Gibbardian Collapse and Trivalent Conditionals
aa r X i v : . [ m a t h . L O ] J un Gibbardian Collapse and Trivalent Conditionals
Paul Égré * Lorenzo Rossi † Jan Sprenger ‡ Abstract
This paper discusses the scope and significance of the so-called triviality resultstated by Allan Gibbard for indicative conditionals, showing that if a conditionaloperator satisfies the Law of Import-Export, is supraclassical, and is stronger thanthe material conditional, then it must collapse to the material conditional. Gib-bard’s result is taken to pose a dilemma for a truth-functional account of indicativeconditionals: give up Import-Export, or embrace the two-valued analysis. Weshow that this dilemma can be averted in trivalent logics of the conditional basedon Reichenbach and de Finetti’s idea that a conditional with a false antecedent isundefined. Import-Export and truth-functionality hold without triviality in suchlogics. We unravel some implicit assumptions in Gibbard’s proof, and discuss arecent generalization of Gibbard’s result due to Branden Fitelson.
Keywords: indicative conditional; material conditional; logics of conditionals; triva-lent logic; Gibbardian collapse; Import-Export
The Law of Import-Export denotes the principle that a right-nested conditional of theform A → ( B → C ) is logically equivalent to the simple conditional ( A ∧ B ) → C whereboth antecedents are united by conjunction. The Law holds in classical logic for materialimplication, and if there is a logic for the indicative conditional of ordinary language,it appears Import-Export ought to be a part of it. For instance, to use an example from(Cooper 1968, 300), the sentences “If Smith attends and Jones attends then a quorum * Institut Jean-Nicod (CNRS / ENS / EHESS), Département de philosophie & Département d’études cog-nitives, Ecole normale supérieure, PSL University, 29 rue d’Ulm, 75005, Paris, France. ORCID: 0000-0002-9114-7686. Email: [email protected] † Munich Center for Mathematical Philosophy (MCMP), Fakultät für Philosophie, Wissenschaftstheorieund Religionswissenschaft, Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, D-80539 München. ORCID: 0000-0002-1932-5484. Email: [email protected] ‡ Center for Logic, Language and Cognition (LLC), Department of Philosophy and Educational Science,Università degli Studi di Torino, Via Sant’Ottavio 20, 10124 Torino, Italy. ORCID: 0000-0003-0083-9685.Email: [email protected] A , B and C themselves are non-conditional sentences, and theequivalence has been described as “a fact of English usage” (McGee 1989). In a celebrated paper, however, Allan Gibbard (1980) showed that a binary condi-tional connective ‘ → ’ collapses to the material conditional of classical logic ‘ ⊃ ’ if thefollowing conditions hold: (i) the conditional connective satisfies Import-Export, (ii) itis at least as strong as the material conditional ( A → C | = L A ⊃ C ), where | = L is the con-sequence relation of the target logic of conditionals, (iii) it is supraclassical in the sensethat it reproduces the valid inferences of classical logic in conditional form ( | = L A → C whenever A | = CL C ). From (i)–(iii) and some natural background assumptions, Gibbardinfers A ⊃ C | = L A → C . Given (ii), → and ⊃ are thus logically equivalent, accordingto the logic of conditionals ( | = L ) under consideration. Prima facie, the conditional thenneeds to support all inference schemes validated by the material conditional in classicallogic. However, inferences such as ¬ A | = A → C (one of the paradoxes of materialimplication) enjoy little plausibility in ordinary reasoning with conditionals.Gibbard’s result poses a challenge for theories that compete with material implica-tion as an adequate analysis of the indicative conditional. For example, Stalnaker’slogic C2 (Stalnaker 1968) and Lewis’s logic VC (Lewis 1973) are both supraclassicaland make the conditional stronger than the material conditional, but they invalidateImport-Export for that matter.Not all theories make that choice, however. All of the above logics operate in a bivalent logical setting, thus limiting their options. In this paper, we explore how certain trivalent logics of conditionals address Gibbard’s challenge. These logics, which retaintruth-functionality, analyze an indicative conditional of the form “if A then C ” as a conditional assertion that is void if the antecedent turns out to be false, and that takesthe truth value of the consequent C if A is true (Reichenbach 1935; de Finetti 1936;Quine 1950; Belnap 1970). This analysis assigns a third truth value (“neither true norfalse”) to such “void” assertions, and gives rise to various logics that combine a truth-functional conditional connective with existing frameworks for trivalent logics (e.g.,Cooper 1968; Farrell 1979; Milne 1997; Cantwell 2008; Baratgin, Over, and Politzer 2013;Égré, Rossi, and Sprenger 2020a,b).This chapter clarifies the scope and significance of Gibbardian collapse results withspecific attention to such trivalent logics, in which the conditional is undefined when itsantecedent is false. We begin with a precise explication of Gibbard’s result, including Import-Export has been challenged on linguistic grounds, see for instance Khoo and Mandelkern(2019), drawing on examples from Fitelson. The alleged counterexamples are subtle, however, and evenKhoo and Mandelkern accept a version of the law. See also Appendix A. Notable defenders of the material implication analysis are Lewis (1976a), Jackson (1979) and Grice(1989). uniqueness result : we cannot have two conditional connectives that satisfy Import-Exportas well as Conjunction Elimination, where one is strictly stronger than the other, andwhere the weaker (already) satisfies Modus Ponens. We also provide three appendices:Appendix A rebuts a recent attempt at a reductio of Import-Export, Appendix B providesthe proofs of various lemmata stated in the paper, and Appendix C gives a moreconstrained derivation of Gibbardian collapse than his original proof, of particularrelevance for the first trivalent system we discuss. For more in-depth treatment oftrivalent logics of conditionals, we refer the reader to our comprehensive survey andanalysis in Égré et al. (2020a,b).
The Law of Import-Export is an important bridge between di ff erent types of condition-als: it permits to transform right-nested conditionals into simple ones. Import-Export isof specific interest in suppositional accounts of indicative conditionals that assess the as-sertability of a conditional by the corresponding conditional probability (as per Adams’thesis, viz. Adams 1965). Import-Export is then an indispensable tool for providing aprobabilistic analysis of embedded conditionals. However, when Adams’ Thesis, orig-inally limited to conditionals with Boolean antecedent and consequent, is extended tonested conditionals, Import-Export creates unexpected problems. For example, a fa-mous result by David Lewis (1976b) shows that combining this latter equation with theusual laws of probability and an unrestricted application of Import-Export trivializes theprobability of the indicative conditional. Gibbard establishes a second di ffi culty with The unrestricted version of Adams’s equation is often called Stalnaker’s Thesis (going back to Stalnaker1970) or simply “The Equation”, with the latter name being prevalent in the psychological literature. Adamsdefends it in his 1975 monograph, too. On the reasons to defend Import-Export in relation to probabilities of conditionals, see McGee (1989)and Arló-Costa (2001). A discussion of the links between Gibbardian collapse and Lewisian triviality liesbeyond the scope of this paper, but we refer to Lassiter (2019) for a survey of Lewisian triviality results | = CL for classical consequence, and ≡ L for the conjunction of | = L and itsconverse. Under (v) we mean that ⊃ obeys a classical law whenever it obeys a classicalinference or a classical metainference. Theorem 1 (Gibbard) . Suppose L is a logic whose consequence relation | = L is at leasttransitive, with ⊃ and → two binary operators, obeying principles (i)-(v) for every formulaeA , B , C. Then → and ⊃ are provably equivalent in L.(i) A → ( B → C ) ≡ L ( A ∧ B ) → C Import-Export(ii) A → B | = L ( A ⊃ B ) Stronger-than-Material(iii) If A | = CL B, then | = L A → B Supraclassicality(iv) If A ≡ L A ′ then A → B ≡ L A ′ → B Left Logical Equivalence(v) ⊃ obeys classical laws in L Classicality of ⊃ Proof. ( A ⊃ B ) → ( A → B ) ≡ L (( A ⊃ B ) ∧ A ) → B by (i)2. (( A ⊃ B ) ∧ A ) ≡ L ( A ∧ B ) by (v) (classical inferences)3. ( A ⊃ B ) ∧ A ) → B ≡ L ( A ∧ B ) → B by 2 and (iv)4. ( A ∧ B ) → B ≡ L ( A ⊃ B ) → ( A → B )
1, 3 and the transitivity of | = L A ∧ B | = CL B Conjunction Elimination6. | = L ( A ∧ B ) → B | = L ( A ⊃ B ) → ( A → B )
4, 6, and the transitivity of | = L ( A ⊃ B ) → ( A → B ) | = L ( A ⊃ B ) ⊃ ( A → B ) by (ii)9. | = L ( A ⊃ B ) ⊃ ( A → B )
7, 8 and the transitivity of | = L A ⊃ B | = L A → B by 9 and (v) (classical metainference) (cid:3) This is not the only proof of Gibbard’s result. In particular Fitelson (2013) andKhoo and Mandelkern (2019) give more parsimonious derivations. But it closelymatches the structure of his original argument: first Gibbard shows that ( A ⊃ B ) → and their treatment in a trivalent framework. An inference is a relation between (sets of) formulae: for instance the relation between ( A ⊃ B ) ∧ A and A ∧ B ; a metainference is a relation between inferences, for example the relation between A | = B and | = A ⊃ B . A → B ) is a theorem of L (step 1–7), from that he derives | = L ( A ⊃ B ) ⊃ ( A → B ) (step8–9) and finally, he infers A ⊃ B | = L A → B (step 10).With Gibbard we can grant that the assumptions (ii) and (iii) introduced alongsideImport-Export are fairly weak. Stronger-than-Material is shared by all theories thatclassify an indicative conditional with true antecedent and false consequent as false. Supraclassicality, a restricted version of the principle of Conditional Introduction, meansthat deductive relations are supported by the corresponding conditional. Even thatcould be weakened by just assuming the conditional to support conjunction eliminationas in step 6. In section 6 we discuss more general conditions for Gibbardian collapseproposed by Branden Fitelson (2013).Assumptions (iv) and (v), on the other hand, are stronger than meets the eye.While the substitution rule LLE was taken for granted by Gibbard, likely on groundsof compositionality, it raises issues in relation to counterpossibles and other forms ofhyperintensionality (see Nute 1980; Fine 2012). However, even if one is inclined togive up principle (iv), one may not find fault with applying it in this particular case.Similarly, (v) implies that the material conditional supports classical absorption laws(step 2 of the proof) and (meta-inferential) Modus Ponens (step 10) in L — two propertiesnot necessarily retained in non-classical logics.Gibbard’s result also leaves a number of questions unanswered. One of them con-cerns the implication of the mutual entailment between → and ⊃ . Does the collapseimply that the two conditionals can be replaced by one another in all contexts, for ex-ample? The answer to this question is in fact negative, as we proceed to show usingtrivalent logic in the next section. From his result, Gibbard drew the lesson that if we want the indicative conditional to bea propositional function, and to account for a natural reading of embedded indicativeconditionals, then the function must be ‘ ⊃ ’, namely the bivalent material conditional. Wedisagree with this conclusion: trivalent truth-functional accounts of the conditional cansatisfy Import-Export and yield a reasonable account of embeddings without collapsingto the material conditional. We now explain why one may want to adopt such anapproach, and then, in the next two sections, how they deal with Gibbard’s result.Reichenbach and de Finetti proposed to analyze an indicative conditional “if A , then The name MP is sometimes used for this principle, see Unterhuber and Schurz (2014), orKhoo and Mandelkern (2019) who call it Modus Ponens. We find more appropriate to use ‘Stronger-than-Material’ since Modus Ponens is strictly speaking a two-premise argument form. The two principlesare not necessarily equivalent: in the system
DF/TT for instance, Stronger-than-Material holds but notModus Ponens (in the form A → B , A | = B ). → DF /
01 1 / / / / / / / / f → CC /
01 1 / / / / / / Table 1:
Truth tables for the de Finetti conditional (left) and the Cooper-Cantwell conditional (right). C ” as an assertion about C upon the supposition that A is true. Thus the conditionalis true whenever A and C are true, and false whenever A is true and C is false. Whenthe supposition ( = the antecedent A ) turns out to be false, there is no factual basis forevaluating the conditional statement, and therefore it is classified as neither true norfalse. This basic idea gives rise to various truth tables for A → C . Two of them arethe table proposed by Bruno de Finetti (1936) and the one proposed independently byWilliam Cooper (1968) and John Cantwell (2008) (see Table 1). In both of them the value / can be interpreted as “neither true nor false”, “void”, or “indeterminate”. There ismoreover a systematic correspondence and duality between those tables: whereas deFinetti treats “not true” antecedents ( <
1) in the same way as false antecedents ( = >
0) in the same way as true ones( = f ¬ / / f ∧ /
01 1 / / / /
00 0 0 0 f ⊃ /
01 1 / / / / Strong Kleene truth tables for negation, conjunction, and the material conditional.
One way to define the other logical connectives is via the familiar Strong Kleenetruth tables (see Table 2). Conjunction corresponds to the “minimum” of the twovalues, disjunction to the “maximum”, and negation to inversion of the semantic value.In particular, beside the indicative conditional A → C , the trivalent analysis also admitsa Strong Kleene “material” conditional A ⊃ C , definable as ¬ ( A ∧ ¬ C ) (see again Table2). To make a logic, however, we also need a definition of validity. This question isnon-trivial in a trivalent setting since preservation of (strict) truth is not the same aspreservation of non-falsity. Like Cooper and Cantwell, and based on independentarguments, we opt for a tolerant-to-tolerant ( TT- ) consequence relation where non- All other consequence relations come with problematic features (Fact 3.4 in Égré et al. 2020a): they A | = C is valid if, for any evaluation function (of theappropriate kind) v from the sentences of the language to the values { / , 1 } , whenever v ( A ) ∈ { / , 1 } , then also v ( C ) ∈ { / , 1 } . This choice yields two logics depending on howthe conditional is interpreted: the logic DF/TT based on de Finetti’s truth table, and thelogic
CC/TT based on the Cooper-Cantwell table. Both logics make di ff erent predictions, but they agree on a common core, and theygive a smooth treatment of nested conditionals. In particular both DF/TT and
CC/TT satisfy the Law of Import-Export. We now investigate how they deal with Gibbardiancollapse.
DF/TT and
CC/TT
We first consider Gibbard’s triviality result in the context of
DF/TT with its indicativeand material conditionals.
DF/TT is contractive, reflexive, monotonic and transitive.An inspection of the principles (i)–(v) in Theorem 1 shows that: • Assumption (i) holds. In particular, both sides of the Law of Import-Export receivethe same truth value in any DF -evaluation. • Assumption (ii) also holds: if there is a DF -evaluation v such that v ( A ⊃ B ) = v ( A ) = v ( B ) =
0, but then v ( A → B ) = A → B tolerantly true. • Assumption (iii) holds in
DF/TT . We prove this in Appendix B. • Assumption (iv) fails in DF/TT . In fact, A | = DF / TT B and B | = DF / TT A if, for any DF -evaluation v , one of the following is given:(a) v ( A ) = = v ( B ) (c) v ( A ) = v ( B ) = / (b) v ( A ) = / = v ( B ) (d) v ( A ) = / ; v ( B ) = v ( C ) =
0, cases (c) and (d) provide counterexamples since either A → C = DF / TT B → C or B → C = DF / TT A → C . A concrete example is thefollowing: p ∨ ¬ p | = DF / TT ( p → ¬ p ) ∨ ( ¬ p → p )( p → ¬ p ) ∨ ( ¬ p → p ) | = DF / TT p ∨ ¬ p but [( p → ¬ p ) ∨ ( ¬ p → p )] → ( p ∧ ¬ p ) = DF / TT ( p ∨ ¬ p ) → ( p ∧ ¬ p ) either fail the Law of Identity (i.e., = A → A ), or they license the inference from a conditional to its converse(i.e., A → C | = C → A ). The system
CC/TT actually matches Cantwell’s system. Cooper’s logic, called OL rests on a di ff erentchoice of truth tables for conjunction and disjunction, and restricts valuations to two-valued atoms. Assumption (v) fails in general of ⊃ in DF/TT . In particular, step 2 of Gibbard’sproof fails: ( A ⊃ B ) ∧ A = DF / TT A ∧ B , assuming v ( A ) = / and v ( B ) = DF/TT irrelevant forthe discussion of his result. But this is not so: despite assumptions (iv) and (v) failingfor
DF/TT ’s indicative conditional and material conditional, the two conditionals turnout to be equivalent. More precisely,
DF/TT validates the equivalence of A ⊃ B and A → B , as a reciprocal entailment ( ≡ DT / TT ), as a material biconditional (denoted by ⊃⊂ ),and as an indicative biconditional (denoted by ↔ ). Lemma 2.
For every A , B ∈ For ( L ) :A ⊃ B ≡ DF / TT A → B | = DF / TT ( A ⊃ B ) ⊃⊂ ( A → B ) | = DF / TT ( A ⊃ B ) ↔ ( A → B ) This result in not a coincidence. As it turns out, Gibbard’s result can be derived onlyusing principles (i), (ii), (iii), (v) and structural assumptions on logical consequence, insuch a way that all uses of (v) are
DF/TT sound. This result directly follows from theversion of Gibbard’s result established by Khoo and Mandelkern (2019), as we prove inAppendix C. We also give a sequent-style proof of the collapse in Appendix B, makinguse of the system presented in our Égré et al. (2020b).However, such an extended form of equivalence between the indicative and thematerial conditional in
DF/TT does not mean that the two conditionals are identifiedwith each other or indistinguishable. In fact, they obey very di ff erent logical principles,such as the following connexive law: A → B | = DF / TT ¬ ( A → ¬ B ) but ¬ A ∨ B = DF / TT ¬ ( ¬ A ∨ ¬ B ) .This shows that indicative and material conditional cannot be validly replaced in com-plex formulae in DF/TT . Put di ff erently, DF/TT fails the classical principle of replacementof equivalents.What is, then, the import of
DF/TT ’s equivalences between di ff erent conditionals?Not much, one might argue. A look at the DF semantics and the status of the premisesof Gibbard’s Theorem in DF/TT shows that such equivalences are largely a byproductof (i) the fact that the DF truth table assigns value 0 to an indicative conditional in thesame cases in which it assigns value 0 to a material conditional, and (ii) the fact that thetolerant-tolerant consequence relation does not distinguish between value 1 and / .Notably, things are di ff erent when we move to CC/TT , keeping the tolerant-tolerantnotion of consequence fixed, but moving to a truth-table for the conditional which8ssigns value 0 to the indicative conditional in more cases. Like
DF/TT , CC/TT iscontractive, reflexive, monotonic and transitive. Moreover: • Assumption (i) and (ii) hold in
CC/TT for the same reasons as
DF/TT . • Assumption (iii) fails in CC/TT . For example, A ∧ ¬ A | = CL B , but = CC / TT ( A ∧ ¬ A ) → B . A CC -evaluation v s.t. v ( A ) = / and v ( B ) = • Assumption (iv) holds in
CC/TT . As in the
DF/TT case, we have that A | = CC / TT B and B | = CC / TT A if, for any CC -evaluation v , one of the following is given:(a) v ( A ) = = v ( B ) (c) v ( A ) = v ( B ) = / (b) v ( A ) = / = v ( B ) (d) v ( A ) = / ; v ( B ) = / in CC -truth tablesof the indicative conditional. Therefore, whenever one of (a)–(d) holds, for everyformula C , we have that v ( A → C ) = v ( B → C ) , proving the claim. • Assumption (v) fails in CC/TT , for the same reason it fails in
DF/TT .One of (i)–(iv) thus fails for
CC/TT as it does for
DF/TT , and (v) fails in both. Thefailure of assumption (iii), supraclassicality, is irrelevant for blocking the proof sincethe only classically valid inference required for the proof is Conjunction Elimination( A ∧ B | = A ). This inference is also validated by CC/TT . The proof is thus blockedexclusively by the failure of assumption (v): ⊃ does not behave classically in CC/TT (i.e., step 2 in our reconstruction of Gibbard’s proof fails). Unlike
DF/TT , CC/TT avoidsGibbardian collapse: it declares both conditionals materially equivalent, but neitherlogically equivalent nor equivalent according to the indicative biconditional:
Lemma 3.
For every A , B ∈ For ( L ) :A → B | = CC / TT A ⊃ B but A ⊃ B = CC / TT A → B | = CC / TT ( A ⊃ B ) ⊃⊂ ( A → B ) = CC / TT ( A ⊃ B ) ↔ ( A → B ) In general, the indicative conditional of
CC/TT is strictly stronger than its materialcounterpart: A → B entails A ⊃ B , but is not entailed by it. And this is, by the light of alogic of indicatives, a welcome result: the paradoxes of material implication consist, forthe most part, of conditional statements that are clearly unacceptable, but are declaredvalid by the material conditional analysis. The Cooper-Cantwell analysis validates fewer conditional principles (‘fewer’ in the sense of inclusion), and avoids the mostproblematic paradoxes. 9ltogether, DF/TT and
CC/TT avert Gibbardian triviality in di ff erent ways. In bothof them the material conditional is not fully classical, but an extensional collapse takesplace in DF/TT anyway; this, however, does not make the material conditional alwaysreplaceable by the indicative in
DF/TT . On the other hand, the indicative conditional of
CC/TT is more remote from its material counterpart: not only does it validate di ff erentconditional principles (removing the most pressing paradox of material implication), itis also extensionally distinct from the material conditional within CC/TT itself.Summing up, while Gibbardian collapse is avoided more markedly in
CC/TT thanin
DF/TT , in neither logic does it constitute a form of “triviality”: even when indicativeand material conditionals are declared to be equivalent, they are firmly set apart bytheir inferential behavior. This concludes our study of Gibbard’s original collapse resultin trivalent logics based on Strong Kleene connectives. In the next section, we expandthe scope of our analysis and look at trivalent logics of conditionals with a di ff erentsemantics for the standard logical connectives. QCC/TT
The logics
DF/TT and
CC/TT solve a large set of problems related to the indicativeconditional, but they also have important limitations. First, both
CC/TT and
DF/TT validate the Linearity principle ( A → B ) ∨ ( B → A ) for arbitrary A and B . This schemawas famously criticized by MacColl (1908): neither of “if John is red-haired, then Johnis a doctor” and “if John is a doctor, then he is red-haired” seems acceptable in ordinaryreasoning. So it is unclear on which basis we should accept, or declare as true, thedisjunction of both sentences. Imagine, for example, that John is a black-haired doctoror a red-haired carpenter.In a similar vein, some highly plausible conjunctive sentences can never be true on DF/TT or CC/TT . The schema ( A → A ) ∧ ( ¬ A → ¬ A ) (“if A, then A; and if ¬ A, then ¬ A”) is always classified as neither true nor false, although each of the conjuncts isa
DF/TT - and
CC/TT -theorem. Likewise, an ensemble of conditional predictions ofthe form ( A → B ) ∧ ( ¬ A → C ) will always be indeterminate or false (Bradley 2002,368–370). However, a sentence such as:(1) If the sun shines tomorrow, Paul will go to the o ffi ce by bike; and if it rains, hewill take the metro.seems to be true (with hindsight) if the sun shines tomorrow and Paul goes to the o ffi ceby bike.A principled reply to these challenges consists in modifying the truth tablesfor trivalent conjunction and disjunction, as proposed by Cooper (1968) (see also We are indebted to Paolo Santorio for this example. ′∧ /
01 1 1 0 / /
00 0 0 0 f ′∨ /
01 1 1 1 / /
00 1 0 0 f ′⊃ /
01 1 0 0 / /
00 1 1 1Table 3:
Truth tables for trivalent quasi-conjunction and quasi-disjunction and the material conditionalbased on quasi-disjunction, as advocated by Cooper (1968).
Dubois and Prade 1994 and Calabrese 2002). In these truth tables, reproduced in Table3, the conjunction of value 1 and value / is value 1, and vice versa for disjunction.This is coherent with the idea that a conditional assertion with two components (e.g., inBradley’s examples) should be classified as true if one of the assertions came out true,and the other one void. Notably, the material conditional A ⊃ C (definable as ¬ A ∨ B or as ¬ ( A ∧ ¬ B ) ) of a TT- logic based on these quasi-connectives blocks the paradoxesof material implication ( ¬ A = A ⊃ C , C = A ⊃ C ), in line with the failure to validateDisjunction Introduction.Adopting “quasi-conjunction” and “quasi-disjunction” (the terminology is due toAdams 1975) invalidates Linearity and gives non-trivial truth conditions for ensemblesor partitions of conditional assertions. In particular, ( A → A ) ∧ ( ¬ A → ¬ A ) is alwaystrue, and so is ( A → B ) ∧ ( ¬ A → C ) when one of its conjuncts is true. We call theresulting logics QDF/TT and
QCC/TT . However, when paired with
DF/TT , quasi-conjunction leads to a violation of Import-Export, but not so in
CC/TT . So the system ofinterest for us in this section is
QCC/TT .How does
QCC/TT then fare with respect to the five premises of Gibbard’s proof? • Assumption (i) holds since both sides of the Law of Import-Export receive thesame truth value in any
QCC -evaluation. • Assumption (ii) fails since the (quasi-)material conditional is strictly stronger thanthe indicative conditional. The valuation v ( A ) = v ( B ) = / is a model of A → B , but not of A ⊃ B , which takes the same truth values as ¬ A ∨ B . • Assumption (iii) and (v) fail with the same countermodels as in
CC/TT . • Assumption (iv) holds: it is independent of the interpretation of the standardconnectives and the proof for
CC/TT can be transferred.In
QCC/TT , two steps of Gibbard’s proof are blocked, corresponding to the failureof assumptions (ii) and (v). Like before, the failure of (iii) is inessential since theproof just requires Conjunction Elimination instead of the more general property ofSupraclassicality. QCC/TT is almost identical to Cooper’s logic OL , except that Cooper requires valuations to be bivalent ≡ ? ↔ ? ⊃⊂ ? DF/TT ✓ ✓ ✓ ✓ ✗ ✗ ✓ ✓ ✓ ✓
CC/TT ✓ ✓ ✗ ✓ ✓ ✗ ✓ ✗ ✗ ✓
QCC/TT ✓ ✗ ✗ ✓ ✓ ✗ ✓ ✗ ✗ ✗
Table 4:
Overview of which premises of Gibbard’s proof are satisfied by the logics
DF/TT , CC/TT and
QCC/TT . CE = conjunction elimination ( = a su ffi cient surrogate for (iii)), TRM = transitivity, monotonocityand reflexivity of the logic. ≡ , ↔ , ⊃⊂ concern whether logical, indicative, or material equivalence holdsbetween ⊃ and → . Since the material conditional is strictly stronger than the indicative in
QCC/TT , Gib-bardian collapse does not happen, and moreover, neither the material nor the indicativeconditional declares the two connectives equivalent:
Lemma 4.
For every A , B ∈ For ( L ) :A ⊃ B | = QCC / TT A → B but A → B = QCC / TT A ⊃ B = QCC / TT ( A ⊃ B ) ⊃⊂ ( A → B ) = QCC / TT ( A ⊃ B ) ↔ ( A → B ) In QCC/TT , the connectives are thus more distinct than in
DF/TT (where they arelogically and materially equivalent) and
CC/TT (where they are not logically, but stillmaterially equivalent). The way out provided by
QCC/TT is notable for another reason,too. Most theorists react to Gibbardian collapse either by giving up or restricting Import-Export (e.g., Stalnaker, Kratzer), or by endorsing a material implication analysis of theindicative conditional (e.g., Grice, Lewis, Jackson). Denying that ⊃ satisfies the classicallaws in a logic of conditionals—the road taken by CC/TT —is already less common.However, Cooper’s original approach is probably unique in entertaining the possibilityof an indicative conditional that is strictly weaker than the material conditional. Theexplanation is probably that bivalent logic has been the default framework for formalwork on conditionals and the material conditional represents, in that framework, theweakest possible conditional connective. The logic
QCC/TT thus shows an original andsurprising way of defining the relationship between the two connectives.
Our rendition of Gibbard’s original argument has revealed that one of the premises—namely that | = A → C whenever A classically implies C — is stronger than needed: on atomic formulae.
12e only require that ( A ∧ C ) → C be a logical truth. On the other hand, Gibbard’sargument uses some properties of classical logic and the material conditional, such asthe fact that A ∧ ( A ⊃ C ) is logically equivalent to A ∧ C . Gibbard’s result can thus begeneralized along two dimensions: first, use premises only as strong as we need themfor the proof of the collapse result; second, make explicit the classicality assumptions(compare Section 2) and extend the result to other logics than just classical logic withthe material conditional.Branden Fitelson (2013) has provided one such generalized result. It concerns therelation between two binary connectives represented by the symbols → and in anarbitrary logic L , whose consequence relation we denote with | = L . Letting A , B and C stand for arbitrary formulae of L , and | = L for some consequence relation defined forthe language of L , Fitelson states eight conditions su ffi cient to derive a general collapseresult: (1) | = L ( A ∧ B ) A (Conjunction Elimination for )(2) | = L ( A ∧ B ) → A (Conjunction Elimination for → )(3) | = L A ( B C ) if and only if | = L ( A ∧ B ) C (Import-Export for )(4) | = L A → ( B → C ) if and only if | = L ( A ∧ B ) → C (Import-Export for → )(5) If | = L A → B , then | = L A B ( → implies )(6) If | = L A B , then A | = L B (Conditional Elimination for ).(7) If A ≡ L B and | = L A → C , then also | = L B → C (Left Logical Equivalence)(8) If A | = L B and A | = L C , then A | = L B ∧ C (Conjunction Introduction)In short, Fitelson’s result concerns the relationship between two conditionals whichsatisfy both Conjunction Elimination (1 +
2) and Import-Export (3 + → , is supposed to represent the indicative conditional. Moreover, it isassumed that the weaker connective satisfies Conditional Elimination relative to thelogic | = L (6), and that one can substitute | = L -equivalents in the premises of → -validities(7). Finally, it assumes Conjunction Introduction (8), a very natural property: if twopropositions follow from a third, then so does their conjunction.Fitelson shows that these axioms are logically independent from each other and thatthey are su ffi cient to show that the two connectives → and are logically equivalent: Our notation swaps the meaning of the symbols → and in Fitelson’s work to make it consistentwith the rest of our paper. What we call Conditional Elimination is the converse of Conditional Introduction. The two propertiestogether are known as the Deduction Theorem. Conditional Elimination corresponds to (meta-inferential)Modus Ponens. heorem (Fitelson 2013): From conditions (1)–(8) it follows that A B | = L A → B and A → B | = L A B As Fitelson emphasizes, this should not be taken to imply that the connective → col-lapses to the material conditional, or that the indicative conditional “If A, then C” shouldbe interpreted as “not A or C”. Fitelson’s result is interpretation-neutral and concerns any two connectives with the said properties; specifically, it does not presuppose that theweaker connective corresponds to the material conditional ⊃ . Whether the materialconditional A ⊃ C (i.e., ¬ A ∨ C ) satisfies the properties of (i.e., conditions (1), (3), (5)and (6)) will depend on which logic we choose to interpret | = L , and we will soon seethat it need not in a trivalent setting. What Fitelson shows is rather that if a conditionalconnective satisfies Conjunction Elimination, Import-Export and Modus Ponens, thenin any logic with Conjunction Introduction, there cannot be a strictly stronger condi-tional connective that satisfies these conditions as well as axiom (7)—the substitution ofequivalents in the premises of its theorems. In this sense, Fitelson proves the existence ofan upper bound for the strength of a conditional that satisfies these intuitively desirablelogical properties. Moreover Fitelson shows that such a connective must also validatesome central intuitionistic principles. What does Fitelson’s result mean for trivalent logics when his two connectives → and are identified with the indicative and the material conditional? Keeping the tolerant-to-tolerant character of the logical consequence relation fixed (see Section 4 for why),we have to assign values to the following parameters: • the truth table for the indicative conditional (de Finetti or Cooper-Cantwell); • the truth table for conjunction and disjunction (Strong Kleene operators orCooper’s quasi-conjunction and disjunction); • which connective in Fitelson’s result represents the indicative conditional, andwhich connective represents the material conditional.This leaves us with eight di ff erent logics, characterized by the choice of the truth tablefor the indicative conditionals ( DF or CC ), the truth tables for conjunction and dis-junction (Strong Kleene or Cooper), and the assignment of conditionals to Fitelson’sconnectives ( → and ). Fitelson suggests that the stronger connective → stands for theindicative conditional. However, the properties of , which include Modus Ponens,Conjunction Elimination and Import-Export, could also square well with the indicative14onditional. Moreover, the indicative conditional can be weaker than the material condi-tional in QCC/TT . Thus, we have to carefully examine all ways of distributing Fitelson’sconnectives to truth tables.As noticed in the previous section,
QDF/TT does not satisfy Import-Export for theindicative conditional and so we set it aside (either condition (3) or condition (4) willfail). All the other logics satisfy conditions (1)–(4) and also condition (8). Thus ourdiscussion will be limited to those logics and the more controversial properties (5), (6)and (7). Actually, we see that none of our trivalent logics satisfies all of these principles: DF / TT with → = → DF Satisfies (5)—material and indicative conditional are DF -equivalent—, but neither (6) nor (7). For (6), consider | A | = / , | B | =
0, andfor (7), consider | A | = / , | B | =
1, and | C | = DF / TT with → = ⊃ Satisfies (5), but neither (6) and (7). Consider the same examples asabove. CC / TT with → = → CC Satisfies (5) and (7), but not (6). Consider again | A | = / and | B | = CC / TT with → = ⊃ Q Satisfies (6), but neither (5) nor (7). For (5), consider | A | = / and | B | =
0; for (7) consider | A | = / , | B | =
1, and | C | = QCC / TT with → = → CC Satisfies (6) and (7), but not (5). Consider | A | = | B | = / . QCC / TT with → = ⊃ Q Satisfies (5) and (6), but not (7). The counterexample is | A | = | C | = / and | B | = / Logic
DF/TT CC/TT QCC/TT
Assignment of Symbols Indicative = ? → → → Material = ? → → → (5): → implies ✓ ✓ ✓ ✗ ✗ ✓ (6): Conditional Elimination for ✗ ✗ ✗ ✓ ✓ ✓ (7): Substitution of Equivalents ( → ) ✗ ✗ ✓ ✗ ✓ ✗ Collapse strongly blocked? ✓ ✓ ✗ ✗ ✗ ✗
Table 5:
Overview of the satisfaction / violation of Fitelson’s conditions (5)–(7) in di ff erent trivalent logics. Table 5 summarizes our findings. As we see, none of our trivalent candidate logicsfor the indicative conditional obeys all of these axioms. Since there are no obviousalternatives to the (various forms of the) material conditional as the second connective inFitelson’s theorem, Gibbardian collapse is blocked for the entire range of trivalent logicsthat we study. In particular, since at least one of the axioms fails for all configurations15e have looked at, the connective → must also fail one the principles of the intuitionisticconditional (this is, as mentioned above, a consequence of satisfying conditions (1)–(8)). In order to better assess the distinct ways in which Fitelson’s collapse is blocked intrivalent logics, we introduce a useful distinction. We say that a logic of indicativeconditionals L blocks the collapse strongly if at least one of conditions (1)–(8) is notsatisfied by letting → = → ind , where → ind is the connective that, in L , is taken to modelthe indicative conditional. We say that the L blocks the collapse weakly if → = → ind and = ⊃ , where ⊃ is the material conditional in L . In other words, L blocks Fitelson’scollapse strongly if some of Fitelson’s premises fails in L once → is interpreted as L ’scandidate for the indicative conditional, regardless of how the other conditional isinterpreted. On the other hand, L blocks Fitelson’s collapse only weakly if some ofFitelson’s premises fails in L once → is interpreted as L ’s candidate for the indicativeconditional and is interpreted as L ’s material conditional. In the former case, L ’sindicative conditional is non-trivial (in the sense of the collapse) by itself , whereas in thelatter case it is non-trivial only if we assume (at least some of) the features of ⊃ in L forthe other conditional.A glance at our findings shows that Fitelson’s collapse result is blocked stronglyfor the DF/TT -logics, and only weakly for all (Q)CC/TT -logics. The failure of collapsein the (Q)CC/TT -logics is due to both features of the Cooper-Cantwell conditional ina TT -consequence relation and the choice of the material conditional as the interpre-tation of the weaker connective . Does this show that the indicative conditional ofthe (Q)CC/TT -logics is “trivial”, or in some sense uninteresting? Not really. All Fitel-son’s result can be used to argue for is that, given (1)–(8), the indicative conditional of (Q)CC/TT -logics is L -equivalent to (i.e., inter- L -inferrable with) an unspecified condi-tional which: (i) cannot be the material conditional of L (since (Q)CC/TT -logics weaklyblock the collapse), and (ii) satisfies conditions (1), (3), (5), and (6), over a backgroundlogic which satisfies (8). Now, not only are these properties unproblematic—by them-selves, they do not give rise to any paradox of implication—, they are indeed desirable.Hence, it should actually be a welcome result that an indicative conditional is equivalentto a conditional with such properties.In summary, since the trivalent logics we have examined block Fitelson’s collapseresult systematically, we do not find ourselves in the dilemma of having to sacrificeImport-Export, or another plausible condition to avoid triviality. To us, the most rea-sonable construal of Fitelson’s theorem is as a uniqueness result : it is impossible to have Respectively: Conjunction Elimination (1), Import-Export (3), being entailed by indicative conditionals(5), Modus Ponens (6), and Conjunction Introduction (8)
QCC/TT , in which the material conditional is stronger than the indicativeconditional. For all other combinations there is a tension between the relative strengthof the connectives (as codified by (5)) and the fact that the weaker connective shouldsatisfy Conditional Elimination (namely (6)).
This paper has given a precise reconstruction of Gibbard’s informal argument that anyindicative conditional that satisfies Import-Export and is supraclassical and strongerthan the material conditional must collapse to the material conditional. Specifically,we have seen that Gibbard’s argument requires additional premises (e.g., structuralassumptions on the underlying logic L ) and that the premises are not tight either (e.g.,supraclassicality can be replaced without loss of validity by Conjunction Elimination).We have then explored how a family of trivalent logics, all based on the idea that aconditional is void when its antecedent turns out false, fare with respect to Gibbardiancollapse. The logics we have examined all block an important premise of Gibbard’sproof, namely the classical behavior of the material conditional ⊃ , as well as one addi-tional premise (di ff erent for each logic). Nonetheless, in DF/TT —the tolerant-to-tolerantlogic based on de Finetti’s truth table for the indicative conditional—Gibbardian collapseoccurs, but this does not mean that both conditionals obey the same logical principles.In contrast, Cantwell’s logic
CC/TT and Cooper’s logic
QCC/TT , based on their commontruth table for the indicative, avoid Gibbardian collapse altogether. This shows us thatthe apparent lesson from Gibbard’s result— that one has to give up Import-Export orendorse the material analysis of the conditional — is mistaken.We confirmed that diagnosis by looking at these logics in the context of thestrengthening of Gibbard’s result proposed by Fitelson (2013). Specifically, we havere-interpreted Fitelson’s result as showing the impossibility of having two distinct con-nectives that both satisfy a set of characteristic properties (Conjunction Elimination,Import-Export), and where the weaker one already satisfies Conditional Elimination. Alogic of indicative conditionals does not have to choose between forswearing Import-Export and embracing the material conditional analysis: trivalent logics of conditionalso ff er a simple, yet articulate and fully truth-functional alternative that avoids both prob-lems. To be sure, one might still have objections to Import-Export but, whatever theyare, they cannot be supported by Gibbard-style collapse arguments.17 eferences Adams, Ernest W. (1965). The Logic of Conditionals.
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A Import-Export Revisited
Our case study on trivalent logics shows that it is possible to have Import-Export withoutrestriction in a conditional logic without running into undesirable results of collapse tothe material conditional, or to other connectives that are clearly too weak. Specifically,even if a conditional connective → validates Import Export, the schema ( A ∧ B ) → A ,and is stronger than the material conditional, it need not be logically equivalent to thelatter.This observation raises the suspicion that the scope of Gibbardian collapse resultsmay have to do with the absence and presence of bivalence. Note that the third truthvalue has been essential to constructing suitable counterexamples to Fitelson’s con-ditions (1)–(8), and to blocking a generalized collapse theorem. In other words, we19onjecture that Gibbardian collapse is a characteristic feature of conditional connectiveswith Import-Export in bivalent logic .This conjecture shall now be probed by studying a recent reductio argument againstImport-Export. Matthew Mandelkern (2020) argues that Import-Export, when con-joined with other plausible principles, leads to absurd conclusions (compare alsoMandelkern 2019). Specifically, for a logic ( L , | = L ) with formulae A , B and C and aconnective → representing the indicative conditional, Mandelkern considers (and de-fends) the following three principles:If A | = L B then | = L A → B (Conditional Introduction)If | = L A → B then A → ( B → C ) ≡ L A → C (Nothing Added)If A → C ≡ L B → C then A ≡ L B (Equivalence)where ≡ L means, as before, that both | = L and its converse hold. Conditional Introductionis valid in all trivalent logics we considered, whereas Nothing Added and Equivalencehold in (Q)CC/TT , but not in (Q)DF/TT . Mandelkern requires another premise, re-stricted to atom-classical formulae A (i.e. such that all propositional variables have aclassical value) , but without restrictions on B :For atom-classical A : | = L ( A ∧ ¬ A ) → B (Quodlibet)Quodlibet too holds in the trivalent logics we surveyed. From these four principlesMandelkern derives the following intermediate result:For atom-classical A : A | = L ¬ A → B (Intermediate)Intermediate also holds in CC/TT , and plausibly so: If A holds then any conditionalassertion with ¬ A as a premise is void, and thus valid in a logic with a tolerant-to-tolerant consequence relation. Intermediate is equivalent to ¬ A | = L A → B , from whichMandelkern derives:For atom-classical A : ¬ ( A → B ) | = L A (Ex Falso)The lesson Mandelkern takes from this is:[Intermediate] is clearly false [...]. For this conclusion entails that the falsityof ¬ A → B entails the falsity of A ; more succinctly (given classical negation,which is not in dispute here), the falsity of A → B entails the truth of A .(Mandelkern 2020, symbolic notation changed) Countermodel for Nothing Added in (Q)DF/TT : v ( A ) = v ( B ) = / , v ( C ) =
0. Countermodel forEquivalence in (Q)DF/TT : v ( A ) = v ( C ) = v ( B ) = / .
20x Falso is definitely an unacceptable principle for a theory of indicative conditionals.As it turns out, it is invalidated in the trivalent logics, including CC / TT (Consider v ( A ) = CC / TT because trivalentnegation is no longer classical. In particular, TT -consequence does not obey Contraposi-tion. This feature suggests a tradeo ff : the trivalent logics of conditionals we consideredvalidate Import-Export without restriction, and they do not fall prey to Mandelkern’sreductio. However, they no longer validate Contraposition without restriction, andbecause CC / TT satisfies the full Deduction Theorem, the associated conditional failscontraposition too. For indicative as well as for counterfactuals, contraposition is moot,however, in that regard the way in which Mandelkern’s reductio is blocked here doesnot appear problematic. B Technical appendix
In this appendix, we first prove that assumption (iii) of Gibbard’s Theorem holds in
DF/TT . Then, we give a syntactic proof of the mutual
DF/TT -entailments of A → B and A ⊃ B (cf. Lemma 2), in the three-sided sequent calculus for DF/TT from (Égré et al.2020b). The remaining claims of the Lemma are then immediate. The calculus issound and complete for
DF/TT , so the proof immediately establishes the correspondingsemantic claims, but we believe that a syntactic proof provides a good illustration ofhow one can, rather naturally, reason in trivalent logics. Similar proofs are available forthe corresponding claims in
CC/TT . Lemma 5.
Supraclassicality holds in
DF/TT .Proof.
We prove the contrapositive. Suppose = DF / TT A → B . Then there is a DF -evaluation v : For ( L ) / , 1 } s.t. v ( A ) = v ( B ) =
0. We then claim that, inthis case, then there is always a classical evaluation v cl : For ( L )
0, 1 } s.t. for every C ∈ For ( L ) , if v ( C ) =
1, then v cl ( C ) = v ( C ) =
0, then v cl ( C ) =
0, thus showingthat A = CL B . We prove this by induction on the logical complexity ( cp ) of A and B : • cp ( A ) = cp ( B ) =
0. Then, A → B has the form p → q , and v ( p ) = v ( q ) = v cl is any classical evaluation which agrees with v on p and q , so clearly p = CL q . • cp ( A ) = m + cp ( B ) =
0. Then A → B has the form C → q , for C a logicallycomplex sentence. We assume the claim as IH up to m , and reason by cases: Mandelkern does not dispute the validity of Import-Export for simple right-nested conditionals whereit looks very compelling; he just thinks that Import-Export has less than general scope. Specifically, hehas doubts about the application of Import-Export to compound conditionals with left-nesting, such as A → (( B → C ) → D ) . Naturally, it is very di ffi cult to find reliable empirical data or expert intuitions onhow such sentences are, or should be, interpreted. C is ¬ D . Then v ( ¬ D ) = v ( q ) =
0, and v ( D ) =
0. By IH, then, there is aclassical evaluation v cl s.t. v cl ( D ) = v ( q ) =
0, so that C = CL q . – C is D ∨ E . Then v ( D ∨ E ) = v ( q ) =
0. There are several cases, allsimilar between them, where at least one of the disjunct receives value 1:* v ( D ) = v ( E ) = v ( D ) = v ( E ) = / * v ( D ) = v ( E ) = v ( D ) = / and v ( E ) = v ( D ) = v ( E ) = X be the (or ‘a’) disjunct which receives value 1 by v . By IH, v cl ( X ) = v cl ( D ∨ E ) = v cl ( q ) =
0, hence C = CL q – The case where C has the form D ∧ E is similar to the above one. – C is D → E . Then v ( D → E ) = v ( q ) =
0, and therefore v ( D ) = v ( E ) =
1. By IH, then, v cl ( D ) = v cl ( E ) =
1, hence C = CL q . • The cases where cp ( A ) = cp ( B ) = n +
1, and where cp ( A ) = m + cp ( B ) = n + (cid:3) Notice that, in this proof, a DF -evaluation for the language including the conditionalis mapped to a classical evaluation for the same language, i.e. a classical evaluation whichalso interpret formulae of the form A → B . However, the proof does not specify howformulae of the form A → B are classically interpreted—that is, A → B may or may notbe interpreted as a classical material conditional. We also note that an attempted proofalong the lines of the above one would fail for CC/TT exactly because the conditionsunder which an indicative conditional receives value 0 under a CC -evaluation strictlyexceed the conditions under which a material conditional receives receives value 0under a classical evaluation, unlike in a DF -evaluation. Lemma 6.
Let Γ ⊢ DF / TT ∆ indicate that there is a derivation of the three-sided sequent Γ | ∆ | ∆ in the calculus developed in Égré et al. (2020b), §§3.1-3.2. Then, for every A , B ∈ For ( L ) :A ⊃ B ⊢ DF / TT A → B and A → B ⊢ DF / TT A ⊃ BProof.
We write ¬ ( A ∧ ¬ B ) for A ⊃ B , as the two formulae are definitionally equivalentin DF/TT . The following derivation establishes that A ⊃ B ⊢ DF / TT A → B :22 Ref A | A , B | A → B , A → - / ∅ | A → B | A → B , A SRef A , B | A , B | A SRef A , B | A , B | B → -1 A , B | A , B | A → B → - / B | A → B | A → B ¬ -1 ∅ | A → B | A → B , ¬ B ∧ -1 ∅ | A → B | A → B , A ∧ ¬ B ¬ -0 ¬ ( A ∧ ¬ B ) | A → B | A → B We now show that A → B ⊢ DF / TT A ⊃ B . First, let D be the following derivation: SRef A , ¬ B | A | A , A SRef A , ¬ B | ¬ B | A , ¬ B SRef A , ¬ B | A , ¬ B | A ∧ - / A , ¬ B | A ∧ ¬ B | A Second, let D be the following derivation: SRef A , ¬ B , B | A | A SRef A , ¬ B , B | ¬ B | ¬ B SRef A , B | A , B | B ¬ - / A , B | A , ¬ B | B ¬ -1 A , ¬ B , B | A , ¬ B | ∅ ∧ - / A , ¬ B , B | A ∧ ¬ B | ∅ Finally, combining D and D yields the desired result: D A , ¬ B | A ∧ ¬ B | A D A , ¬ B , B | A ∧ ¬ B | ∅ → -0 A , ¬ B , A → B | A ∧ ¬ B | ∅ ∧ -0 A ∧ ¬ B , A → B | A ∧ ¬ B | ∅ ¬ -1 A → B | A ∧ ¬ B | ¬ ( A ∧ ¬ B ) ¬ - / A → B | ¬ ( A ∧ ¬ B ) | ¬ ( A ∧ ¬ B ) (cid:3) C Gibbardian collapse without Left Logical Equivalence
Khoo and Mandelkern (2019, 489) prove Gibbard’s collapse result using Reasoning byCases. They do not use Left Logical Equivalence (as in our reconstruction of Gib-bard’s original proof) and explicitly refer to principles (i)–(iii) only (i.e., Import-Export,Stronger-than-Material and Supraclassicality). However, like Gibbard, they actuallymake use of more assumptions, in particular (v): the classicality of ⊃ . Their proof canbe formalized thus: Theorem 7.
Let L be a reflexive, monotonic, and transitive consequence relation, with ∨ satisfying Reasoning by Cases. Then if (i), (ii), (iii) and (v) hold in L, ⊃ entails → , that is, forany A , B ∈ For ( L ) , A ⊃ B | = L A → B. ¬ A ∧ A | = CL B , classical logic2. | = L ( ¬ A ∧ A ) → B , by 1 and (iii)3. | = L ¬ A → ( A → B ) , by 2 and (i)4. ¬ A → ( A → B ) | = L ¬ A ⊃ ( A → B ) , by (ii)5. | = L ¬ A ⊃ ( A → B ) , by 3, 4 and Transitivity6. ¬ A | = L ¬ A ⊃ ( A → B ) , by 5 and Monotonicity7. ¬ A | = L ¬ A , by Reflexivity8. ¬ A | = L A → B , by 6, 7 and (v), using (meta) Modus Ponens for ⊃ B ∧ A | = CL B , classical logic10. | = L ( B ∧ A ) → B , by 9 and (iii)11. | = L B → ( A → B ) , by 10 and (i)12. B → ( A → B ) | = L B ⊃ ( A → B ) , by (ii)13. | = L B ⊃ ( A → B ) , by 11, 12, and Transitivity14. B | = L B ⊃ ( A → B ) , by 13 and Monotonicity15. B | = L B by Reflexivity16. B | = L A → B , by 14, 15, (v), using (meta) Modus Ponens17. ¬ A ∨ B | = L A → B , by 8, 16 and Reasoning by Cases18. A ⊃ B | = L A → B , by 17 and (v)This version does not use the replacement principle (iv) of Gibbard’s original proof,making it particularly interesting, in particular in relation to DF/TT . Indeed, Reason-ing by Cases is valid in
DF/TT and
CC/TT , as are structural assumptions on logicalconsequence. We know that
CC/TT fails Supraclassicality and so step 2 and 10 of theproof are blocked. Interestingly, however, all steps of the proof here are sound in