An Axiomatic System for Concessive Conditionals

. According to the analysis of concessive conditionals suggested by Crupi and Iacona, a concessive conditional p (cid:2) → q is adequately formalized as a conjunction of conditionals. This paper presents a sound and complete axiomatic system for concessive conditionals so understood. The soundness and completeness proofs that will be provided rely on a method that has been employed by Raidl, Iacona, and Crupi to prove the soundness and completeness of an analogous system for evidential conditionals.


Overview
Concessive conditionals, which are typically indicated by the expression 'even if', exhibit distinctive logical features that set them apart from ordinary indicative conditionals.Imagine that the sentences (1) and (2) below are used in a situation in which Glen intends to go out for a walk and hopes for a sunny day: (1) If the weather is good, Glen will go out (2) Even if the weather is not good, Glen will go out In this case, (2) differs from (1) in at least three important respects.First, from (2) one can reasonably infer that Glen will go out, because (2)  seems to imply that Glen will go out no matter whether the weather is good.By contrast, (1) does not convey a similar claim, for it leaves unspecified what Glen will do in case of bad weather.Second, (2) seems to imply that 'the weather is not good' does not support 'Glen will go out'.It would be inappropriate to paraphrase (2) by saying that if the weather is bad, that is a reason for thinking that Glen will go out, or that if the weather is bad, then as a consequence Glen will go out.By contrast, such a paraphrase is first developed by Raidl in a general manner and then employed by Raidl, Iacona, and Crupi for a similar system for evidential conditionals. 4he structure of the paper is as follows.Section 2 introduces two languages, L > and L → , which differ only in that the former includes > in addition to the usual sentential connectives while the latter includes →.Section 3 defines two functions that guarantee the intertranslatability between L > and L → .Section 4 presents the well known system VC in L > , and some derivable principles.Section 5 presents an axiom system in L → that we call CC.Section 6 draws attention to some principles that are derivable in CC.Finally, Sections 7 and 8 prove the soundness and completeness of CC by relying on the soundness and completeness of VC.
Let L → be a language whose alphabet is constituted by the same 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 β are formulas, then (α ⊃ β), (α ∧ β), (α ∨ β), (α → β) 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.We adopt the convention of not writing outer brackets in formulas.We will use the notation α ≡ β to abbreviate (α ⊃ β) ∧ (β ⊃ α), and we will use to refer to any classical propositional tautology, and ⊥ to refer to any classical propositional antilogy.
The semantics of L > and L → will be given in terms of systems of spheres, along the lines suggested by Lewis. 5 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: any two spheres in O w are such that one of them includes the other.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.
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.Let us start with L > .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 > : Definition 3.
1. w > α iff V (α, w) = 1, for any sentence letter α; Note that, given clauses 2 and 6, the necessity operator is definable in L > as follows: α = ¬α > α.To see why, assume that O w ∩ [¬α] > = ∅, which is the truth condition for α relative to w.Then, w > ¬α > α by the first disjunct of clause 6.Conversely, assume that w > ¬α > α.Then the first disjunct of clause 6 must hold, that is, O w ∩ [¬α] > = ∅, given that the second cannot hold by clause 2. So, from now on we will take for granted that α abbreviates ¬α > α, and we will write for the dual ¬ ¬.Now let us consider L → .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 → : Definition 4.
1. w → α iff V (α, w) = 1 for any sentence letter α; Clauses 1-5 of Definition 4 are exactly like clauses 1-5 of Definition 3.This means that, as far as L is concerned, Definitions 3 and 4 yield the same results: Fact 5.For every model, every world w, and every formula χ of L, w > χ iff w → χ.
Proof.The proof is by induction on the complexity of χ.
The key difference between Definitions 3 and 4 lies in clause 6.While clause 6 of Definition 3 requires that the closest worlds in which α is true are worlds in which β is true, clause 6 of Definition 4 adds to this conditionexpressed by (a)-two further conditions, (b) and (c).(b) requires that the closest worlds in which ¬α is true are worlds in which β is true, and (c) requires that the closest worlds in which ¬β is true are worlds in which α is true.In other words, while clause 6 of Definition 3 expresses the Ramsey Test for α > β, clause 6 of Definition 4 expresses the Ramsey test for α > β augmented by the Chrysippus Test for ¬α β.
As in the case of L > , the necessity operator is definable as follows: Then the first disjunct of (a) must hold, that is, O w ∩ [¬α] → = ∅, given that the second disjunct cannot hold by clause 2. So, from now on we will take for granted that, in L → , α abbreviates ¬α → α, and we will again write for the dual ¬ ¬.
The notation that will be used for validity is the same for both languages.
> α means that α is true in every world in every model according to Definition 3. Similarly, → α means that α is true in every world in every model according to Definition 4.

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 → .Definition 6.Let • be the function from L → to L > such that: Clauses 1-5 entail that, whenever a formula α belongs to L, α • = α.Clause 6 says that, for every formula of the form α → β, there is a formula of L > that translates it, namely, (α The function • is well-behaved in the following sense: Fact 7.For every model, every world w, and every formula Proof.The proof is by induction on the complexity of χ.
Basis.Consider the case in which χ is a sentence letter.In this case χ • = χ.Then, by Fact 5, w → χ iff w > χ • .
Step.Assume that the equivalence holds for any formula of complexity less than or equal to n, and that χ is a formula of complexity n + 1.Then five cases are to be considered, depending on whether the main connective of χ is ¬, ∧, ∨, ⊃, or →.In the first four cases, given the induction hypothesis, w → χ iff w > χ • because clauses 2-5 of Definition 3 are exactly like clauses 2-5 of Definition 4. So, the only case left is that in which χ has the form α → β.In this case, ). Assume that w → χ.This means that conditions (a)-(c) of clause 6 of Definition 4 hold for α and β.By the induction hypothesis, the same conditions hold for α • and β A similar reasoning in the opposite direction shows that if w > χ • , then w → χ.
Definition 8. Let • be the function from L > to L → such that: Clauses 1-5 entail that, whenever a formula α belongs to L, α • = α.Clause 6 says that, for every formula of the form α > β, there is a formula that provides the translation of To grasp the meaning of this clause it suffices to think that, assuming that α • and β • belong to L, and thus simplify to α and β, its right-hand side can be rewritten in , which in turn is equivalent to α > β given Definition 3. 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 our purposes.

The System VC
As is well known, given a language that includes > in addition to the sentential connectives ¬, ⊃, ∧, ∨, Lewis' system VC is sound and complete with respect to his centered sphere semantics. 6The axioms of VC are all the formulas obtained by substitution from propositional tautologies (PT) and all the formulas that instantiate the following schemas: C S The principles expressed by these schemas are respectively Identity, Conjunction of Consequents, Disjunction of Antecedents, Cautious Monotonicity, Rational Monotonicity, Material Implication, and Conjunctive Sufficiency.
The rules of inference of VC are MP -the standard Modus Ponens for ⊃ -plus Left Logical Equivalence and Right Weakening:

RW
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, 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 >.
In the rest of this section we will briefly recall some useful facts about VC.First of all, the following principles are derivable in VC.The principles expressed by the first two facts are Supraclassicality and Right Logical Equivalence.CN says that conditionals with tautological consequents always hold.N says that tautologies are necessary.TI says that truth implies the inner modality , where α = ( > α).Note in fact that in VC, the inner modality collapses with truth, due to MP and CS.The outer modality , instead, is definable in the way explained in section 2, that is, α = (¬α > α).
The next principle, Necessary Consequent, is phrased in terms of the latter definition, as it says that conditionals with necessary consequents always hold.Let us close with the following principle, which may be called Modularity: To understand the name of this principle it suffices to think that, according to Definition 3, (α ∨ β) > ¬β is true when α precedes β in the sense that the closest worlds in which α is true come before the closest worlds in which β is true (unless α is impossible, in which case (α ∨ β) > ¬β says that β is impossible as well).So the principle says that if α precedes β, then either α precedes γ or γ precedes β.This is precisely the order property called Modularity.

The System CC
Now we will present an axiom system in L → called CC, which stands for 'concessive conditional'.The axioms of CC are all the formulas obtained by substitution from propositional tautologies (PT) and all the formulas that instantiate the following schemas: AND is exactly as in VC, and N, WDW, TI, NC, and Mod are derivable in VC, as it turns out from Facts 12-15.Some of these axioms can be seen as weak replacements of principles from VC.For example, WDW is a weak replacement of RW.WDW is derivable from RW, but the reverse is not true. 8imilarly TI is a weak replacement of CS, in that it is derivable from CS, but for the converse we need CM.This is why CS is not derivable in CC.Other principles are obtained by backtranslating principles from VC. Mod is a backtranslation of Mod. 9 The axioms presented so far express properties that → shares with >.The remaining two axioms CT and CSS, instead, express principles that do not hold in VC. 10 CT expresses the intuition illustrated in Section 1 that a concessive conditional implies its consequent.This axiom replaces MI, but it is stronger than MI: if β holds, then α ⊃ β holds, but not the other way round.This, among other things, makes MI derivable in CC, while CT is not derivable in VC.CSS has no obvious intuitive meaning.It is required for technical purposes and encodes the semantic definition of the concessive conditional.Indeed, if we rewrite CSS by replacing first α and β by α • and β • and then applying the backtranslation to the left-hand side, we get the definition of → in terms of >. 11 The rules of inference of CC are MP and the following:

RLE
LLE is as in VC.RLE is weaker than RW, in that it requires that α and β are provably equivalent.

Derivable Principles
This section draws attention to five principles that are derivable in CC, and that will prove useful in the following sections.The first principle (CN) also holds in VC (Fact 11).The second, the third, and the last of these principles are respectively backtranslations of CM, RM, and OR (which we denote by CM*, RM*, OR*).To these five principles, we add three connexive principles-Restricted Aristotle's Thesis (RAT), weak Boethius Thesis (wBT), and Restricted Aristotle's Second Thesis (RAT2)-which we also show to be derivable.At the end of the section, we provide a Table 1 with a list of further (in)valid principles for the concessive → in comparison to the suppositional conditional >. 9 WDW is a backtranslation of DW. 10 Parts of CSS are however valid in semantics for VC.The valid part of CSS is the left to right implication, and the first reverse implication (α > β) ⊃ (α > (¬α ∨ β)).The two other reverse implications are invalid.That is, the following principle survives in VC: 11 According to the terminology adopted in Raidl [9], CSS is the proper axiom of CC.
In CC we could replace N by CN.This principle backtranslates ID.Note that Definition 4 does not validate ID, for conditions (b) and (c) of clause 6 are not satisfied whenever α is not necessary.This is quite plausible, for it would make little sense to take sentences such as 'Even if the weather is not good, the weather is not good' to be valid.

Soundness of CC
Now we will prove that CC is sound by relying on the fact that VC is sound.
The key result we need is the following, where χ is any formula of L → : Proof.The proof is by induction on the length of the proof of χ in CC.Basis.Assume that there is a proof of χ of length 1.In this case χ is an axiom.Nine cases are possible.Case 1: For a general proof of this case, see Lemma 1 in Raidl [8].
In VC, ¬ > holds in virtue of N (Fact 12), ¬¬ > in virtue of ID and LLE, and ¬ > ¬ in virtue of ID.Case 3: χ is an instance of AND.In this case χ • is equivalent to a material conditional with antecedent (α ) and invalid (×) principles for the concessive ( →) and the suppositional (>) conditional Form for And we can reverse the reasoning, by using RW.
Step.Assume that the condition holds for every proof of length less than or equal to n, and consider a proof of χ of length n + 1.Then four cases are possible.
Case 1: χ is an axiom.In this case we know that VC χ

Completeness of CC
In this last section we will prove that CC is complete by relying on the fact that VC is complete.First we will show that • inverts • in CC, namely, that for any formula χ of L → , the backtranslation of the translation of χ, that is, χ •• , is provably equivalent to χ in CC.Then we will show that, for any formula χ of L > , if VC χ, then CC χ • .The combination of these two results yields the converse of Fact 25, which suffices to establish the completeness of CC.
The proof is by induction on the complexity of χ.
Step.Assume that the condition holds for every formula of complexity less than or equal to n, and that χ has complexity n + 1.Then five cases are to be considered, depending on whether the main connective of χ is ¬, ∧, ∨, ⊃, or →.In the first four cases, given the induction hypothesis, we obtain that CC χ •• ≡ χ.
Case 4: χ is obtained by means of MP.In this case χ is preceded by two formulas δ ⊃ χ and δ in the proof.By the induction hypothesis, δ • ⊃ χ • and δ • are provable in VC, so the same goes for χ • , given that VC has MP.Theorem 26.If CC χ, then → χ.Proof.Assume that CC χ.Then, by Fact 25, VC χ • .Since VC is sound, it follows that > χ • .By Fact 7, this entails that → χ.