and so are other key propositions in his argument, in a way that vitiates that argument, as well as my Gaunilo-style counterargument.15 We can bring this difficulty out by moving from the modal operators “
” and “♦” to explicit quantifications over worlds.16 Let us adopt the following notation:17 xw: proposition x holds in world w Hw : the conjunction of the laws and past history of w @: the actual world For example: D@: determinism holds in the actual world Hw w: the conjunction of the laws and past of w holds in w Hw w∗ : the conjunction of the laws and past of w holds in w∗ Now (11) is ambiguous between: (11a.) ∀w∀x(Dw ⊃ (xw ⊃ ∀w∗ (Hw w∗ ⊃ xw∗ ))) (11b.) ∀w∀x(Dw ⊃ (xw ⊃ ∀w∗ (Hw∗ w∗ ⊃ xw∗ ))) and (11c.) ∀w∀x(Dw ⊃ (xw ⊃ ∀w∗ (H@ w∗ ⊃ xw∗ ))).18 10 MICHAEL KREMER Of these, we can note that it is clearly (11a) which is intended, rather than (11b) or (11c) – although (11c) is what one would have expected (11) to represent, on the basis of the standard convention of the univocity of propositional letters in modal contexts. (11c) has the unacceptable consequence that in any deterministic world, all true propositions are strictly implied by the laws and past history of our world. On the other hand, since the laws and past history of w∗ automatically hold in w∗ , the condition Hw∗ w∗ in (11b) is vacuous and can be dropped. Therefore (11b) amounts to (11d.) ∀w∀x(Dw ⊃ (xw ⊃ ∀w∗ (xw∗ ))) which asserts that in any deterministic world, all truths are logically necessary truths. But this is certainly not the case – the past history, initial conditions, or even the laws themselves might be logically contingent, for example. If we now turn this disambiguating apparatus on the argument of Section 3 above we can locate the difficulty in that argument. From (11), together with the thesis that determinism is possibly true, we concluded that (19.) ∀x(x ⊃
(H ⊃ x)) is true in some world w. We now see that this must be read as: (19a.) ∀x(xw ⊃ ∀w∗ (Hw w∗ ⊃ xw∗ )) rather than (19b.) ∀x(xw ⊃ ∀w∗ (Hw∗ w∗ ⊃ xw∗ )). Then we took (21) as a representation of (20) and (24) as a representation of (22): (20.) P is true and there’s nothing which is logically possible that even might result in ∼P (21.) P & ∀x(♦x ⊃
((x & H) ⊃ P)) (22.) P is true and there’s nothing that which is logically possible that would definitely result in ∼P (24.) P & ∀x∼(♦x &
((x & H) ⊃∼P)). HOW NOT TO ARGUE FOR INCOMPATIBILISM 11 But again, (21) and (24) are ambiguous. In particular, if we are evaluating (21) and (24) at some world w, the ambiguity to consider is that between: (21a.) Pw & ∀x(∃w∗ (xw∗ ) ⊃ ∀w∗ ((xw∗ & Hw w∗ ) ⊃ Pw∗ )) and (24a.) Pw & ∀x∼(∃w∗ (xw∗ ) & ∀w∗ ((xw∗ & Hw w∗ ) ⊃∼Pw∗ )) on the one hand, and (21b.) Pw & ∀x(∃w∗ (xw∗ ) ⊃ ∀w∗ ((xw∗ & Hw∗ w∗ ) ⊃ Pw∗ )) and (24b.) Pw & ∀x∼(∃w∗ (xw∗ ) & ∀w∗ ((xw∗ & Hw∗ w∗ ) ⊃∼Pw∗ )) on the other.19 If we read (21) and (24) as (21a) and (24a) respectively, then the argument of Section 3 goes through to show that if determinism is possibly true and necessarily, some propositions are contingent, then the inference from (21a) to (24a) is invalid.20 However, if we read (21) and (24) as (21b) and (24b), there is no parallel argument to establish that the inference from (21b) to (24b) is invalid, unless we read (19) as (19b), which we saw above to be implausible. The question now is which of (21a)–(24a) and (21b)–(24b) is the appropriate representation of the premiss (20) and the conclusion (22) of (∗∗ ). If, as I shall argue, the answer is (21b)–(24b), then we have located the source of the difficulty with the argument of Section 3 – the argument is guilty of a subtle equivocation between (21a)–(24a) and (21b)–(24b), made possible by the shifting use of “H” in modal contexts, as employed by Warfield. That is, the argument aims to show that under certain assumptions, the inference from (20) to (22) is invalid; it aims to show this by showing that the inference from (21) to (24), taken as representing (20) and (22) is invalid; but (21) and (24) are ambiguous between a reading ((21a)–(24a)) on which the inference is shown to be invalid, but does not represent that from (20) to (22), and a reading ((21b)–(24b)) which does represent the inference from (20) to (22), but on which the inference is not shown to be invalid. In order to complete this diagnosis of the argument of Section 3, it is only necessary to argue that the appropriate representation of (20) and (22) is more along the lines of (21b)–(24b), rather than (21a)–(24a). To this end, let us consider the question of how to properly represent: (25.) Q might result in P, 12 MICHAEL KREMER and (26.) Q would definitely result in P. The issue is essentially which of the following more appropriately represents (25) and (26)’s being true in a world w: (27a.) ∼ ∀w∗ ((Qw∗ & Hw w∗ ) ⊃∼Pw∗ )) and (28a.) ∀w∗ ((Qw∗ & Hw w∗ ) ⊃ Pw∗ )) on the one hand, and (27b.) ∼ ∀w∗ ((Qw∗ & Hw∗ w∗ ) ⊃∼Pw∗ )) and (28b.) ∀w∗ ((Qw∗ & Hw∗ w∗ ) ⊃ Pw∗ )). If the appropriate representation of (25) and (26) is (27a) and (28a), then the appropriate representation of (20) and (22) is (21a) and (24a); similarly if the appropriate representation of (25) and (26) is (27b) and (28b), then the appropriate representation of (20) and (22) is (21b) and (24b). If we take (27a) and (28a) as the appropriate representations of (25) and (26), however, then whether w is a deterministic world or not, we will get the following result: If Q is any proposition which is not compatible with the past and laws of w, even if Q is logically possible, then: (29.) ∀x∼(Q might result in x), and (30.) ∀x(Q would definitely result in x) are both true in w. These results hold, for instance, if Q states initial conditions compatible with the laws of w, but distinct from those which actually obtained in w. In particular, even if P describes a state of the universe which is entailed by Q and the laws of w, and R describes a state of the universe which is incompatible with Q and the laws of w, we would still have: (31.) ∼(Q might result in P), HOW NOT TO ARGUE FOR INCOMPATIBILISM 13 and (32.) Q would definitely result in R; indeed we would have: (33.) ∼(Q might result in Q), and (34.) Q would definitely result in ∼Q. This is counter-intuitive however. The difficulty is that when considering what would or might result from a logical possibility Q which is itself incompatible with the laws and past history of w, it is wrong-headed to restrict consideration only to worlds with those laws and that past history. For this is to restrict consideration to worlds in which Q does not, and cannot, obtain. But if we ask what might have occurred or would have occurred if, counterfactually, Q had obtained, we are directed to consider worlds in which Q does obtain, and this is to consider worlds with different laws-and-past-history from w. This is what is allowed for by taking (27b) and (28b) as appropriate representations of (25) and (26). Since by definition the laws and past history of any world w∗ hold in w∗ , in fact, (27b) and (28b) are tantamount to: (27c.) ∼ ∀w∗ (Qw∗ ⊃∼Pw∗ ) and (28c.) ∀w∗ (Qw∗ ⊃ Pw∗ ) or in other “words” (with no ambiguity!) (27d.) ∼
(Q ⊃∼P) and (28d.)
(Q ⊃ P). One might object here that this analysis of (25) and (26) as, in effect, stating mere compatibility of Q and P, and strict implication of P by Q, is inadequate. In fact, I think this is correct, and I would argue that 14 MICHAEL KREMER the most appropriate representation of (25) and (26) must actually use counterfactuals. Thus (25) is best represented by (35) and (26) by (36): (35.) Q♦→P (36.) Q
→ P. An analysis closely related to that provided by (27b) and (28b) is obtained if we follow David Lewis in holding (roughly) that (36) is true in a world w, just in case P is true in all of the Q-worlds (worlds where Q is true) most similar to w, and (35) is true in a world w just in case P is true in some of the Q-worlds most similar to w (or equivalently, (35) is true in w just in case ∼(Q
→∼P) is true in w). This analysis, while different from that provided by either (27a) and (28a), or by (27b) and (28b), is like the latter in that it allows us to consider worlds with different laws-and-pasthistory when considering what might, or would, result from a proposition Q which is incompatible with the actual laws and past history. Because of this, a Lewis-style counterfactual analysis of (25) and (26) will block the argument of Section 3 as well. Such an analysis has the additional advantage of allowing us to respond not only to the intuitions involved in the choice of (27b) and (28b) as analyses of (25) and (26), but also those involved in the choice of (27a) and (28a). For we can insist that worlds which share the past-history-andlaws of w are to count as more similar to w than worlds which do not share w’s past- history-and-laws, in evaluating (35) and (36). The effect of this will be that if Q is compatible with the laws and past history of w, (35) and (36) will be evaluated by considering worlds which share those laws and past history – just as (27a) and (28a) are. Thus, if Q is “historically possible” in w, to determine whether P might, or would, result from Q, we need to consider worlds which share w’s past history and laws, in which Q is true, as the analysis through (27a) and (28a) would have it. But if Q is not historically possible in w, but is logically possible, then we need to consider worlds in which Q is true, and which therefore do not share w’s past history and laws, as the analysis through (27b) and (28b) would have it. 5. ANALYSIS OF THE ERROR IN WARFIELD ’ S ARGUMENT We are now in a position to appreciate the error in Warfield’s argument for incompatibilism: It is precisely the same error as that involved in the argument of Section 3, which we have just diagnosed. As we saw, Warfield HOW NOT TO ARGUE FOR INCOMPATIBILISM 15 argues in two stages. In the first stage, he argues from (11) and (10) to the incompatibilist CONCLUSION, with the result that the compatibilist must deny (10): (10.)
∀x∀s(Fsx ⊃ ♦(H & x)) (11.)
∀x(D ⊃ (x ⊃
(H ⊃ x))) ∴CONCLUSION:
∀x∀s((D & x) ⊃∼Fs∼x) We have already seen that in this argument (11) is ambiguous, and must be interpreted as: (11a.) ∀w∀x(Dw ⊃ (xw ⊃ ∀w∗ (Hw w∗ ⊃ xw∗ ))). Similarly, (10) is ambiguous. If, following the conventions of Section 4, we write “Fsxw” for “s is free in w to make it the case that x”, the important ambiguity is that between: (10a.) ∀w ∀x∀s(Fsxw ⊃ ∃w∗ (Hw w∗ & xw∗ )) and (10b.) ∀w ∀x∀s(Fsxw ⊃ ∃w∗ (Hw∗ w∗ & xw∗ )). In order for Warfield’s first-stage argument to go through, given that (11) must be read as (11a), (10) must also be read as (10a) rather than (10b). What the compatibilist must deny, then, is this: If a subject is free to make some proposition be the case, that proposition must be compatible with the laws and past history of the world in which the subject has that freedom. The compatibilist must deny this because the compatibilist admits that subjects can be free to perform actions which are ruled out by the laws and past history. Warfield’s second-stage argument was then that denying (10) results in denying the validity of the inference (∗ ): (∗ ) P is true and there’s nothing anyone is free to do in the circumstances that even might result in ∼P; therefore, P is true and there’s nothing anyone is free to do in the circumstances that would definitely result in ∼P. This argument relied on the paraphrase of the premiss (12) and conclusion (14) of (∗ ) as: (13.) P & ∀s∀x(Fsx ⊃
((x & H) ⊃ P)) 16 MICHAEL KREMER and (16.) P & ∀s∀x∼(Fsx &
((x & H) ⊃∼P)). But here, of course, the same ambiguities as we saw in Section 4 rear their heads. For (13) and (16) are ambiguous between (13a.) Pw & ∀s∀x(Fsxw ⊃ ∀w∗ ((xw∗ & Hw w∗ ) ⊃ Pw∗ )) and (16a.) Pw & ∀s∀x∼(Fsxw & ∀w∗ ((xw∗ & Hw w∗ ) ⊃∼Pw∗ )) on the one hand, and (13b.) Pw & ∀s∀x(Fsxw ⊃ ∀w∗ ((xw∗ & Hw∗ w∗ ) ⊃ Pw∗ )) and (16b.) Pw & ∀s∀x∼(Fsxw & ∀w∗ ((xw∗ & Hw∗ w∗ ) ⊃∼Pw∗ )) on the other. The validity of Warfield’s argument demands that (13) and (16) should be interpreted as (13a) and (16a) rather than as (13b) and (16b). But one can object, as in Section 4, that this places an unreasonable interpretation on (12) and (14). The key question, as in Section 4, is how we are to understand (25.) Q might result in P, and (26.) Q would definitely result in P, particularly in the case where Q is something which a subject is free to make the case, even though Q is incompatible with the past laws and history of the world under consideration. A compatibilist about freedom and determinism will surely insist, in line with our discussion in Section 4, that in evaluating what might or would result from eventualities which are incompatible with the past history and laws of the world, we must consider counterfactual situations, other possible worlds, in which those laws and past history are not held fixed. As we saw in Section 1, a core idea of compatibilist accounts of HOW NOT TO ARGUE FOR INCOMPATIBILISM 17 freedom is that freedom of action can be analyzed in something like the following fashion:21 S is free to do A iff (if S were to will to do A, then S would do A). But here compatibilists insist that the right side of this equivalence can be non-trivially true even in cases where the antecedent “S wills to do A” is itself ruled out by past history and the laws of nature. Clearly this requires that in determining what would or might result were S to will to do A, we must consider worlds with different past history and laws of nature than obtain in the actual world. Consequently, to read (25) and (26) as (27a.) ∼ ∀w∗ ((Qw∗ & Hw w∗ ) ⊃∼Pw∗ )) and (28a.) ∀w∗ ((Qw∗ & Hw w∗ ) ⊃ Pw∗ )) rather than (27b.) ∼ ∀w∗ ((Qw∗ & Hw∗ w∗ ) ⊃∼Pw∗ )) and (28b.) ∀w∗ ((Qw∗ & Hw∗ w∗ ) ⊃ Pw∗ )), as is required if we take (13a) and (16a) as our representations of (12) and (14), is really to beg the question against the compatibilist. Warfield’s argument, like that of Section 3, is guilty of a subtle equivocation between (13a)–(16a)–(27a)–(28a) and (13b)–(16b)–(27b)–(28b), made possible by the shifting use of “H” in modal contexts. That is, the argument aims to show that given compatibilism, the inference from (12) to (14) is invalid; it aims to show this by showing that the inference from (13) to (16), taken as representing (12) and (14) is invalid; but (13) and (16) are ambiguous between a reading ((13a)–(16a)) on which the inference is invalid, but which the compatibilist would reject as not representing that from (12) to (14), and a reading ((13b)–(16b)) which a compatibilist might accept as representing the inference from (12) to (14), but on which the inference is not shown to be invalid. Of course, the compatibilist may insist that the best way to represent (25) and (26) is, as suggested in Section 4, neither through (27a)–(28a) nor through (27b)–(28b), but by using counterfactuals: (35.) Q♦→P 18 MICHAEL KREMER (36.) Q
→ P, combined with the suggestion that worlds which share the past-historyand-laws of w are to count as more similar to w than worlds which do not share w’s past-history-and-laws, in evaluating (35) and (36). On this analysis, (12) and (14) can be represented (with no ambiguity!) as: (37.) P & ∀s∀x(Fsx ⊃∼(x ♦ → P)) and (38.) P & ∀s∀x(Fsx ⊃∼(x
→ P)). Furthermore, as long as we assume that whatever anyone is free to make the case must be logically possible: (39.)
∀s∀x(Fsx ⊃ ♦x) then on this counterfactual analysis (12), represented as (37), will imply (14), represented as (38). Thus we obtain a nice account of the validity of (∗ ) independent of the outcome of the free-will determinism debate.22 The argument depends on the theorem of the logic of counterfactuals that two counterfactuals with the same, possible, antecedent, and contradictory consequents, cannot both be true: (40.) ♦Q ⊃∼((Q
→ P) & (Q
→∼P)) or equivalently (41.) [♦Q & ∼(Q ♦ → P)] ⊃∼(Q
→ P).23 Now, given (37), and assuming that Q is any proposition which s is free to make the case, FsQ, we will have that Q is logically possible, ♦Q by (39), and ∼(Q ♦ → P) by (37). Hence ∼(Q
→ P) by (41). Since Q was arbitrary, it follows that (38) is true.24 6. CONCLUDING REMARKS I will conclude by drawing three morals from our analysis of the error in Warfield’s argument for incompatibilism. None of these points should be news – all three are as old as possible-worlds semantics for modal logic. HOW NOT TO ARGUE FOR INCOMPATIBILISM 19 Yet, if Warfield’s paper is any indication, modern modal metaphysicians could perhaps stand to be reminded of them from time to time. The first moral is a simple one: Anyone who thinks to use the apparatus of modal logic in order to draw conclusions concerning substantive metaphysical debates must exercise considerable caution. As Warfield’s argument shows, the use of such apparatus can do as much to obscure the central issues involved as to clarify them. In particular, paradoxically, in this case the use of a modal-logical Begriffsschrift introduced ambiguities, rather than eliminating them, in such a way as to produce a crucial equivocation in the argument. On the other hand, and this is my second moral, correcting and monitoring such errors in the use of logical apparatus itself requires a careful application of that same apparatus. For example, use of explicit quantifiers over possible worlds rather than propositional modal operators can allow crucial distinctions to be brought to light.25 In any particular case the advantages of a modal language – lower complexity, closer connection to natural language, and so on – have to be weighed against the advantages in explicitness and lack of ambiguity of quantification over possible worlds. Finally, and this my third moral, a careful analysis along these lines can in the end help to refocus attention on the real philosophical issues involved in the underlying metaphysical debates. In the present case, the real issues surround the question, which we saw Warfield to implicitly beg against the compatibilist, whether we can make reasonable sense out of the idea of what “would”, non-trivially, result from something which itself is a historical impossibility (as is required by compatibilism). This is an issue which modal logic itself cannot resolve. APPENDIX : NELKIN AND RICKLESS ON WARFIELD (Nelkin and Rickless 2002) attack Warfield’s argument on quite different grounds than those presented here. In this appendix I briefly discuss their critique, noting that their objections leave Warfield with plausible replies, and cut less deep than those developed here. Nelkin and Rickless first object that Warfield’s CONCLUSION is not equivalent to incompatibilism, which they think ought to be represented by
∀x∀s(D ⊃∼Fsx). They admit that if “an individual’s not being free to act otherwise than she does entails that she is not free to do what she does”,
∀x∀s(∼Fs∼x ⊃∼Fsx), Warfield’s CONCLUSION will entail (and so be equivalent to) their formulation of incompatibilism. (Nelkin and Rickless 2002, p. 105).26 Nelkin and Rickless claim, though, on the authority of Harry Frankfurt, that “it is possible for a person to be free to act as she 20 MICHAEL KREMER does even though she is not free to do otherwise” (Nelkin and Rickless 2002, p. 105).27 But this is at least a debatable claim; and in any case their objection would leave Warfield having ruled out all forms of compatibilism which deny it. Nelkin and Rickless’ second objection is that the proper representation of (12) is not (13) but (in their numbering) (16∗ ) P & ∼ ∃s∃x(Fsx & ♦((x & H) ⊃∼P)). (Nelkin and Rickless 2002, p. 106.) They argue that on this analysis of (12), the validity of (∗ ) is secured by the basic modal-logical fact that
p entails ♦p, since (∗ ) amounts to: P & ∼ ∃s∃x(Fsx & ♦((x & H) ⊃∼P)) therefore P & ∼ ∃s∃x(Fsx &
((x & H) ⊃∼P)). Since (∗ ) is valid in any case, no additional assumption such as the denial of (10) can undercut its validity.28 This analysis, however, is predicated on the view that “x might result in P” ought to be represented by “♦((x & H) ⊃ P)” rather than “♦((x & H) & P)”. But this analysis has counterintuitive consequences, which make it unacceptable. For example it implies that the following (obviously invalid) inference is valid: ∼x, therefore x might result in P. (Proof: ∼x entails (x & H) ⊃ P by propositional logic, and so entails ♦((x & H) ⊃ P) by the T-axiom p ⊃ ♦p.) Thus Warfield has possible replies to both Nelkin and Rickless’ objections. Furthermore, Nelkin and Rickless never question the modal-logical framework of Warfield’s paper, and so fail to diagnose the ambiguities which that framework generates. Both their objections are constructed using the apparatus introduced by Warfield. Thus their critique of Warfield is less radical than that offered in the present paper.29 ACKNOWLEDGEMENTS Thanks are due to Tim Bays for valuable conversations, to Dana Nelkin, Sam Rickless, and two anonymous referees for helpful comments on an earlier version of this paper, and especially to Fritz Warfield for generous discussions. HOW NOT TO ARGUE FOR INCOMPATIBILISM 21 NOTES 1 The sort of use of modal logic made in Warfield’s paper is not uncommon in the literature on the free will debate. This literature (as indeed much metaphysical literature in general) typically employs S5 modal logic, which allows an interpretation of necessity as truth in all possible worlds, with no relation of accessibility or relative possibility. One source of this can be found in Plantinga’s influential arguments that, for “broadly logical” or “metaphysical” modality, there is no sensible notion of “relative possibility”, so that S5 is the correct logic (Plantinga 1974, pp. 51–54). Warfield’s paper adheres to this view, as can be seen in the absence of references to accessibility relations in his informal glosses on modal claims. In the body of the paper, I generally follow this tradition, taking up in footnotes points at which acceptance of a weaker modal logic based on a notion of relative possibility might be thought to affect the argument. Furthermore, throughout this paper “♦” and “
” refer to “broadly logical” or “metaphysical” possibility and necessity unless otherwise noted (in particular, see note 10). 2 A quite different criticism of Warfield is found in Nelkin and Rickless (2002), which was brought to my attention after I had substantially completed work on this paper. I discuss their critique in the appendix. 3 This is a very brief survey of a highly complex and developed literature. For a good introduction and sampling of representative works, see Kane (2002), on which I draw heavily in this section. Readers familiar with this literature can safely skip this section. 4 For an introductory discussion, see Kane (2002, pp. 10ff, 80ff). 5 It is worth noting that Warfield’s objection to Van Inwagen does not have much force. For Van Inwagen’s argument can be carried out for any possible world in which determinism is true, as long as for every potential agent in that world, there is a time before that agent exists. (If there is a first moment of time and an individual existing at that time, Van Inwagen’s argument would not apply to that individual; hence Van Inwagen’s argument leaves open the possibility of a determinist world in which such an individual at such a time – but only such an individual at such a time – has free will.) This generalization does not depend, as Warfield seems to think it would, on acceptance of the necessitations “
NP0 ” and “
NL” of Van Inwagen’s premises “NP0 ” and “NL”. Indeed, if “P0 ” and “L” refer to the past and laws of the actual world, these claims of necessity are entirely implausible. In the generalization of Van Inwagen’s argument suggested here, “P0 ” and “L” will have to be chosen to suit the determinist world in question, with a different “P0 ” and “L” chosen for each world. Since Van Inwagen’s argument can be generalized to all determinist possible worlds in this way, it actually establishes almost completely (modulo the special exception noted above) the modal claim that Warfield desires. Hence, the evaluation of the force of Van Inwagen’s argument has to turn on the numerous other objections that have been raised in the literature. In my view some of the most interesting and serious objections are those that raise doubts about principle (β), for example those of McKay and Johnson (1996). 6 I summarize here the argument of Warfield (2000, pp. 172–177). 7 I have modified the numbering of propositions for this paper. 8 To anticipate, the crucial ambiguity in Warfield’s argument occurs here. The crux of the matter is this: Does “‘H” in propositions like (10) stand for the conjunction of the past and the laws of nature in the actual world, or in some other world (implicitly picked out by the modal operators in (10))? – and if the latter is the case, in which world? As will be seen in detail below, this kind of ambiguity leads Warfield to equivocate in his argument. 22 MICHAEL KREMER 9 Nelkin and Rickless (2002) argue that compatibilists need not reject the CONCLUSION. See the appendix. 10 Nelkin and Rickless (2002) dispute the correctness of (13) as a representation of (12), but accept the correctness of (15)/(16) as a representation of (14). See the appendix. An anonymous referee suggests instead representing (14) as P & ∼∃s∃x(Fsx &
(x ⊃∼P)) thereby eliminating the pesky “H” which will cause so much trouble below. Presumably the suggestion would be to eliminate “H” from some or all of its other occurrences as well. If the “
” here represents broadly logical or metaphysical necessity, following out this suggestion for (11), (13), and (16) results in: (11′ .)
∀x(D ⊃ (x ⊃
x)). (13′ .) P & ∀s∀x(Fsx ⊃
(x ⊃ P)) (16′ .) P & ∀s∀x∼(Fsx &
(x ⊃∼P)). This would in fact correspond to readings (11b) of Section 4 and (13b) and (16b) of Section 5 (see the discussion of (27b), (28b), (27c), (28c), (27d), and (28d) in Section 4 below). Arguments parallel to those of Sections 4 and 5 concerning (11b), (13b), and (16b) then show that while (13′ ) and (16′ ) might be acceptable to the compatibilist as representations of (12) and (14), (11′ ) would not be acceptable as a representation of a determinist thesis. Determinists need not hold that all truths are broadly logically necessary (the initial conditions might be metaphysically contingent). However, the suggestion might be to read some occurrences of “
” as representing a more restricted form of necessity – perhaps nomological necessity (truth in all worlds which share the same laws as the world at which necessity is evaluated), or perhaps historical necessity (truth in all worlds which share the same laws and the same past as the world at which necessity is evaluated – here relative also to a time-instant). This suggestion would build into the modal notation, through the accessibility relation introduced with non-logical modalities, restrictions on which possible worlds are relevant – restrictions that Warfield tries to handle through the explicit condition “H”. Following this suggestion out a bit, let’s use “
N ” for nomological necessity, and “
H ” for historical necessity, and let’s consider: (11N .)
∀x(D ⊃ (x ⊃
N x)). (13N .) P & ∀s∀x(Fsx ⊃
N (x ⊃ P)) (16N .) P & ∀s∀x∼(Fsx &
N (x ⊃∼P)), on the one hand, and (11H .)
∀x(D ⊃ (x ⊃
H x)). (13H .) P & ∀s∀x(Fsx ⊃
H (x ⊃ P)) (16H .) P & ∀s∀x∼(Fsx &
H (x ⊃∼P)) HOW NOT TO ARGUE FOR INCOMPATIBILISM 23 on the other. For Warfield’s argument to go through, one would need to choose employ either all of (11N ), (13N ), and (16N ), or all of (11H ), (13H ), and (16H ). However, on the one hand, only (11H ) is plausible as capturing a determinist principle – determinism does not rule out that the past is nomologically contingent. On the other hand, only (13N ) and (16N ) would be acceptable to the compatibilist as representations of (12) and (14), since (13H ) and (16H ) would be open to objections much like those raised against (13a) and (16a) in Section 5 below. (Taking (13H ) and (16H ) to represent (12) and (14) involves reading “P would definitely result in Q” and “P might result in Q” as “
H (P ⊃ Q)” and “∼
H (P ⊃∼Q)” respectively; this requires that in considering what would or might result from a free action which does not in fact occur, we take into account only worlds which share the same laws and past as the actual world; but compatibilists would not accept this claim.) Thus the introduction of more restricted forms of modality, far from shoring up Warfield’s argument, only serves to highlight in another way its basic flaw, here shown as an equivocation on the force of “
”. 11 As we will see this is the crucial point in the conjuring trick. 12 Strictly speaking, the detour through the Gaunilo-style objection of this section and the next is unnecessary to establish the incorrectness of Warfield’s argument. However, the argument of these two sections does help to make clear both that there must be something wrong with his argument, and precisely what is wrong, especially at what level of generality the error occurs. Consideration of the Gaunilo-style objection, in particular, shows that the fundamental problem with Warfield’s argument has nothing to do with the proper analysis of incompatibilism – the issue on which Nelkin and Rickless focus in their first (and more successful) response to Warfield, discussed in the appendix below. My analysis of the difficulty in Warfield’s argument gains in power in that it simultaneously refutes analogous fallacious arguments. 13 The argument of this section does not depend on any modal principles other than those of the weak modal logic K. In the context of the modal logic S5, which is usually assumed in contemporary metaphysical uses of modal logic, the objection can be strengthened to show that, if Warfield’s style of reasoning is correct, the mere possibility of determinism, together with the hypothesis of the possible existence of contingent propositions, entails the invalidity of (∗∗ ). For under S5, every world is accessible to every other world. It follows that if it is possible that there are contingent propositions, it is necessary that there are contingent propositions. Hence, under S5, Warfield’s style of reasoning would in fact show that the mere possibility of determinism entails the necessary collapse of modality. 14 We have assumed that contingent propositions necessarily exist. Hence there is some R which is contingent in w. Then ∼R is also contingent in w, and one or the other of R and ∼R is true in w. 15 Nelkin and Rickless have no qualms about (11) and see Warfield’s argument to CONCLUSION as simply correct. 16 In the following analysis I do not make any use of accessibility relations between worlds. Hence the analysis operates at the level of S5 modal logic. This fits with Warfield’s argument and with much of the literature to which it is a contribution; it also simplifies the exposition of my argument. Introducing accessibility relations into the picture would complicate the presentation of my objection to Warfield, but not weaken its force. For some brief related remarks see footnote 10 above. 17 The logical grammar of this notation calls for some comment. Warfield does not explicitly distinguish between sentential and individual variables of quantification, but such a distinction is implicit in his use of “x” and “s” in formulae such as: 24 MICHAEL KREMER
∀x∀s(Fsx ⊃ ♦(H & x)). Here “x” occupies a sentential position, whereas “s” occupies an individual position. Moreover, “F” in this formula must be understood as a functor taking arguments of mixed type, one sentential and the other objectual, and yielding a value of sentential type. The present notation diverges from Warfield’s in that no variables of sentential type are required. “x” is now a variable ranging over propositions, which can for present purposes be thought of as constituting a special kind of objects (without trying to prejudge ultimate questions of the ontology of propositions); similarly, “w” is a variable ranging over possible worlds, which can again be thought of as a special kind of objects (without trying to prejudge ultimate questions of the ontology of possible worlds). “H” then represents a function from possible worlds to propositions, so that “x” and “Hw ” are expressions of essentially the same logical type (proposition-object). Finally, concatenation of an expression picking out a proposition and an expression picking out a possible world is used to express that the proposition holds in the world; thus “xw” and “Hw w∗ ”, as opposed to “x” and “Hw ”, are expressions of sentential type. In a more explicit notation, one might introduce a predicate “Holds(x, w)” for this purpose. I have avoided doing this because it seemed to detract rather than contribute to the clarity of my argument. Later, I introduce the expression “Fsxw” to express that subject s is free to make proposition x true in world w. One can think of “F” here as functioning as a three-place predicate; or one can think of “F” as functioning as a two-place functor yielding an expression for a proposition, which is then said to hold (or not) in w (that is, one can think of “Fsxw” as “Holds(Fsx, w)”): My original intention was that “Fsxw” should be read in the former way, but it would not affect my argument if the latter reading were adopted. 18 One could use an enriched modal language to express the distinctions drawn here, and later in the paper, using explicit quantification over possible worlds. This would require combining two ideas. The first idea is to treat sentential letters like “H” non-rigidly, as expressing different propositions at different worlds. More formally, where a typical modal model assigns a truth-value to each sentence letter in each world, here a model would assign a function from worlds to truth-values, or a set of worlds, understood as representing a proposition, to each sentence letter in each world. The second idea is to employ special indexed actuality operators to allow reference to worlds other than the actual world in fixing the proposition expressed by a non-rigid sentential letter. A formal development of this idea could be adapted from the treatment of indexed actuality in Stephanou (2001). Stephanou’s indexed actuality operators refer back to indexed modal operators to determine the worlds at which sentences are to be evaluated as true or false; in the present application, the indexed actuality operators would have to refer back to indexed modal operators for a different purpose, that of determining the proposition expressed by a sentential letter. As an example of the disambiguating power of such an enriched modal language, our (11a)–(11c) might be expressed as follows (using α1 , α2 , . . . as indexed actuality operators in the sense described above, and following Stephanou’s convention that a “free” indexed actuality operator is understood as referring back to the actual world): (11a′ .)
1 ∀x(D ⊃ (x ⊃
2 (α1 H ⊃ x))) (11b′ .)
1 ∀x(D ⊃ (x ⊃
2 (α2 H ⊃ x))) (11c′ .)
1 ∀x(D ⊃ (x ⊃
2 (α3 H ⊃ x))). HOW NOT TO ARGUE FOR INCOMPATIBILISM 25 As Stephanou’s treatment of indexed modal operators as binding indexed actuality operators “in a manner analogous the binding of variables by quantifiers” (Stephanou 2001, p. 358) helps make clear, such a treatment would essentially be a notational variant of our method using explicit quantifiers over possible worlds. Hence, an analysis of Warfield’s argument using such a language would yield the same results as the analysis in the main body of the paper. Nonetheless, if only for reasons of familiarity, the use of quantification over possible worlds in the body of the paper seems to make the point more clearly. 19 Nelkin and Rickless (private correspondence) object that while “(10) and (11) are indeed ambiguous” this is because “both of these sentences contain TWO modal operators, one embedded in the scope of the other”, while in contrast (21) and (24) “contain ONE modal operator each, and so do not suffer from the relevant sort of ambiguity”. They add “you say that (21) and (24) (each of which contains two modal operators, but neither embedded in the scope of the other) are ambiguous, but you are careful to qualify this as being conditional on ‘evaluating (21) and (24) at some world’. This is correct, but all it really shows is that the MODALIZATIONS of (21) and (24) are ambiguous. No reason is given (and none could be) for thinking that (21) and (24) themselves are ambiguous”. However, their admission that there is an ambiguity when we are “evaluating (21) and (24) at some world w” is all I need, since the argument of Section 3 precisely turns on evaluating (21) and (24) at some world w. That is, the argument of Section 3 aims to show that (21) does not entail (24), by constructing a world w in which (21) is true and (24) is false, and it is in considering the truth-values of (21) and (24) in w that the ambiguity arises. To put matters another way: Nelkin and Rickless suggest that the ambiguity can only arise in the case of embedded modal operators. But what the argument of Section 3 aims to prove is that ♦((21) & ∼(24)) – and this latter claim does involve modal operators (in (21) and (24)) embedded within the outer “♦”. 20 Assume that determinism is possibly true, so that for some world w, (19a) is true; and assume that necessarily, some propositions are contingent. Let P be any proposition which is true in w, but not necessarily true. By (19a), ∀w∗ (Hw w∗ ⊃ Pw∗ ) is true in w. Letting Q be any proposition whatsoever, we then have that ∀w∗ ((Qw∗ & Hw w∗ ) ⊃ Pw∗ ), and so ∃w∗ Qw∗ ⊃ ∀w∗ ((Qw∗ & Hw w∗ ) ⊃ Pw∗ ). As Q was arbitrary, (21a) is true. On the other hand, ∀w∗ ((∼Pw∗ & Hw w∗ ) ⊃∼Pw∗ ) is clearly true in w. Also ∃w∗ ∼Pw∗ is true in w, so ∼(∃w∗ ∼Pw∗ & ∀w∗ ((∼Pw∗ & Hw w∗ ) ⊃∼Pw∗ )) is false in w, so ∀x∼(∃w∗ (xw∗ ) & ∀w∗ ((xw∗ & Hw w∗ ) ⊃∼Pw∗ )) is false in w, so (24a) is false in w. 21 Of course, sophisticated compatibilist accounts do not remain content with this formulation, but it remains at the core of such views. 22 Nelkin and Rickless also provide an explanation of the validity of (∗ ); but their explanation has counterintuitive consequences which mine lacks. See the appendix. 23 Recall that, following Lewis, we are taking “P ♦ →Q” to be equivalent to “∼(P
→∼Q)”. 24 Warfield claims that even if we understand (12) and (14) “using subjunctives rather than strict conditionals” we “could construct an argument parallel to the one in the text reaching the same conclusion”. He “leave[s] the task of constructing the argument to those attracted to the subjunctive interpretation . . . ” (Warfield 2000, p. 179). However, at least if subjunctives are analyzed along the lines of Lewis” theory of counterfactuals, no such argument is to be found. 25 This is not to say that only a use of explicit quantification over possible worlds can do this work. As seen in notes 10 and 18 above, similar analyses could be carried out using enriched versions of the modal language employed by Warfield. 26 MICHAEL KREMER 26 Clearly their formulation of incompatibilism entails Warfield’s CONCLUSION. On the other hand, suppose that CONCLUSION holds and assume
∀x∀s(∼Fs∼x ⊃∼Fsx), or equivalently
∀x∀s(Fsx ⊃ Fs∼x). Consider any world w in which D holds. Since Warfield’s CONCLUSION is equivalent to
(D ⊃ ∀x∀s(x ⊃∼Fs∼x)), ∀x∀s(x ⊃∼Fs∼x) holds in w. Now suppose that Fsx holds in w; then also Fs∼x holds in w by hypothesis. But either x holds in w or ∼x holds in w. In the first case we have (in w) x ⊃∼Fs∼x, and so ∼Fs∼x – a contradiction. In the second case we have (in w) ∼x ⊃∼Fsx and so ∼Fsx – again a contradiction. Hence by reductio ∼Fsx in w; and since s, x and w were arbitrary, we have proved Nelkin and Rickless’ form of incompatibilism. 27 The reference is to Frankfurt (1969); for discussion see Kane (2002) especially Part II. 28 This account of the validity of (∗ ) should be contrasted with that offered at the end of Section 4 above. 29 Thanks to Fritz Warfield for bringing their paper to my attention. REFERENCES Frankfurt, Harry: 1969, ‘Alternative Possibilities and Moral Responsibility’, Journal of Philosophy 66, 829–839. Frankfurt, Harry: 1971, ‘Freedom of the Will and the Concept of a Person’, Journal of Philosophy 68, 5–20; reprinted in R. Kane (ed.): 2002, Free Will, Blackwell Publishing, Oxford. Kane, Robert (ed.): 2002, Free Will, Blackwell Publishing, Oxford. McKay, Thomas J. and David Johnson: 1996, ‘A Reconsideration of an Argument against Compatibilism’, Philosophical Topics 24, 113–122. Nelkin, Dana K. and Samuel C. Rickless: 2002, ‘Warfield’s New Argument for Incompatibilism’, Analysis 62, 104–107. Plantinga, Alvin: 1974, The Nature of Necessity, Clarendon Press, Oxford. Stephanou, Yannis: 2001, ‘Indexed Actuality’, Journal of Philosophical Logic 30, 355– 393. Van Inwagen, Peter: 1983, An Essay on Free Will, Clarendon Press, Oxford. Excerpted in Kane (ed.): 2002, Free Will, Blackwell Publishing, Oxford. Warfield, Ted A.: 2000, ‘Causal Determinism and Human Freedom are Incompatible: A New Argument for Incompatibilism’, Philosophical Perspectives 14: Action and Freedom, pp. 167–180. Wolf, Susan: 1987, ‘Sanity and the Metaphysics of Responsibility’, in F. Schoeman (ed.), Responsibility, Character and Emotions, Cambridge University Press, Cambridge. Reprinted in Kane (ed.): 2002, Free Will, Blackwell Publishing, Oxford. Wolf, Susan: 1990, Freedom within Reason, Oxford University Press, Oxford. Department of Philosophy University of Chicago 1010 E. 59th St. Chicago, IL 60637 USA E-mail: [email protected] Manuscript submitted 15 April 2002 Final version received 26 November 2002