THE LOGIC OF THE EVIDENTIAL CONDITIONAL

Abstract In some recent works, Crupi and Iacona proposed an analysis of ‘if’ based on Chrysippus’ idea that a conditional holds whenever the negation of its consequent is incompatible with its antecedent. This paper presents a sound and complete system of conditional logic that accommodates their analysis. The soundness and completeness proofs that will be provided rely on a general method elaborated by Raidl, which applies to a wide range of systems of conditional logic.

from the actual world if compared with those in which α and are both true. This is the Ramsey Test as understood by Stalnaker and Lewis: in the closest worlds in which α is true, must be true. 2 The second, which may be called Reverse Ramsey Test, requires that the worlds in which α is true and is false are distant from the actual world if compared with those in which α and are both false: in the closest worlds in which is false, α must be false. Crupi and Iacona call Chrysippus Test the conjunction of the Ramsey Test and the Reverse Ramsey Test. 3 The Chrysippus Test can easily be illustrated by means of an example. Consider the following conditional: (1) If it's pure cashmere, it will not shrink Insofar as one accepts (1), one will find that 'It's pure cashmere', and 'It will shrink' do not go well together. According to the evidential account, this amounts to saying that (1) passes the Chrysippus Test: in the closest worlds in which its antecedent is true, its consequent is also true, and in the closest worlds in which its consequent is false, its antecedent is also false. Now consider the following sentence, which is definitely less compelling: (2) If it's pure cashmere, the EU will not collapse next week In this case there is no incompatibility between 'It's pure cashmere' and 'The EU will collapse next week', which amounts to saying that (2) does not pass the Chrysippus Test. While the Ramsey Test is satisfied, the Reverse Ramsey Test is not satisfied. Even if the closest worlds in which the antecedent is true and the consequent is false may well be distant, they are no more distant than the closest worlds in which the antecedent and the consequent are both false. 4 The evidential account significantly differs from the traditional Stalnaker-Lewis account, which is based solely on the Ramsey Test. This emerges clearly from the examples above, since both (1) and (2) pass the Ramsey Test. However, there is an interesting link between the two accounts. Let > stand for the Stalnaker-Lewis conditional. As long as the truth conditions of α > are defined in terms of the Ramsey Test, and the truth conditions of α are defined in terms of the Chrysippus Test, it can be shown that any sentence of the form α is equivalent to a complex sentence whose only conditional symbol is >, and any sentence of the form α > is equivalent to a complex sentence whose only conditional symbol is . This link is the cornerstone of the reasoning articulated in what follows.
We will start from a well known fact: given a language that includes > in addition to the usual sentential connectives ∼, ⊃, ∧, ∨, Lewis' system VC is sound and complete with respect to his centered sphere semantics. The axioms of VC are all the formulas obtained by substitution from propositional tautologies and all the formulas that instantiate the following schemas 5 : Given that VC is sound and complete with respect to Lewis' centered sphere semantics, and that and > are related in the way explained, it is provable that there is a system for which is sound and complete with respect to a similar semantics. Of course, this system will not have exactly the same properties as VC, which is understandable, given that it rests on a different account of conditionals. But the principles it displays will be reducible to principles that hold in VC, modulo the relation between and >. This is a specific application of a more general method elaborated by Raidl, which can be applied to various kinds of conditionals. 6 The structure of the paper is as follows. Section 2 introduces two languages, L > and L , such that the former includes > in addition to the usual connectives ∼, ⊃ , ∧, ∨, while the latter includes . Section 3 defines two functions that guarantee the intertranslatability between L > and L . Section 4 presents an axiom system in L called EC. Section 5 draws attention to some principles that are derivable in EC. Finally, sections 6 and 7 prove the soundness and completeness of EC by relying on the soundness and completeness of VC. §2. The languages L > and L . Let L > be a language whose alphabet is constituted by a set of sentence letters p, q, r, ... , the connectives ∼, ⊃, ∧, ∨, >, and the brackets (, ). The formation rules of L > are as follows: the sentence letters are formulas; if α is a formula, then ∼α is a formula; if α and are formulas, then α ⊃ , α ∧ , α ∨ , α > are formulas. Let L be a language whose alphabet is constituted by the same sentence 5 Compare Lewis [8]. The form of the axioms and rules comes from the conditional logic of Chellas [1], most of the labeling from the non-monotonic reasoning tradition, see Kraus, Lehmann, and Magidor [7]. 6 Raidl [10]. letters p, q, r, ... , the connectives ∼, ⊃, ∧, ∨, , and the brackets (, ). The formation rules of L are as follows: the sentence letters are formulas; if α is a formula, then ∼α is a formula; if α and are formulas, then α ⊃ , α ∧ , α ∨ , α are formulas. Basically, L differs from L > only in that its alphabet includes instead of >. Their shared fragment, call it L, is a classical propositional language.
From the semantic point of view, L > and L are alike in some respects, while they differ in one important respect, namely, the interpretation of the non-material conditional symbol. First let us see what they have in common.
Definition 1. Given a non-empty set W, a system of spheres O over W is an assignment to each w ∈ W of a set O w of non-empty sets of elements of W -a set of spheres around w-such that: Clause 1 says that O w is nested. This condition is essential, otherwise we could have two spheres S, S and two worlds w , w such that w ∈ S but w / ∈ S and w ∈ S but w / ∈ S. Clause 2 implies that O w is centered on w. Since {w} ∈ O w , because we assume spheres to be non-empty, by clause 1 we have that, for every S ∈ O w , {w} ⊆ S. This means that w belongs to every sphere around w. Clause 3 states the limit assumption, according to which, for every antecedent which is true in some sphere, there is a smallest sphere in which the antecedent is true somewhere: getting closer and closer, we eventually reach a limit. 7 Definition 2. A model for L > and L is an ordered triple W, O, V , where W is a non-empty set, O is a system of spheres over W, and V is a valuation function such that, for each sentence letter α and each w ∈ W , V (α, w) ∈ {1, 0}.
L > and L have the same models, so they are exactly alike in this respect. The part of the semantics in which they differ is the definition of truth of a formula in a world. The truth of a formula α of L > in a world w, which we will indicate as w > α, is defined as follows, where [α] > is the set of worlds in which α is true according to > : We will use the notation > α to say that α is valid, that is, true in every world in every model.
The truth of a formula α of L in a world w, which we will indicate as w α, is defined as follows, where [α] is the set of worlds in which α is true according to : As in the case of L > , we will use the notation α to say that α is valid, that is, true in every world in every model.
Clauses 1 -5 are exactly like clauses 1-5. This means that, as far as L is concerned, Definitions 3 and 4 yield the same results: Proof. The proof is by induction on the complexity of .
Basis. Consider the case in which is a sentence letter. In this case w > iff w because both hold exactly when V ( , w) = 1.
Step. Assume that the equivalence holds for any formula of complexity less than or equal to n, and that has complexity n + 1. Then there are four possible cases, depending on whether the main connective of is ∼, ∧, ∨, or ⊃. In each of these cases, given the induction hypothesis, it follows that w > iff w .
The key difference between Definitions 3 and 4 is that the former includes clause 6, which expresses the Ramsey Test, while the latter includes clause 6 , which expresses the Chrysippus Test. Clause 6, just as condition (a) of clause 6 , requires that if α is true in some sphere, then the closest worlds in which it is true are worlds in which is true. Instead, condition (b) of clause 6 requires that if is false in some sphere, then the closest worlds in which it is false are worlds in which α is false. §3. Translation and backtranslation. The link between and > can be stated in precise terms by defining a translation function • that goes from L to L > and a backtranslation function • that goes from L > to L .
Clauses 1-5 entail that, whenever a formula α belongs to L, α • = α. In particular, it entails that, if is defined as any propositional tautology and ⊥ = ∼ , then • = and ⊥ • = ⊥. Clause 6 is the crucial one. What it says is that for every formula of the form α , there is a formula of L > that translates it, namely, The function • is well-behaved in the following sense: Fact 2. For every model, every world w, and every formula of L , w iff w > • .
Proof. The proof is by induction on the complexity of .
Basis. Consider the case in which is a sentence letter. In this case • = . By fact 1 it follows that w iff w > • .
Step. Assume that the equivalence holds for any formula of complexity less than or equal to n, and that is a formula of complexity n + 1. Then five cases are to be considered, depending on whether the main connective of is ∼, ∧, ∨, ⊃, or . In the first four cases, given the induction hypothesis, we have that w iff w > • because clauses 2-5 of Definition 3 are exactly like clauses 2 -5 of Definition 4. So, the only case to be considered is that in which has the form α . In this case, . This means that conditions (a) and (b) of clause 6 of Definition 4 hold for α and . By the induction hypothesis it follows that the same conditions hold for α • and • , with ] > replacing ] . So, w > α • > • and w > ∼ • > ∼α • , which entails that w > • . A similar reasoning in the opposite direction shows that if w > • , then w .
Definition 6. Let • be a function from L > to L such that: Clauses 1-5 entail that, whenever a formula α belongs to L, α • = α. In particular, it entails that • = and ⊥ • = ⊥. Clause 6 says that, for every formula of the form α > , there is a formula that provides the translation of α > into L , namely, The function • is well-behaved exactly in the same sense in which • is well-behaved, although we will not prove this fact here, given that it is not necessary for the soundness and completeness results to be established. §4. The system EC. Now we will present an axiom system of conditional logic in L . We call this system EC, for 'evidential conditional'. The axioms of EC are all the formulas obtained by substitution from propositional tautologies and all the formulas that instantiate the following schemas: ) ∧ (α ∨ (α (α ∧ )))) ⊃ (α ) D ID, AND, CM, and MI, are exactly as in VC, so they express properties that shares with >. OR * and RM * , the ugliest axioms in this list, are counterparts of OR and RM modulo Definition 6, that is, they are obtained by applying the function • to OR and RM and adding some elementary transformations. IA stands for Impossible Antecedent. This principle does not occur as an axiom in VC, although it is derivable in VC. Since VC ⊥ ⊃ , by RW it follows that VC (α > ⊥) ⊃ (α > ). Finally, C and D characterize as distinct from >. C stands for Contraposition. Although there is no widespread agreement on this principle, and the Stalnaker-Lewis account provides a clear rationale for thinking that > is not contrapositive, C turns out valid insofar as the truth conditions of are defined in terms of incompatibility in the way suggested here. 8 D is more complex and has no overt intuitive meaning, although it is needed for technical reasons. 9 The rules of inference of EC are MP and the following, where EC indicates provability in EC: LLE is as in VC. RLE is weaker than RW, in that it requires that α and are provably equivalent. §5. Derivable principles. In this section we will draw attention to some well known principles of conditional logic that are derivable in EC, in order to provide further clarifications about the properties of .
Let us start with OR. As we have seen, OR occurs as an axiom in VC, while EC includes its ugly counterpart OR * . However, OR is derivable in EC. A second principle, which is derivable both in VC and in EC, is Super-Classicality: Proof. Assume that EC α ⊃ . Then, EC ≡ ( ∨ α). Since by ID, we get ( ∨ α) by LLE. Moreover, α α by ID, and consequently α ∨ (α α). Since EC α ≡ (α ∧ ), we get α ∨ (α (α ∧ )) by RLE. Given D, we obtain α .
A third principle, which is derivable both in VC and in EC, is Necessary Consequent: Proof. Assume that ∼ ⊥. Then ∼ ∼α by IA. Given C, ∼∼α ∼∼ , which yields α by LLE and RLE. 8 Stalnaker [14], p. 39, and Lewis [9], p. 35, motivated their rejection of Contraposition by means of examples. These have been widely debated in the subsequent literature on conditionals, see for example Tichý [15] and Gomes [6]. Overall, putative counterexamples to Contraposition seem to be more controversial than the well known counterexamples to Monotonicity or Transitivity. Crupi and Iacona [2] provides a more detailed discussion of C. 9 Raidl [10] calls C and D "proper axioms," and explains their role in more general terms.
Three further principles, which have been widely discussed in the literature on connexive logics, are Restricted Abelard' Thesis, Restricted Aristotle's Thesis, and Restricted Aristotle's Second Thesis.
Proof. Suppose α and α ∼ . Then α ⊥ by AND and RLE, contrary to the first conjunct of the antecedent.
Proof. Suppose α ∼α. Since α α by ID, we get α ⊥ by AND and RLE, contrary to the antecedent.
Proof. Suppose that α and ∼α . Then by OR and LLE. Given C, from this we get ∼ ⊥, contrary to the first conjunct of the antecedent.
While RBT and RAT hold both for and for >, RAST characterizes as distinct from >. 10 We will end with two principles which hold in EC and are weaker than RW and RM respectively.
Basis. Assume that there is a proof of of length 1. In this case is an axiom. Ten cases are possible.
Case 1: is obtained by substitution from a propositional tautology α, where α ∈ L. In this case is the result of replacing in α a set of sentence letters p 1 , ... , p n with a set 1 , ... , n of formulas of L , and it is provable that • is the formula obtained by replacing 1 , ... , n with • 1 , ... , • n . It follows that • is an axiom of VC. Case 2: is an instance of ID. In this case • is a conjunction (α > α) ∧ (∼α > ∼α), and both conjuncts are provable in VC in virtue of ID.
Step. Assume that the condition holds for every proof of length less than or equal to n, and consider a proof of of length n + 1. Then four cases are possible.
Case 1: is an axiom. In this case we know that VC • , given what has been said in the basis.
Case 3: is obtained by means of RLE. This case is analogous to case 2.
Case 4: is obtained by means of MP. In this case is preceded by two formulas ⊃ and in the proof. By the induction hypothesis, • ⊃ • and • are provable in VC, so the same goes for • , given that VC has MP. Theorem 1. If EC , then .
Proof. Assume that EC . Then, by fact 11, VC • . Since VC is sound, it follows that > • . By fact 2, this entails that . §7. Completeness of EC. In this last section we will prove that EC is complete by relying on the fact that VC is complete. First we will show that • inverts • in EC, namely, that for any formula of L , the backtranslation of the translation of , that is, •• , is provably equivalent to in EC. Then we will show that, for any formula of L > , if VC , then EC • . The combination of these two results yields the converse of fact 11, which suffices to establish the completeness of EC.