Abstract
The expression conditional fallacy identifies a family of arguments deemed to entail odd and false consequences for notions defined in terms of counterfactuals. The antirealist notion of truth is typically defined in terms of what a rational enquirer or a community of rational enquirers would believe if they were suitably informed. This notion is deemed to entail, via the conditional fallacy, odd and false propositions, for example that there necessarily exists a rational enquirer. If these consequences do indeed follow from the antirealist notion of truth, alethic antirealism should probably be rejected. In this paper we analyse the conditional fallacy from a semantic (i.e. model-theoretic) point of view. This allows us to identify with precision the philosophical commitments that ground the validity of this type of argument. We show that the conditional fallacy arguments against alethic antirealism are valid only if controversial metaphysical assumptions are accepted. We suggest that the antirealist is not committed to the conditional fallacy because she is not committed to some of these assumptions.
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Notes
The philosophical problem of the conditional fallacy was apparently introduced by Shope [22].
In Lewis-Stalnaker conditional logic, schema I is valid only if q is possible.
Briefly, a chameleon is in the dark and is actually green. This creature is very shy and very perceptive: if another being were to spot it from a sufficiently close distance, the chameleon would instantaneously notice it and blush bright red. If we suppose that the concept of green is response-dependent and is defined in terms of \(\textsc {sc}\), this is a case in which (p) the chameleon is actually green but (q□→ ¬p): if the chameleon were observed, it would not be green any longer. Schema I licenses the false conclusion that (¬ p) the chameleon is actually not green. Note that Johnston’s chameleon example is also interpretable as a case of conditional fallacy of type II, for p = ‘the chameleon is red’.
For an overview of the diverse conditional fallacy schemata and a general assessment of them see Bonevac et al. [2].
Wright’s original proof and Plantinga’s original proof (which we consider below) rely on exclusively classical principles that are unacceptable for the intuitionist, so an antirealist following Dummett’s work could easily reject them. However, Brogaard and Salerno ([3]: 131) have provided an intuitionistically respectable version of Plantinga’s proof, which contains an embedded version of Wright’s proof and is based on the theorem that \(\Box \big (p\equiv (q \Box {\kern -0.9pt} \rightarrow r)\big )\) and \(\Box ({q}\supset {p})\) jointly entail \(\Box p\). In this paper we focus on Brogaard and Salerno’s reconstructions of Wright’s and Plantinga’s original proofs.
It is worth stressing, however, that Plantinga is not arguing against antirealism—though he can certainly be reinterpreted to be doing so. Plantinga takes the conclusion of his proof to show only that the antirealist is committed to some form of theism (i.e. to the thesis that, roughly, there is necessarily an omniscient rational enquirer/community). Wright [27] and Brogaard and Salerno [3] respond that this is a misguided interpretation of his finding, which should instead be seen as an instance of the general problem of the conditional fallacy that plagues counterfactual analyses. As we find this response plausible, we will not question it in our paper.
Plantinga apparently obtains a stronger result because he assumes that the accessibility relation is transitive, which is not presupposed by Wright.
Brogaard and Salerno’s proof improves upon the proof made by Rea [19]. Like Plantinga, Rea takes his proof to show that the antirealist is committed to some form of theism.
The notion of the end of enquiry and even the notion of approximating the end of enquiry, is not perspicuous and might prove incoherent under close scrutiny. Interestingly, Wright himself ([26]: 44–48) has argued for this conclusion.
An equivalent (and perhaps more precise) formulation of p is this: ‘for some statement x, epistemic conditions are suitable for evaluating whether x is true’.
The main reason why we also focus on III, rather than on just IV, is that we cannot exclude a priori that once we allow statements to have individual truth-conditions, there might exist a statement s similar to the one used by Brogaard and Salerno that is intuitively false, or not known to be true, but such that \(\Box \big (Q(s)\supset T(s)\big )\). If this were the case, one could use III to produce a conditional fallacy argument against non-Peircean forms of antirealism. Suppose for instance P is a property that cannot possibly be detected. Also suppose that \(s =\) ‘this object has property P’. It seems impossible that \(Q(s)\)—i.e. that epistemic conditions are suitable for evaluating whether s is true. Hence, \(\Box \big (Q(s)\supset T(s)\big )\). Thus, on III, we should conclude that s is true, which appears unjustified. Although the antirealist could dismiss the idea of a property like P and a statement like s as incoherent or absurd, there might be other statements capable of engendering similar problems that the antirealist cannot easily dismiss. As we show in Sections 4 and 5, the validity of relation containment together with some background assumptions is sufficient and necessary to validate III, and thus to get the antirealist into trouble. In Section 6 we provide metaphysical reasons to doubt that relation containment is satisfied.
A binary relation S is a strict partial order if and only if S is:
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1.
Irreflexive: \(\neg \exists x Sxx\);
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2.
Antisymmetric: \(\forall x, y: (Sxy \, \& \, Syx) \Rightarrow x = y\);
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3.
Transitive: \(\forall x, y, z: (Sxy \, \& \, Syz) \Rightarrow Sxz\).
It is a well-known fact in logic that conditional logic cannot express the strictness of the similarity relation. We could have chosen to work with weak partial orders \(\leq \), but instead we chose strict partial orders as they seem to be more commonly used by logicians.
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1.
A binary relation S is well-founded on a set U if and only if for every non-empty subset \(U'\) of U, there exists an element \(x\in U'\) such that, for every \(y\in U'\), it is false that \(Syx\).
When the limit assumption is accepted, \(\varphi \Box {\kern -1.5pt} \rightarrow \psi \) proves false at w in \(\mathfrak {M}\) just in case there is a world u minimal in \(W_w\) at which \(\varphi \) is true but \(\psi \) false in \(\mathfrak {M}\). When this condition is satisfied the more general condition for falsity of \(\varphi \Box {\kern -0.6pt} \rightarrow \psi \) is also satisfied.
This is true in both generalised and special semantics for counterfactuals.
That is to say, V makes p true at any world x in W such that u is not in the zone of entertainability of x, and also V makes q true at u.
We assume that u is in \(W_w\)—and thus that u is a world relevant for the truth-value of \(q\Box {\kern -0.6pt} \rightarrow r\)—even if it is false that \(Rwu\). In Section 6 we give arguments in support of the general acceptability non-uncontroversial assumptions of this type.
That is to say, V makes p true at any world x in W such that v is not in the zone of entertainability of x, and also V makes q true at u.
Note that both proofs of Theorems 3 and 4 would work almost identically with further assumptions on \(\mathfrak {F}\), such as the limit assumption, strong centering or connectedness of \(<\), as it can easily be checked. For instance, in the case of connectedness of \(<\), there would be no world \(k'\) such that \(u\not \in W_{k'}\). Thus \(K'\) would be empty, but the valuation V we have chosen would remain the same otherwise. Hence, even though we have phrased our theorems with very minimal conditional assumptions, the same proofs would work for more restrictive classes. This again demonstrates the generality of relation containment.
Consider for instance the frame \(\mathfrak {F}\) with \(W = \{w, v, u\}\), weak centering, R reflexive and such that \(Rwv\), \(u\in W_v\) and \(Rvu\) but not \(Rwu\). \(\mathfrak {F}\) satisfies relation containment but not quasi-transitivity.
This argument and the next one parallel a well-known objection to Lewis [11]’s thesis that all counterfactuals with impossible antecedents are vacuously true. For instance, Mares [12] lists counterfactuals with impossible antecedents that appear true and counterfactuals with impossible antecedents that appear false. A difference between Mares’ objection and the arguments we put forward here (on the antirealist’s behalf) is that ours are meant to apply to counterfactuals with locally impossible antecedents—i.e. antecedents that are false at any world accessible from a given possible world—whereas Mares’ objection is meant to apply to counterfactuals with absolutely impossible antecedents—i.e. antecedents that are false at any possible world whatsoever.
Or that counts as closest to u, if we use the special counterfactual semantics.
An objection has been made to us that the antirealist would still commit a conditional fallacy if there were some statement s with epistemic conditions \(Q(s)\) satisfiable only at worlds accessible from the actual world. This would be so because for any such s, the semantics of the counterfactual in \(\Box \big ({T(s)}\equiv \big ({Q(s)}{\Box {\kern -0.6pt} \rightarrow {R(s)}}\big )\big )\) would not require the antirealist to consider models with worlds inaccessible from the actual one, thus relation containment would be satisfied. In other words, for any such s, the semantics of \(\Box \big ({T(s)}\equiv \big ({Q(s)}{\Box {\kern -0.6pt} \rightarrow {R(s)}}\big )\big )\) could be contained in a class of frames that do have relation containment. This is an interesting criticism. Notice that for s to satisfy III in this case, it would also have to make \(\Box \big ({Q(s)}\supset T(s)\big )\) true. Furthermore, the conclusion of the relevant instance of III would prove paradoxical or problematic only if s were known to be false, or at least if s were not known to be true. Finally, to have a useful result, s should have individual truth-conditions (see note 11 above). We do not see why the antirealist who rejected relation containment as valid generally should be committed to the existence of statements of this sort. Also note that, from a logical point of view, this objection seems to assume that there is a dependence between the propositional valuation and the accessibility and similarity relations in models, which imposes restrictions on class of frames. But this is nowhere to be found in the literature.
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Acknowledgments
We are very grateful to Franz Berto, Fred Kroon, Hannes Leitgeb, Chris Tucker, Jonathan McKeown-Green, Olivier Roy, Jeremy Seligman, Elia Zardini, Imogene Paterson, audience at the Munich Center for Mathematical Philosophy, and various referees for valuable discussions and important criticisms concerning drafts of this paper. We are grateful to Antti Keskinen and Ralph Wedgwood for drawing our attention to papers relevant for the discussion in Section 6.
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Girard, P., Moretti, L. Antirealism and the Conditional Fallacy: The Semantic Approach. J Philos Logic 43, 761–783 (2014). https://doi.org/10.1007/s10992-013-9288-0
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DOI: https://doi.org/10.1007/s10992-013-9288-0