Conditional Excluded Middle

In this essay I renew the case for Conditional Excluded Middle (CXM) in light of recent developments in the semantics of the subjunctive conditional. I argue that Michael Tooley’s recent backward causation counterexample to the Stalnaker-Lewis comparative world similarity semantics undermines the strongest argument against CXM, and I offer a new, principled argument for the validity of CXM that is in no way undermined by Tooley’s counterexample. Finally, I formulate a simple semantics for the subjunctive conditional that is consistent with both CXM and Tooley’s counterexample.

1. I will use the corner ‘ > ’ to represent the subjunctive conditional connective.

2. See, for example, Stalnaker (1981).

3. I take the standard Stalnaker-Lewis semantics to include additional assumptions about comparative world similarity, namely the six conditions given at Lewis (1973, p. 48).

4. The ambiguity in our use of ‘ > ’ will always be resolved by context. The two uses of ‘ > ’ are related as follows: if p is the proposition expressed by sentence A and q is the proposition expressed by B, then p > q is the proposition expressed by the sentence A > B.

5. Tooley provides arguments against the Stalnaker-Lewis approach in both Tooley (2002) and Tooley (2003). I will focus exclusively on the argument in Tooley (2002) because I take it to be more convincing than the argument in Tooley (2003), which renews and re-evaluates the sort of objection found in Fine (1975).

6. By ‘worlds like’ w ₁ (or w ₂) I mean worlds that fit the obviously incomplete description of w ₁ (or w ₂) given in the table.

7. Cross (2008, p. 106).

8. Cross (2008, p. 119).

9. See Edgington (2004, p. 18–20).

10. I will leave aside the issue of precisely how causal independence itself should be analyzed.

11. For a rather different take on Morgenbesser’s Coin and related matters, see Bennett’s treatment in Bennett (2003, pp. 232–239).

12. I should mention another possible source of resistance to the intuition that Morganbesser’s Coin would have come up heads had I bet on it, namely, the statistical fact that a fair coin comes up heads only half the time. To conclude from this that the coin might have come up differently had I bet on it, however, is to confuse the actual coin toss with hypothetical alternative coin tosses. In supposing that I had bet on heads, I move to a counterfactual situation in which the same coin toss occurred, not a situation in which the coin is tossed ‘again’. This is not a new point, but it is, I think, one that bears repeating.

13. We assume, for now, that there are no truth-value gaps. Later we will relax this assumption and investigate how this argument must be modified in light of that change.

14. Here we assume that the set of counterfactual consequents of a proposition is closed under entailment, where a set X of propositions entails a proposition p iff \(\,\bigcap\{q:\: q\in X\}\subseteq p.\)

15. Similarly, monadic predicates—even vague ones—always purport to divide the class of individuals exhaustively into instances and non-instances.

16. Supertruth at a world i is the property of being true at i under all resolutions of vagueness. Superfalsity at i is the property of being false at i under all resolutions of vagueness. Supervaluation theory allows gappiness because a statement can be true at i under one resolution of vagueness and false at i under another resolution of vagueness.

17. Note that in arguing from Maximal Preservation to the validity of CXM we do not make an inference within the object language of counterfactuals, since neither Maximal Preservation nor the validity of CXM is an object language claim. Our argument will not, therefore, invoke minimal localist validity. Still, our argument will have the flavor of an application of minimal localist validity because Supervaluational Maximal Preservation is framed in terms of truth under a full resolution of vagueness.

18. Our definition of entailment does not change. Given our change in the definition of a proposition, entailment now means preservation of truth at a world and a specification point.

19. Failures of causal independence can give rise to an indeterminacy that is similar to the vagueness in the Bizet-Verdi example. Consider a variant on the Morgenbesser’s Coin case in which placing a bet on heads would have started an indeterministic process that could have changed the outcome of the coin toss. Does the outcome of heads in this case survive the counterfactual supposition that a bet is placed? It may appear that in this example, as in Morgenbesser’s Coin, Maximal Preservation implies that the actual outcome does survive, since the counterfactual supposition of a bet may appear not to rule out the coin’s coming up heads. Maximal Preservation does not apply in this variant case, however, because here, as in the Bizet-Verdi case, the identity of c _i (Bet-on-heads) is indeterminate, and Maximal Preservation does not apply at the level at which there are (super)truth-value gaps. Thus, it is wrong to say that the supposition of a bet on heads would not have ruled out the coin’s coming up heads. Instead we should say that, because of the indeterministic process that would have been initiated by a bet on heads, there is no fact of the matter about whether the supposition of a bet on heads would have ruled out the coin’s coming up heads. As in the Bizet-Verdi example, the result is that certain sentences have truth-value gaps, and in this case the gappy sentences include Bet-on-heads >  Outcome-of-heads, Bet-on-heads \( > \neg\) Outcome-of-heads, and \(\neg\) (Bet-on-heads \( > \neg\) Outcome-of-heads). The gappiness in this case can be interpreted supervaluationally, but I am not convinced that the case is properly described as vagueness.

20. I do not rule out strengthening this list of conditions. For example, I would have no objection to replacing f1 with the principle that if i ∈ p, then f(p,i) = {i}. Still, that principle’s old label (Strong Centering) seems inappropriate in the semantics offered here.

21. These are a.0.7 and a.0.8 in Lewis (1971).

22. See Theorem 1, Cross (2008, pp. 104–105).

I thank the anonymous referee for raising issues that led to improvements in this essay.

Authors and Affiliations

1. Department of Philosophy, University of Georgia, Athens, GA, 30602-1627, USA

   Charles B. Cross

Corresponding author

Correspondence to Charles B. Cross.

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