This article provides a survey of classic and recent work inconditional logic. We review the problems of a two-valued analysis andexamine logics based on richer semantic frameworks that have beenproposed to deal with conditional sentences of the form “ifA,B,” including trivalent semantics,possible-world semantics, premise semantics, and probabilisticsemantics. We go on to examine theories of conditionals involvingbelief revision, and highlight recent approaches based on the ideathat a conditional is assertable provided the truth of its antecedentmakes a relevant difference to that of its consequent. We close with abrief discussion of systems accounting for the interaction ofconditionals with modalities and speech act operators.

• 1. Introduction
• 2. Three-Valued Conditionals
  • 2.1 Motivations
  • 2.2 Some trivalent logics
  • 2.3 Open issues
• 3. Possible Worlds Models
  • 3.1 Strict conditionals
  • 3.2 Variably strict conditionals
    • 3.2.1 Main semantics
    • 3.2.2 Frame correspondence properties
  • 3.3 Logics for conditionals
  • 3.4 Nonmonotonic logic and preferential models
  • 3.5 Or-to-If, Import-Export, Simplification of Disjunctive Antecedents
• 4. Premise Semantics
  • 4.1 Motivations
  • 4.2 Premise frames
  • 4.3 Correspondence with ordering semantics
• 5. Probabilistic Logics
  • 5.1 Adams’s logic
  • 5.2 Lewis’s triviality theorem
  • 5.3 Cardinality and imaging
    • 5.3.1 Cardinality
    • 5.3.2 Imaging
  • 5.4 Probabilities for compound conditionals
• 6. Belief Revision Approaches
  • 6.1 Belief revision models and the Ramsey test
  • 6.2 Gärdenfors’s triviality theorem
• 7. Relevance and Difference-Making
• 8. Modals and Speech Acts
  • 8.1 Modalities
  • 8.2 Beyond assertion
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

Logics of conditionals deal with inferences involving sentences of theform “ifA, (then)B” of naturallanguage. Despite the overwhelming presence of such sentences ineveryday discourse and reasoning, there is surprisingly littleagreement about what the right logic of conditionals might be, or evenabout whether a unified theory can be given for all kinds ofconditionals. The problem is not new, but can be traced back todebates between Megarian and Stoic logicians (see Sextus Empiricus,Outlines of Scepticism II, 110–112; Kneale & Kneale1962; Sanford 1989; Weiss 2019). Famously, Philo of Megara proposedthat a conditional of the form “ifA thenB” is true exactly when it is not the case thatA is true andB is false. The same definition is atthe core of the modern treatment of the conditional as so-called“material implication” in the two-valued propositionalframework of Frege (1879) and Whitehead and Russell (1910).

This understanding of the conditional has considerable virtues ofsimplicity, and in that regard the material conditional analysisprovides a benchmark for other theories. Probably its main virtue isthat it lends itself to a truth-functional treatment (the truth valueof a conditional is a function of the truth values of antecedent andconsequent). A related one is that it makes the conditionalinterdefinable with Boolean negation, disjunction, and conjunction. Athird, among the driving motivations for Frege, Russell and Whitehead,is that it appears adequate to regiment mathematical proofs involvingconditional sentences. Despite that, as granted by Frege himself intheBegriffsschrift, it fails to do justice to some of theways in which conditionals are ordinarily understood and used innatural language.

An oddity pointed out early by MacColl (1908) is the observation thatof any two sentences of the form “notA orB” and “notB orA”, atleast one must be true. Assuming the equivalence with the materialconditional, this implies that either “if John is a physician,then he is red-haired” or “if John is red-haired, then heis a physician” must be true. Intuitively, however, one may beinclined to reject both conditionals. Similar complications, known astheparadoxes of material implication, concern the fact thatfor any sentencesA andB, “ifA thenB” follows from “notA”, but alsofrom “B”, thereby allowing true and falsesentences to create true conditionals irrespective of their content(C. I. Lewis 1912). Another peculiarity looms large: the negation of“ifA thenB” is predicted to be“A and notB”, but intuitively one maydeny that “if God exists, all criminals will go to heaven”without committing oneself to the existence of God (cited in Lycan2001).

A fourth complication is that conditional sentences in naturallanguage are not limited to indicative conditionals (“if Istrike this match, it will light”), but also include subjunctiveconditionals used to express counterfactual hypotheses (“if Ihad struck this match, it would have lit”). All counterfactualconditionals would be vacuously true if analyzed as materialconditionals with a false antecedent, as pointed out by Quine (1950),an obviously inadequate result, suggesting that the interplay ofgrammatical tense and grammatical mood should also be of concern tounderstand the logic of conditionals.

To a large extent, the development of conditional logics over the pastcentury has thus been driven by the quest for a more sophisticatedaccount of the connection between antecedent and consequent inconditionals. A number of conditional logics have thus been developedwithin the framework ofmodal logic, to capture the idea thatconditional sentences express a necessary connection of sort betweenantecedent and consequent, whether metaphysical, epistemic, or deontic(see C. I. Lewis 1912; Stalnaker 1968; D. Lewis 1973; Kratzer 2012). Arelated wave of work encompasses the so-called cotenability theoriesof conditionals, in which the idea is that a conditional is assertableif its antecedent, together with suitable (co-tenable) premises,entails its consequent (see Chisholm 1946; Goodman 1955; Rescher1964).

Another strand of influential logics has been motivated by the ideathat conditionals have probabilities or express a probabilisticdependence between their antecedents and their consequents (Ramsey1931; de Finetti 1936; Adams 1965; Edgington 1995; Douven 2016). Avery influential source of this is a passage by Ramsey (1931), inwhich he writes:

If two people are arguing “If \(p\), then \(q\)?” and areboth in doubt as to \(p\), they are adding \(p\) hypothetically totheir stock of knowledge and arguing on that basis about \(q\); sothat in a sense “If \(p\), \(q\)” and “If \(p\),\(\neg q\)” are contradictories. We can say that they are fixingtheir degree of belief in \(q\) given \(p\). If \(p\) turns out false,these degrees of belief are renderedvoid. If either partybelievesnot \(p\) for certain, the question ceases to meananything to him except as a question about what follows from certainlaws or hypothesis. [Notation adapted]

Importantly, the so-called Ramsey Test (adding the antecedenthypothetically to one’s beliefs) has inspired a number ofapproaches that stand as some of the cornerstones of conditionallogics. This includes Stalnaker’s modal logic of conditionalsbased on the idea that the antecedent of a conditional selects ahypothetical possible world (Stalnaker 1968). It also includesAdams’s approach based on studying formally the idea that theprobability of (non-nested) conditionals is given by the correspondingconditional probability (Adams 1965, 1975), as well as morequalitative theories of the acceptability of conditionals in terms of(non-probabilistic) belief revision policies (Gärdenfors 1986).Finally, Ramsey’s idea that conditionals with a false antecedentare void underlies the main motivation behind three-valued accounts ofconditionals, developed independently by various authors (inparticular by de Finetti 1936).

In this entry, our presentation of conditional logics will bestructured by the centrality of those various frameworks. Our goal inparticular will be to allow readers to orient themselves easilybetween those frameworks, and also to show how they communicate. Westart with a presentation of three-valued logics of conditionals insection 2.Section 3 surveys the main analyses of the conditional in modallogics (possible worlds semantics). Insection 4, we discuss so-called premise semantics, which formalize theco-tenability idea, and which shed a different light on standard modalanalyses.Section 5 moves on to probabilistic logics of conditionals, andsection 6 to qualitative approaches in terms of belief revision.Section 7 andsection 8, finally, will present more recent developments or refinements: first,logics based on the idea that the antecedent ought to make a relevantdifference to the consequent, then some extensions of propositionallogics of conditionals addressing the interaction of conditionals withmodalities as well as with questions and commands.

One caveat is that in this entry we will try to keep a unifiedperspective on both indicative and subjunctive conditionals. Althoughsome authors, notably David Lewis, have insisted on treating themdifferently, differences can be handled parametrically relative to acommon semantic core (viz., Stalnaker 1975 or Pearl 2009). We willpoint to differences when relevant, but refer the reader toStarr’s (2019) SEP entry on counterfactuals that deals with thetopic rather extensively. On some aspects our entry also overlaps withEdgington’s (2020) SEP entry on indicative conditionals, andreaders are invited to consult both of those entries forcomplements.

In what follows, we will represent the conditional operator by thesymbol \(>\). We distinguish three main layers of syntacticcomplexity. Let \(L_0\) be a propositional language capable ofbuildingfactual sentences from atoms with the help of theusual Boolean connectives (\(\neg\), \(\wedge\), \(\vee\), \(\supset\)and \(\equiv\) for negation, conjunction, disjunction, materialconditional and material equivalence, respectively). Then \(L_1\) isthe extension of \(L_0\) withflat conditionals, such that if\(A\) and \(B\) are in \(L_0\) then \(A>B\) is in \(L_1\). And\(L_2\) is the extension of \(L_0\) and \(L_1\) allowing freecombinations of all connectives, including arbitrary nestings ofconditionals.

2. Three-Valued Conditionals

2.1 Motivations

Logics of indicative conditionals, in so far as they are based ontruth evaluations, all agree that a conditional is false if itsantecedent is true and its consequent is false. This is captured bythe two-valued analysis. Less obvious is the assumption that aconditional is true when both antecedent and consequent are true, forthis predicts that \(A\wedge B\) entails \(A >B\). And even morecontroversial is the proviso that a conditional is true whenever itsantecedent is false, which indeed is one of the paradoxes of thematerial conditional.

C. I. Lewis’s (1912) response to the paradoxes of the materialconditional is his proposal of a strict conditional (aka “strictimplication”, section3). The strict conditional is no longer truth-functional, however. Acloser kin to standard two-valued logic in this regard is the familyof three-valued logics originating from the work of Łukasiewicz(1920), which retain truth-functionality. Łukasiewicz’smain motivation was not to deal specifically with conditionals,however, but with the notion of possibility. Łukasiewicz’sconditional is shown inTable 2. Łukasiewicz defines validity to be the preservation of the value1 from premises to conclusion, as in two-valued logic. His operatorretains the law of Identity \(A>A\) and various classic schemata(such as Modus Ponens), but it also fails some classical properties(it is non-contractive, for example, failing the inference from\(A>(A>B)\) to \(A>B\)). It agrees with the two-valuedconditional on classical input, and it does not block the paradoxes ofthe material conditional.

Łukasiewicz’s work was a source of inspiration toReichenbach, and subsequently to de Finetti, both of whom put forwardthe idea expressed earlier by Ramsey that a conditional must beindeterminate when its antecedent is false (see Reichenbach 1935,1944; de Finetti 1936). Reichenbach deemsmeaningless theevaluation of a conditional \(A>B\) when \(A\) concerns somemeasurement rendered impossible by some other measurement (aspredicted in quantum physics). For de Finetti, the idea is that aconditional sentence is like a bet. When someone bets that the cardwill be a king if it is a club, the bet should simply be called off ifthe card turns out to be of a different suit. On that basis, deFinetti proposed a table (sometimes called the “defective”conditional, see Baratgin et al. 2013, Over & Baratgin 2017) forwhat he calls the logic of tri-events, where a tri-event refers to“the event \(B\) conditional upon event \(A\)”. Thetri-event is true when both events are true, false when \(A\) is trueand \(B\) false, andnull if \(B\) is false.

AsTable 3 shows, there is now a third truth value ½ (for“null” or “void” or“indeterminate”) beyond the classical values 1 (for“true”) and 0 (for “false”). This is why theselogics are calledtrivalent. The conditional is no longerclassically valued for classical input. Although Reichenbach and deFinetti considered the interplay of this conditional with othertrivalent connectives, they did not investigate the logics resultingfrom that choice. Investigations of logics resulting from deFinetti’s table spread in a scattered way, often oblivious ofthe work of Reichenbach and de Finetti. One path was paved by remarksdue to Quine (1950), crediting the work of Ph. Rhinelander on the ideathat the assertion of a sentence of the form “if \(A\) then\(C\)” is commonly felt “less as the assertion of aconditional than as a conditional assertion of the consequent”(see Jeffrey 1963; Cooper 1968; Belnap 1970, 1973; Manor 1975; Farrell1979, 1986; Olkhovikov 2002; Huitink 2008). This view is known morebroadly as the suppositional view (Adams 1975; Edgington 1995).Another line of work was pursued in connection with algebraictreatments of probability, much in the spirit of de Finetti’soriginal work, who viewed trivalence as a way of representing thenotion of conditional probability more qualitatively (see Schay 1968;Calabrese 1991; Dubois & Prade 1994; McDermott 1996; Milne 1997;Cantwell 2008; Rothschild 2014; Lassiter 2020).

2.2 Some trivalent logics

Amongst trivalent logics based on variations of de Finetti’stable, some systems stand out. First, there is Cooper’s logicOL and a close variant of it, Cantwell’s logicCC/TT (as named in Égré, Rossi, &Sprenger 2021, after Cooper and Cantwell), itself a fragment of thelogicLImp of Olkhovikov (2002), which alsoincorporates a modal necessity operator.OL and CC/TT aredefined for \(L_2\), the full language allowing nested conditionals.Like de Finetti, Cooper (1968) handles the conditional asindeterminate when the antecedent is false (0). However, theconditional takes the value of the consequent when the antecedent isnot false (either 1 or ½). Hence, instead of grouping½ with 0, as de Finetti does, it groups ½ with 1.Table 4 was rediscovered independently by Olkhovikov (2002) and by Cantwell(2008), with very similar motivations in mind. It was also proposed byBelnap (1973), who presents it as a suitable modification from Belnap(1970), who first used the same table as De Finetti’s.

Cooper, Olkhovikov, and Cantwell define an inference to be validif the conclusion is not false whenever all the premises are notfalse, for all trivalent valuations of the language. A central featureof their systems is that the full deduction theorem holds (a featurealso stressed by Jeffrey 1963 and Belnap 1973):

One difference between their systems is that Cooper defines validityrelative to trivalent valuations that are atom-classical; another isthat Cooper uses so-called quasi-conjunction and disjunction (seebelow), whereas Olkhovikov and Cantwell use the common min and maxrules for conjunction and disjunction, respectively. For Cooper andCantwell, in particular, and similarly for Belnap (1973), a centralmotivation for this conditional is the validation of the principle ofNegation Commutation for the conditional, namely the conjunctionof:

Principle EX in particular is characteristic of connexive logics (seeSection7 below). Another important feature of the resulting logics is thatwhereas Modus Ponens is a valid argument schema, Modus Tollens andContraposition fail in general (though they remain valid over atomicvariants of the schemata in Cooper’sOL):

Since \(A\not \vDash \neg A >B\), the Cooper conditional blocks oneof the paradoxes of the material conditional, but it retains anotherone, namely \(A\vdash B>A\). Relatedly, although Cooper andCantwell or Olkhovikov use different semantics for conjunction,on their approaches, conjunction entails the conditional. Thisprinciple is calledConjunctive Sufficiency:

Cooper provides a complete axiomatization for his logic, using asystem of inference rules close to a natural deduction system (systemOL). Using tableaux and sequent calculi,Égré, Rossi, and Sprenger (2021) compare theCooper-Cantwell logicCC/TT with the logicDF/TT that results from de Finetti’s table andthe same notion of validity (preservation of non-zero values). ThelogicDF/TT is connexive, too, and supports NegationCommutation. A central difference, however, is that de Finetti’stable fails to support MP.

Another system worthy of consideration is the trivalent logic of“conditional objects” investigated by Dubois and Prade(1994). This system is itself closely related to a three-valuedtreatment of conditionals presented in Adams (1986). The system restson de Finetti’s original table, but only formulae from the flatfragment \(L_1\) are admitted. Validity in Dubois and Prade’sapproach is defined as follows. Assume \(\Gamma\) is a set of flatconditionals of the form \(A_i>B_i\). Then define \(\Gamma \modelsA>B\) iffeither \(A\) classically entails \(B\)or for every trivalent valuation the value of thequasi-conjunction of some subset of conditionals in \(\Gamma\) is lessthan or equal to the value of \(A>B\) (for the usual ordering oftruth-values: \(0<½<1\)). Quasi-conjunction—aconnective proposed by Cooper (1968: 305) and in a different guise byAdams (1966: 306), also proposed by Belnap (1973:60), and earlier bySobociński 1952 in relation to a different trivalent conditional—differs from min conjunction in that the conjunction ofclassical values is classical, and the conjunction of ½ with aclassical value is that classical value. Quite remarkably, thisrelation of logical consequence can be axiomatized by the systemP of preferential entailment of Kraus, Lehmann andMagidor, which is discussed in detail below. Thus, the associatednotion of consequence does not support Monotonicity, Transitivity, andContraposition of the conditional (seesection 3.2).

2.3 Open issues

Despite its simplicity, the use of trivalence raises several issues.One open problem is how to deal with counterfactualconditionals—the problem is not without options, but trivalenceso far has been used mostly to deal with indicative conditionals. Arelated issue is how to make three-valued conditionals interact withmodalities and other operators more generally.

Another problem concerns the expressiveness of a finitely-valuedtruth-functional account. McGee (1981) shows that the common“flat” \(L_1\)-fragment of the logics of conditionals ofAdams, Stalnaker and Lewis, i.e., the systemP,cannot be characterized in terms of the preservation of a designatedset of truth-values in a finite-valued logic. Prima facie, thisresults appears in tension with Adams’s as well as Dubois andPrade’s trivalent characterization ofP. Aspointed out by M. Schulz (2009), however, the trivalentcharacterization proposed by Adams—or indeed Dubois andPrade—differs from the one entertained by McGee (it asks formore than the preservation of a fixed set of designated values, andallows quantification over subsets of the premises).^[1] As discussed by Schulz, whether a similar characterization can begiven for the non-flat logics of Stalnaker and Lewis is an openquestion.

3. Possible Worlds Models

Trivalent accounts suppose that the truth value of a conditionaldepends only on the truth values of its parts. However, thisassumption does not account for a connection of the contents of theantecedent and the consequent, a point granted by Frege (1879)himself. In this section we review accounts that preserve the ideathat conditionals are truth-evaluable, but assume that their truthvalue cannot be settled just by the truth values of the antecedent andthe consequent at the actual world. Instead, the antecedent has acertain modal force, it makes its consequent in some sense necessary.Possible-worlds models give an explication of the necessityinvolved.

3.1 Strict conditionals

The first step toward an intensional treatment of conditionals wasmade with C. I. Lewis’s introduction of the strict conditional.Although Lewis’s treatment was axiomatic, its semantic contentcan be made transparent using possible worlds semantics. OnLewis’s approach, “If \(A\), then \(C\)” means thatthe material conditional is not just true, but necessarily true:

From a semantic view point, reference is made here not just to theactual world, but also to other possible worlds, in fact to all“accessible” worlds. Thus, the meaning of this conditionalcannot be explained by a simple truth table.

In modal logic, aKripke frame is a pair \(\langle W,R\rangle\) where \(W\) is a non-empty set ofpossible worlds(possible states, possible situations or possible ways the world mightbe) and \(R\) is a binary relation on \(W\). \(R\) is called theaccessibility relation of the model, and intuitively \(Rwv\)means that world \(v\) is (in some sense) possible from the point ofview of \(w\). Amodel is a triple \(\langle W, R, V\rangle\)such that \(\langle W, R\rangle\) is a frame and \(V\) is a valuationfunction assigning truth values to all propositional variables of thelanguage at all possible worlds in \(W\). We write \(M, w \Vdash A\)for “\(A\) is true at world \(w\) in model \(M\)” and let\(\lvert A\rvert^M\) denote the set \(\{w\in W: M, w \Vdash A\}\) ofpossible worlds at which \(A\) is true. Let \(R(w) := \{v\in W:Rwv\}\). The truth conditions for the Boolean connectives are theusual ones for propositional logic, extending \(V\) to complexsentences. Those for the strict conditional are as follows:

Validity (relative to a class of frames) is defined astruth-preservation at all worlds of all models (based on any frame inthe class). It is easy to see that the strict conditional invalidatesthe paradoxes of the material conditional, relative to models built onarbitrary Kripke frames. The strict conditional retains Modus Ponensand a number of classic properties, such as Monotonicity,Transitivity, and Contraposition (see below). Starting with Goodman(1955), however, such inferences have been regarded with suspicion,both for indicative conditionals (see Adams 1965) and forcounterfactuals conditionals (see Stalnaker 1968, Lewis 1973). Forinstance, in the case of Transitivity, we have the followingcounterexamples:

(1)

If Brown wins the election, Smith will retire to private life. IfSmith dies before the election, Brown will win it. #Therefore, ifSmith dies before the election, then he will retire to private life.(Adams 1965, p. 166)

(2)

If J. Edgar Hoover were today a communist, then he would be atraitor. If J. Edgar Hoover had been born a Russian, then he wouldtoday be a communist. #Therefore, if J. Edgar Hoover had been born aRussian, he would be a traitor. (Stalnaker 1968, p. 106)

Such examples inspired Adams to propose a probabilistic treatment ofconditionals. They inspired Stalnaker, Lewis, Nute, and a number ofothers to weaken the strict conditional analysis. But recently therehas been a revival of the strict conditional analysis, based on theidea that many phenomena can be accounted for pragmatically byconstruing the accessibility relation that determines which possibleworlds are relevant as massively context-dependent (Daniels &Freeman 1980; von Fintel 2001; Gillies 2007; Starr 2014a, andStarr’s entry on counterfactuals).

3.2 Variably strict conditionals

The basic idea behind “variably strict conditionals” (anexpression due to D. Lewis 1973) is that the standard of necessityrelevant to evaluate a conditional sentence depends on the antecedent.Roughly speaking, in order to evaluate a conditional \(A>B\), onehas to extend the reach of the accessibility relation to the“closest” or “most similar” worlds in whichthe antecedent is true. In what follows we present three related waysof stating the truth conditions of the variably strictconditional.

3.2.1 Main semantics

Relative modalities

Though not historically the first, arguably the simplest way ofexpressing the idea of a strict conditional whose necessity variesdepending on the antecedent can be found in Chellas (1975) andSegerberg (1989), who describe a conditional \(A>B\) as a relativemodality, namely a sententially indexed modality \(\Box_{A}B\), withthe corresponding accessibility relation \(R_{\lvert A\rvert^{M}}\)for every sentence \(A\). A relative-modality frame orrm-frame can be defined as a pair \(\langle W, R\rangle\)where \(R\) is a ternary relation on \(W\times W\times \cP(W)\). Inpractice we can write \(R_Uvw\) instead of \(RvwU\), and see \(R\) asassigning a binary relation over worlds relative to a proposition\(U\). Anrm-model is a triple \(M = \langle W, R, V\rangle\)such that \(\langle W, R\rangle\) is an rm-frame and \(V\) a valuationfunction. Given an rm-model \(M\), the truth condition for theconditional is this:

Note that \(R\) could be defined on \(W\times W\times L_2\), letting\(R_A(w)\) depend on the syntactic form of \(A\) (instead of theproposition \(\lvert A\rvert^M\) expressed by \(A\)). The main effectof this modification is to block the substitution of equivalentantecedents, a feature called for by several theorists on separategrounds (seesection 3.5).

Selection functions

An equivalent definition is in terms of selection functions (Stalnaker1968; Nute 1980). Ansf-frame is a pair \(\langle W,f\rangle\) where \(W\) is a non-empty set ofpossible worldsand \(f: W \times \cP(W) \to \cP(W)\) is a function taking a world anda proposition (i.e., a set of worlds) and yielding a proposition.\(f\) is called theselection function of the model, andintuitively \(v \in f(w,U)\), where \(U\subseteq W\), means that world\(v\) is a world in \(U\) that is (in some sense)closest ormost similar to \(w\), as compared to all the other worlds in\(U\). The correspondence with the previous approach isstraightforward, if we let \(f(w,\lvert A\rvert^M)\) coincide with\(R_{\lvert A\rvert^M}(w)\). Ansf-model is a triple \(M =\langle W, f, V\rangle\) such that \(\langle W, f\rangle\) is ansf-frame and \(V\) is a valuation function. The truth condition forthe variably strict conditional becomes:

As with rm-frames, the definition of an sf-frame can be modified tolet \(f\) depend on \(w\) and \(A\), instead of \(\lvertA\rvert^M\).

Similarity orderings

There is a closely related alternative to this set-up which usescomparative similarity orderings in place of selection functions (D.Lewis 1973; Burgess 1981). Here, ano-frame is a pair\(\langle W, \prec \rangle\) where \(W\) is a non-empty set ofpossible worlds and \(\prec\) is a ternary relation on theset of worlds, assumed to be transitive and irreflexive relative toits first argument. The relation \(u \prec_w v\) is supposed to meanthat world \(u\) is (in some sense)more similar orcloser to \(w\) than world \(v\). Let \(R(w) := \{y\in W:\exists z \in W(y\prec_wz)\}\). This corresponds to the set ofaccessible worlds from \(w\). Ano-model is a triple\(\langle W, \prec, V\rangle\) such that \(\langle W, \prec\rangle\)is an o-frame, and \(V\) is a valuation function. We use theabbreviation \(\lvert A\rvert^M_w := \lvert A\rvert^M \cap R(w)\), andlet \(min_{\prec_{w}}(U)\) denote the \(U\)-worlds most similar to\(w\) (such that no \(U\)-world is strictly more similar to \(w\)). Asimple form of the truth-condition for the variably strict conditionalreads as follows:

These truth-conditions, however, make sense only if \(\lvertA\rvert^M_w\) possesses minimal elements under \(\prec_w\), that is,only if we can assume (L), in which we write \(v \preceq_w u\) for \(v\prec_w u\) or \(v = u\):

(L) is a slight generalization of theLimit Assumption ofLewis (1973, 1981). While most researchers are ready to endorse theLimit Assumption, Lewis rejects it. According to him, for acounterfactual such as “If Fred was taller than 2 meters, hewould be in the university basketball team” there is no closestworld that would make the antecedent true. Ordering semantics can dealwith failures of (L) by stating more complex truth conditions.

(VSC)-o

\(M, w \Vdash A >B\quad \) iff for all \(v \in \lvertA\rvert^M_w\) there is a \(u\in \lvert A\rvert^M_w\) such that \(u\preceq_w v\) and for all \(u'\in \lvert A\rvert^M_w\) such that \(u'\preceq_w u,\ M, u' \Vdash B\).

This intuitively means that the \(A\wedge \neg B\) worlds are moreremote from \(w\) than the \(A\wedge B\)-worlds.

Comparisons

First of all, it is easy to see how all three semantics can invalidateinferences like(1) and(2), which assume Transitivity. In an o-model, for example, from

and

it need not follow that

See the three-world model inFig. 1, where the arrows represent \(\prec_w\) so that \(x\!\leftarrow\!y\)means the same thing as \(x\prec_w y\).

Figure 1: Counterexample against Transitivity in ano-model [Anextended description of Figure 1 is in the supplement.]

Similar counterexamples to Monotonicity and Contraposition are easy tocome by.

The correspondence between selection function models and relativemodality models is very direct. The correspondence between selectionfunction models and ordering models is less obvious. How can weexplain it? If we interpret similarity relations as preferencerelations, we can import the insights gained in rational choice theoryto the analysis of conditionals (Sen 1970; Suzumura 1983; Aleskerov,Bouyssou, & Monjardet 2007; applications to nonmonotonic reasoningare in Lindström 1994 and Rott 2001). If the Limit Assumption issatisfied, then we can simply identify \(f(w,U)\) with\(\min_{\prec_w} (U)\), for all \(U\subseteq W\). For the conversedirection, we can use the revealed preferences, defined by

or, if \(f\) can take propositions with two elements,

These preferences “rationalize” the selections made by\(f\). This, however, works only under certain constraints. In thisrespect sf-models are more general than o-models. They are also easierto work with, and giving up the Limit Assumption is generally viewedas inadequate from a linguistic point of view (see Stalnaker 1980;Schlenker 2004; S. Kaufmann 2017 for more on (L)).

3.2.2 Frame correspondence properties

For each semantics, specific conditions can be imposed on frames inorder to secure the validity of particular axioms or argumentschemata. Their effects on the logic of conditionals have been studiedin detail. For illustration we give the examples of two specificaxioms and their correspondence properties in the three frameworks.Identity requires that every \(A\)-closest world be indeed an\(A\)-world. Conditional excluded middle corresponds to a uniquenesscondition: there is at most one closest \(A\)-world.

We refer to Unterhuber and Schurz (2014) for a systematic presentationof frame correspondence properties for rm-frames, to Nute (1980),Girard (2007), and Raidl (2021) on the correspondence properties ofsf-frames, and to Friedman and Halpern (1994) and Herzig (1996) on thecorrespondence properties of o-frames.

3.3 Logics for conditionals

In order to present some central logics of conditionals based on thepossible-worlds framework, we adopt a Frege-Hilbert axiomaticperspective and introduce a list of axioms and rules of inference. Ourpresentation and choice of terminology relies on Nute (1980) as wellas Herzig (1996), Unterhuber and Schurz (2014), and Crupi and Iacona(forthcoming-b).

• As a first layer for all systems, we letPCconsist of all tautologies of the classical propositionalcalculus.
• We also assume the systems below to be closed under material Modus Ponens, but distinguish moreover the following inference rules (Left Logical Equivalence, Right Weakening, and Rule of Conditional K). Those mean that if the premises are theorems, then so is the conclusion. \[\tag*{LLE} \frac{B \equiv C}{(B >A) \equiv(C >A)} \] \[\tag*{RW} \frac{B \supset C}{(A >B) \supset(A >C)} \] \[\tag*{RCK} \frac{(B_1 \wedge \ldots \wedge B_n) \supset C} {(A >B_1) \wedge \ldots (A >B_n) \supset(A >C)\quad (n\geq 0)} \]
• Axioms

  LT	(Logical Truth) \(A>\top\)
  ID	(Identity) \(A>A\)
  AND	(And) \(((A>B)\wedge (A>C))\supset(A>(B\wedge C))\)
  OR	(Or) \(((A>C)\wedge (B>C))\supset((A\vee B)>C)\)
  CCut	(Cautious Transitivity) \(((A>B)\wedge ((A\wedge B)>C))\supset(A>C)\)
  CMon	(Cautious Monotonicity) \(((A>B)\wedge (A>C))\supset((A\wedge B)>C)\)
  Rec	(Reciprocity) \(((A>B)\wedge (B>A)) \supset((A>C) \equiv(B>C))\)
  SM	(Stronger-than-Material) \((B>C)\supset(B\supset C)\)
  CS	(Conjunctive Sufficiency) \((B\wedge C)\supset(B>C)\)
  RMon	(Rational Monotonicity) \(((A>B)\wedge \neg(A>\neg C))\supset((A\wedge C)>B)\)
  CEM	(Conditional Excluded Middle) \((A>B)\vee (A>\neg B)\)

We highlight nine systems of conditional logic that have drawnparticular attention.

SystemCK axiomatizes the truth-conditions laid outin (VSC)-rm.CK has been called the basic conditionallogic by Chellas (1980), as it is meant to be the counterpart to thebasic systemK of modal logic. In particular ityields the same theorems as systemK relative to afixed modality \(\Box_{A}\). Thus, it satisfies conditional analoguesof the axiom K of standard modal logic (viz., the axiom CK: \((A>B)\supset((A > (B\supset C)) \supset(A > C))\)) and of the rule ofNecessitation (viz., the rule CNec: from \(C\) infer \(A>C\)). Itcan be described as the smallest logic closed under LLE and RCK (thelatter entailing LT, RW and AND), and is also called a normalconditional logic because of that. The logicCkcorresponds to its hyperintensional variant, in which LLE isdropped.

SystemB, proposed by Burgess, is the smallestextension ofCK containing ID, OR and CMon. SystemB axiomatizes the truth conditions stated under(VSC)-o assuming \(\preceq\) is a preorder (reflexive and transitive).Another specific interest of systemB is that itsflat fragment corresponds to the systemP ofnonmonotonic logic due to Kraus, Lehmann and Magidor (seesection 3.4). SystemP is also sound and complete for Adams’probabilistic semantics of conditionals (seesection 5), and a three-valued characterization can be given for it too, asmentioned above, see Dubois and Prade (1994). In that sense, systemP is quite central amongst logic of conditionals.

Particularly noteworthy is the fact that none of these logicsvalidates Monotonicity (also known as Strengthening the Antecedent),Transitivity or Contraposition for conditionals:

These principles have in many contexts been considered as paradigmaticinvalidities of conditional logic.

If we add RMon toB we get the systemV of David Lewis. John Pollock (1976) proposed thesystemSS obtained by adding SM and CS toB. Lewis’ “official” axiomatizationof the logic of counterfactuals is the systemVC thatis obtained by adding SM and C toV. The extension ofV with just SM is the systemVW.Finally, systemC2 with the incorporation of CEM intoVC is Stalnaker’s logic of conditionals. AsTable 6 makes clear, principles like CS, CMon, RMon, and CEM in particularare controversial, they have all been subject to philosophicaldiscussions.

James Delgrande (1987) proposed the systemNP as adifferent extension ofCK. The main difference isthatB satisfies Cautious Monotonicity, whereasNP satisfies Rational Monotonicity. InDelgrande’s interpretation, an instance of Rational Monotonicityis the inference from the premises “ravens normally fly”and “it is not the case that normally ravens are notblack” to “black ravens normally fly”. Note thatCautious Monotonicity predicts the inference from “ravensnormally fly” and “ravens normally are black” to“black ravens normally fly”. The existence of systemsB andNP makes clear that strictlyspeaking, RMon and CMon are logically independent. But it is fair tosay that RMon isessentially stronger than CMon, because inmost environments one can infer \(\neg(A>\neg C)\) from \(A>C\).In the ordering semantics, RMon needs weak orders which satisfyModularity (for all worlds \(t, u, v, w \in W\), if \(t\prec_w u\),then \(t\prec_w v\) or \(v\prec_w u\)), while CMon needs onlytransitive relations. But RMon may be satisfied without CMon beingsatisfied if one has a weak order over an infinite set withoutanything like the limit assumption and uses a selection function or aminimal models approach. This is the case in Delgrande (1987).^[2]

Beside the systems mentioned, other intermediate systems betweenCK andC2 have been studied, inparticular by proof theorists. Olivetti, Pozzato, and Schwind (2007)and Poggiolesi (2016) present sequent calculi for the systemsCK,CK+ID,CK+SMandCK+SM+ID. Poggiolesi also investigates systems ofnatural deduction for the same systems.

3.4 Nonmonotonic logic and preferential models

Earlier we mentioned that systemP corresponds to theflat fragment of systemB. Given the centrality ofsystemP, here we say more about the way Kraus,Lehmann and Magidor defined it as a system of nonmonotonic logic. KLMdo not aim at characterizing which formulae are logically valid, butat specifying closure conditions for rational sets of conditionals.They read the conditional \(A>C\) (actually they use the notation\(A\pipsim C\)), with \(A\) and \(B\) only from \(L_0\), as expressinga default of the form “if \(A\) then normally \(C\)” or“\(C\) is a plausible consequence of \(A\)”. They startwith models right away, but their models are slightly different fromthe ones we have seen so far. Importantly, KLM have no use for a truthvalue of a conditional at a single world, but only look at the“consequence relation” determined by a full model. WhatKLM call “states” are more like information states thanlike possible worlds, what they call “worlds” arevaluations, i.e., assignments of truth values to the propositionalvariables of the language. KLM’s preferential models are similarto the o-models discussed above, except there is only one binaryrelation for the full model rather than one for each world in themodel.

Acumulative model \(M\) for a universe \(W\) is a triple\(\langle S, l, < \rangle\) where \(S\) is a set of states, \(l: S\rightarrow \cP(W)\) is a labeling function assigning to each state anon-empty set of worlds from the universe \(W\), and \(<\) is abinary relation on \(S\) satisfying thesmoothness condition:for all \(A\in L_0\) and all \(s\) such that\(s\mathrel|\mathrel{\mkern-3mu}\equiv A\) either \(s\) is minimal in\(\hat{A} = \{s \in S: s\mathrel|\mathrel{\mkern-3mu}\equiv A\}\) orthere is an \(s' < s\) that is minimal in \(\hat{A}\). Here\(s\mathrel|\mathrel{\mkern-3mu}\equiv A\) (read: \(s\) satisfies\(A\)) means that for every world \(w\) in \(l(s)\), \(w \VdashA\).

The smoothness condition is almost the same as condition (L) ofsection 3.2.1 above. This notion of a model allows for a modification of theclassical definition of entailment in such a way that it becomescompatible with the idea of defeasibility or nonmonotonicity:

Given a cumulative model \(M = \langle S, l, < \rangle\), theentailment relation \(\pipsim _M\) defined by \(M\) is given by: \(A\pipsim _{M} B\) iff for all states \(s\) minimal in \(\hat{A}\),\(s\mathrel|\mathrel{\mkern-3mu}\equiv B\).

Thus every single cumulative model defines a preferential entalimentrelation—one with respect to which a sentenceAtypically entails much more than just its classical logicalconsequences.

Kraus, Lehmann, and Magidor (1990) focused on a particular class ofcumulative models, the so-called preferential models. They require thebinary relation to be a strict partial order and the labeling functionto assign just one world to each state. Apreferential modelfor a universe \(W\) is a triple \(\langle S, l, < \rangle\) where\(S\) is a set of states, \(l: S \rightarrow W\) is a labelingfunction assigning to each state a single world from the universe\(W\), and \(<\) is a strict partial order on \(S\) (irreflexive,transitive) satisfying thesmoothness condition.

Lehmann and Magidor (1992) focused on a subfamily of preferentialmodels, the so-called ranked models. Aranked model \(R\) isa preferential model \(\langle S, l, < \rangle\) where the strictpartial order \(<\) satisfies Modularity: for all \(s, t, u \inS\), if \(s<t\), then \(u<t\) or \(s<u\).

These modes are called “ranked” because the property ofModularity of the strict partial order < is equivalent to therebeing a totally ordered set \(\Omega\) (the strict order on \(\Omega\)will be denoted by \(\bangle\)) and a ranking function \(r : S\rightarrow \Omega\) such that \(s < t\) iff \(r(s) \bangle r(t)\).Intuitively, a state of smaller rank is “more normal” thana state of higher rank.

The main representation theorem of Kraus, Lehmann, and Magidorconcerns the systemP in terms of preferentialmodels. Below we repeat the rules of the system, except that hereaxioms come in argument form (the entailment symbol“\(\pipsim\)” is not part of the object language), andinference rules in meta-argument form. The rules in the systems theyconsider are not thought of as validity preserving. They are rathertaken to be closure conditions: any reasonable nonnomotonic inferencerelation \(\pipsim \) is such that when it includes the premises(sometimes, though, the premises areexclusion conditions,see RMon, DRat and NRat below), then it includes the conclusion, too.Kraus, Lehmann and Magidor’s representation results then statethat an inference relations \(\pipsim \) is reasonable in thissyntactically specified sense just in case it can be defined by amodel with certain properties.

In Kraus, Lehmann, and Magidor (1990), rules ID\(_{\pipsim}\),LLE\(_{\pipsim}\), RW\(_{\pipsim}\), CCut\(_{\pipsim}\) andCMon\(_{\pipsim}\) together define the systemC ofcumulative reasoning; AND\(_{\pipsim}\) is a derived rule ofC. If OR\(_{\pipsim}\) is added, one gets systemP ofpreferential reasoning; if one keepsAND\(_{\pipsim}\), CCut\(_{\pipsim}\) can be derived from the otherrules ofP.

Lehmann and Magidor (1992) prove that the systemR ofrational reasoning that is obtained by adding the followingrule of Rational Monotonicity to the above set of rules is completewith respect to ranked models.

This rule has an untypical form in that one of its premises is thedenial of a nonmonotonic implication; such rules are called“non-Horn”. If RMon\(_{\pipsim}\) is added toP, then CM\(_{\pipsim}\) can be replaced by the rulethat infers \(A\wedge B\pipsim \bot\) from \(A\pipsim \bot\).Arló-Costa and Shapiro (1992) proved that the system obtainedby dropping CMon\(_{\pipsim}\) fromR is sound andcomplete with respect to “rough” ranked models in whichthe ordering of the states is no longer subject to the smoothnesscondition.

Other important non-Horn rules in the literature are“Disjunctive Rationality” and “NegationRationality”:

Negation Rationality is weaker than Disjunctive Rationality, which inturn is weaker than Rational Monotonicity (see Lehmann & Magidor1992).

3.5 Or-to-if, import-export, simplification of disjunctive antecedents

To conclude this section we point out three axioms that are valid forthe material conditional but which fail in the extensions ofCK mentioned earlier, even in the strong logicsC2 andVC of Stalnaker and Lewis.^[3] These principles concern the interaction of conditionals withdisjunction and conjunction. They are:

The principle OI is intuitively plausible, but as noted by C. I.Lewis, assuming it leads to one of the paradoxes of the materialconditional. For assuming that \(A\) entails \(A\vee B\) (disjunctionintroduction), and that the latter entails \(\neg A>B\), bytransitivity it should follow that \(A\) entails \(\neg A>B\).Various options exist in the face of the paradox: to deny disjunctionintroduction, to deny transitivity, or indeed to deny the validity ofOI. Stalnaker (1975) for one defends the invalidity of OI, arguingthat OI is apragmatically reasonable principle, but that theinference is context-sensitive.

A similar collapse result concerns the principle IE. Gibbard (1980)shows that provided we have Modus Ponens, IE, LLE, and the principleof Supraclassicality—whereby \(A>C\) must hold if \(A\)classically entails \(C\)—, then if the conditional entails thematerial conditional, the reciprocal also holds. McGee (1989) modifiedStalnaker’s semantics in such a way that IE is valid, but MP isno longer valid. McGee’s rejection of MP is independentlymotivated by natural language examples, in particular by the famousexample of the 1980 US elections, in which Reagan was ahead of thepolls as a Republican, Carter second as a Democrat, and then Andersonas a distant third, but also a Republican. In this context, McGeeargues that the following instance of MP is problematic:

(3)

A Republican will win. If a Republican wins, then if Reagan doesnot win, Anderson will win. Therefore if Reagan does not win, Andersonwill win.

Here the conclusion is problematic, because intuitively, if Reagandoes not win, Carter appears as the more likely winner. RecentlyMandelkern (2020) has argued that IE should not be viewed asunrestrictedly valid. Mandelkern’s argument is in part based onlogical considerations and in part based on natural language examples.Mandelkern points out that in a situation (adapted from McGee’scounterexample) in which Reagan is ahead of Carter and Anderson, butthe relative situation of Anderson and Carter is not known, there is acontrast between:

(4)

If a Republican will win the election, and Anderson will win ifReagan doesn’t win, then both Republicans are currently in astronger position to win than Carter.

(5)

If a Republican will win the election, then if Anderson will winif Reagan doesn’t, then both Republicans are currently in astronger position to win than Carter.

According to Mandelkern, the latter is less acceptable than theformer, because the second conditional antecedent “Anderson willwin if Reagan doesn’t” is contextually equivalent to“a Republican will win”. Mandelkern does find IEacceptable if the nested antecedent is restricted to a non-conditionalsentence, however.

Another argument against IE, put forward in Adams (1975), concerns theequivalence of \(A>(B >A)\) and \((A\wedge B) >A\). Whereasthe latter is a clear logical truth, the former is not. S. Kaufmann(2005) thus argues that the validity or invalidity of IE depends onspecific causal assumptions linking the consequent and theantecedents.

The case of SDA is also a bone of contention between theorists.Chellas (1975) and Fine (1975) first pointed out that SDA seemsintuitively valid, against Lewis’s and Stalnaker’stheories. However, as pointed out by Ellis, Jackson, and Pargetter(1977), SDA and LLE together imply Monotonicity: from \(A>C\), itfollows that \(((A\wedge B) \vee (A\wedge \neg B)) >C\) by LLE, andthus by SDA, \((A\wedge B) >C\).

Since then, various theorists have proposed to abandon LLE (viz., Nute1980; Fine 2012; Ciardelli, Zhang, & Champollion 2018; Santorio2018) in order to endorse SDA without Monotonicity. Santorio (2018) inparticular provides a systematic comparison between pragmatic accountsbased on the idea that SDA might be based on a mechanism akin toscalar implictures (see Klinedinst 2009), and semantic accounts basedon the idea that logically equivalent antecedents can directlygenerate different alternatives. Santorio’s proposal, in anutshell, is that \(A>C\) is true iff the alternative propositions\(A_1,\dots,A_n\) that are ways for \(A\) to be true are such that theclosest worlds in them are \(C\)-worlds.

Importantly, Santorio’s account drops LLE but nevertheless makesSDA only optionally valid. Lassiter (2018) recently suggested thatsome classic counterexamples put forward against SDA by McKay andInwagen (1977), and originally judged inconclusive, can bestrengthened. For example, assuming Jim likes odd numbers but isindifferent between them, and actually bet on 5, one can accept:

(6)

If Jim had bet on 1 or 3, there is an exactly 50% probability thathe would have bet on 3.

SDA would predict:

(7)

a.

If Jim had bet on 1 there is an exactly 50% probability that hewould have bet on 3.

b.

If Jim had bet on 3 there is an exactly 50% probability that hewould have bet on 3.

Both conclusions are problematic under at least one reading. However,we note that there is also a reading that would make them fine, if theevaluation of chance is prior to Jim’s betting.

In summary, we see that although OI, IE and SDA appear intuitivelyvalid, to accept them involves substantive trade-offs regarding otherbasic principles.

4. Premise Semantics

4.1 Motivations

Some early theories of conditionals said that a counterfactual istrue, or assertable, if and only if its antecedent, together withfurther “co-tenable” premises, implies its consequent(Ramsey 1931 commenting on Mill; Chisholm 1946; Goodman 1947, 1955;Mackie 1962). D. Lewis called such theories metalinguistic, the ideabeing that “if \(A\), \(C\)” is true provided there is anargument leading from the sentence \(A\) and additional premises to\(C\). On these accounts, the counterfactual is either a sentencemeaning that some suitable argument backing it exists, or—as inMackie’s version—it is itself an elliptical presentationof such an argument. The principal problem of any such theory is tospecify which “additional premises” are suitable to beconjoined to a given antecedent and which are not. Goodman (1955)views the additional premises as general laws, supplemented byrelevant factual conditions, but the problem is to determine whichfacts are to be assumed, and which facts are to be retracted. Rescher(1964) treats additional potential premises as ordered according totheir position in a system of “modal categories”.

An account of the co-tenability idea based on possible worlds wasfurther elaborated in the work of Veltman (1976) and Kratzer (1979),in what has come to be known aspremise semantics (anexpression coined by D. Lewis 1981). Premise semantics can bepresented either as providing acceptance conditions or as providingtruth conditions. It is compatible with an epistemic as well as withan ontic interpretation of conditionals. The former builds on the ideathat the premises represent essentially an agent’s basicbeliefs; this is the approach of Veltman (1976) who took inspirationfrom Ramsey. The latter takes the premises to represent the basicfacts of a possible world; this is the approach of Kratzer (1979,1981), who was inspired by Rescher and Lewis (D. Lewis 1981) and hasfunctions assigning a premise set to every possible world. In anycase, the idea of premise semantics is this. Relative to a given set\(\Gamma\) of premises (assigned to a possible world \(w\)), aconditional \(A>B\) is accepted (or true at \(w\)) if and only iffor every maximal subset of \(\Gamma'\) of \(\Gamma\) that isconsistent with \(A\) it holds that \(\Gamma' \cup \{A\}\) implies\(B\).

4.2 Premise frames

More formally, apremise frame is a pair \(\langle W,\Gamma\rangle\) where \(W\) is a non-empty set of possible worlds and\(\Gamma: W \to \cP(\cP(W))\) is as a function assigning to any world\(w\) a set of propositions—the premises for \(w\). One may, butneed not stipulate that \(w\) does, and all other worlds do not,belong to every proposition in \(\Gamma(w)\) (Centering). For the sakeof simplicity, we assume that \(W\) is finite. Apremisemodel is a triple \(M = \langle W, \Gamma, V\rangle\) such that\(\langle W, \Gamma\rangle\) is a premise frame and \(V\) a valuationfunction. For any proposition \(F\), let \(\Gamma(w)\bot F\) denotethe set of all subsets of \(\Gamma(w)\) that are consistent with \(F\)but have no proper supersets in \(\Gamma(w)\) that are also consistentwith \(F\). That is, \(\Gamma(w)\bot F\) is the set of subsets of\(\Gamma(w)\) that are maximally consistent with \(F\). The truthconditions for conditionals are:

This means that a (counterfactual) conditional is true at a world\(w\) if each set of basic beliefs or facts associated with this worldthat is maximally consistent with \(A\), taken together with \(A\),implies \(C\).

Kratzer shows how this definition gives the flexibility to either“lump” some facts together, or to keep facts separate fromeach other relative to a conditional antecedent. One of herillustrative examples is the following.^[4] Angelika and Regina must pass a bridge one after the other. Angelikatakes one minute to pass the bridge, and Regina is waiting one minutebefore passing. Assuming Angelika had passed the bridge in 40 seconds,would Regina still be waiting for one minute? Let \(B\) represent“Angelika passes in one minute”, \(C\) “Regina waitsone minute” and \(A\) represent “Angelika passes in 40seconds”, with the proviso that \(A\) and \(B\) areinconsistent. The issue is whether the conditional \(A>C\),“if Angelika had passed in 40 seconds, Regina would have waitedone minute”, is true or false. One option is to let \(\Gamma(w)=\{B,C\}\).^[5] \(\Gamma(w)\cup \{A\}\) is inconsistent, but contains as maximal\(A\)-consistent set only \(\{A,C\}\) and the intersection of this setentails \(C\). This predicts that Regina would have waited one minuteindeed under the assumption that Angelika had passed faster. Anotheroption is to let \(\Gamma(w)=\{B\cap C\}\). This time, \(\Gamma(w)\cup\{A\}\) contains \(\{A\}\) as sole maximal \(A\)-consistent set, andthis set fails to entail \(C\). Although the two premise sets have thesame intersection, the lumping of premises in the second set leads oneto retract more facts when making the counterfactual assumption. Thisflexibility, as emphasized by Kratzer, accounts for thecontext-sensitivity of conditionals.

It is easy to see how premise semantics invalidates Transitivity. Forexample, if

then we get that both \(A>B\) and \(B>C\) are true, but\(A>C\) is false at \(w\) (in fact, \(A>\neg C\) is true; seeFig. 2). There are similar counterexamples to Monotonicity andContraposition.

Figure 2: Counterexample againstTransitivity in a premise model [Anextended description of figure 2 is in the supplement.]

4.3 Correspondence with ordering semantics

David Lewis (1981) proved that the truth conditions of premisesemantics and ordering semantics fundamentally correspond to eachother. One can start with a premise model and define a strict partialordering for an o-model that gives the same results: define \(u\prec_w v\) iff the set of all premises in \(\Gamma(w)\) true at world\(u\) is a proper superset of the set of premises true at world \(v\).And conversely, one can start with an o-model based on a strictpartial ordering and define a premise model that gives the sameresults: define \(\Gamma(w)\) as the set of all propositions of theform

for every \(v\in W\).

Lewis in his paper discusses several refinements of thecorrespondence, depending on which conditions are imposed on orderingframes, and further results can be found in Chemla (2011). From aphilosophical and foundational perspective, the correspondence may betaken to suggest that the choice between an ordering-first approachand a premise-first approach doesn’t matter. Arguably, however,a premise semantics may be judged more explanatory than one in whichthe ordering comes first, for two reasons. First of all, from acognitive perspective, it may be argued that premise semantics iscloser to implementing the Ramsey test idea, since an orderingsemantics only implicitly refers to the idea of making adjustments toone’s belief set. Secondly, in an o-semantics the question isindeed moot what determines similarity between worlds. Of course,which facts to admit or not in premise semantics is also problematic,but using propositions as an ordering source makes clearer thatsimilarity between worlds is a relative rather than an absolutematter. In recent years, several authors have proposed to furtherclarify the problem of premise selection in counterfactuals by lookingat causal models of reasoning (K. Schulz 2007; Briggs 2012; S.Kaufmann 2013; Santorio 2019).

Based on the correspondence between ordering semantics and premisesemantics, Kratzer (1979) mentions that Burgess’s systemB is sound and complete for the latter. Veltman(1985: 108–132) proved that flat conditionals interpreted bypremise semantics can be characterized by the axioms of systemP. Results obtained in the 1990s show that suchconditionals have an additional special property that cannot, however,be expressed as an axiom of conditional logic:^[6]

\((*)\)

If \((A\vee B) >C\) is accepted (or true at \(w\)), then thereare \(D\) and \(E\) such that \(D\wedge E\) entails \(C\), and \(A>D\) and \(B >E\) are both accepted (or true at \(w\)).

Among the recent modifications of premise semantics with causalmodels, Santorio’s (2019) filtering semantics ends up giving upOR.

5. Probabilistic Logics

The idea that the right semantics for conditionals may beprobabilistic is appealing for at least three reasons. First of all,as pointed out by Adams (1965), it is unclear whether it makes senseto declare either true or false conditionals with false antecedents.Adams thus prefers to think of conditionals as having assertabilityconditions, rather than truth conditions, and justified assertabilityis naturally handled in terms of probability. Adams’s scepticismabout truth conditions for conditionals is shared by several authors,in particular Gibbard (1980) and Edgington (1995), and it was furtherfueled by Lewis’s (1976) triviality results. Secondly, manyconditional sentences appear to express only a relation of probableinference between antecedent and consequent. Thus, whereas “if\(a=b\), then \(a+1=b+1\)” expresses a purely deductiverelation, “if you boil this egg, it will become hard”expresses an inductive relation, and “if the light does not turnon, then the lamp must be broken” expresses an abductiverelation (Douven & Verbrugge 2010; Krzyżanowska, Wenmackers,& Douven 2013; Douven 2016). But in the same way in whichinductive and abductive inferences express that a conclusion may beinferred from given premises with only a reasonable probability,putting these inferences into a conditional appears to demand theassignment of a degree of probability to the corresponding sentences.Thirdly, as stressed by Stalnaker (1970),

[a]lthough the interpretation of probability is controversial, theabstract calculus is a relatively well defined and well establishedmathematical theory.

Indeed, based on its mathematical simplicity, the theory offers auniversal framework in the psychology of reasoning, where conditionalsoccupy central stage (Over, Hadjichristidis, Evans, Handley, &Sloman 2007). We open this section with Adams’s logic forconditionals. We then explain Lewis’s triviality results and theproblem they raise for the assignment of probabilities to compounds ofconditionals.

5.1 Adams’s logic

Central to probabilistic treatments of conditionals is the notion ofconditional probability. Given a Boolean formula \(A\) of \(L_0\), anda set \(W\) of possible worlds over which they are interpreted, we let\(\Pr(A)\) denote \(\Pr(\lvert A\rvert)\), namely the probabilityassigned to the worlds in which \(A\) is true. For \(A\) and \(C\) twoBoolean formulae, the conditional probability\(\Pr(C\mkern2mu|\mkern2muA)\) is definable by the Ratio Formula:

• (Ratio Formula)\(\Pr(C\mkern2mu|\mkern2muA)=\displaystyle\frac{\Pr(A\wedgeC)}{\Pr(A)}\), provided \(\Pr(A)>0\).

Now consider the problem of determining the probability that if a fairdie lands even, it will land on six. The probability of the materialconditional “either it is not even, or it is a six” is

This clearly exceeds the probability sought, which is instead capturedin terms of the conditional probability of getting a six assuming itis an even number,

More generally, the probability \(\Pr(A\supset C)\) of any materialconditional is never less than the conditional probability\(\Pr(C\mkern2mu|\mkern2muA)\), a point apparently first stressed byReichenbach (1949, p. 437).

Adams’s central assumption is that for every conditionalsentence \(A>C\) in which \(A\) and \(C\) are Boolean sentences,and for every probability function \(\Pr\), the probability of theconditional equals the corresponding conditional probability:

• (Adams’s Thesis) For every Booleansentences \(A\), \(C\), and every probability function \(\Pr\),\(\Pr(A>C)=\Pr(C\mkern2mu|\mkern2muA)\), provided\(\Pr(A)>0\).

Adams’s Thesis has two sides. From the inner standpoint ofAdams’ logic of conditionals, it may be viewed as adefinition: the conditional probability defines the degree ofprobability assigned to a simple conditional. From the outerstandpoint of ordinary English, Adams’s Thesis makes anempirical claim: namely, that theassertability of a simpleconditional is a function of the conditional probability of theconsequent given the antecedent.

Stipulating that \(\Pr(A>C)=1\) when \(\Pr(A)=0\), Adamsinvestigates several notions of validity for the language \(L_1\) offlat conditionals. His original definition is the following:

• (Probabilistic Validity) An inference isprobabilistically valid if and only if, for any positive\(\varepsilon\), there exists a positive \(\delta\) such that underany probability assignment under which each of the premises has aprobability greater than \(1 - \delta\), the conclusion will have aprobability of at least \(1 - \varepsilon\).

One way of understanding this definition is by contraposing it: anargument is invalid when an arbitrarily high probability can beassigned to the premises, but such that the conclusion falls short ofa high probability.

Adams contrasts this definition with an alternative criterion forvalidity that is perhaps more intuitive:

• (Strict Validity) An inference isstrictlyvalid if and only if its conclusion has probability 1 under anyprobability assignment under which its premises each have probability1.

Every probabilistically valid inference is strictly valid, but notvice versa. Strict validity coincides with classical validity. Inlater work, Adams uses an alternative definition of probabilisticvalidity, based on the notion of uncertainty. Given a probabilityfunction \(\Pr\), the uncertainty U\((A)\) of a sentence \(A\) isdefined as \(1-\Pr(A)\):

• (p-Validity) An inference isp-valid ifand only if for every probability assignment, the uncertainty of theconclusion is no greater than the sum of the uncertainty of thepremises.

Probabilistic validity and \(p\)-validity can be shown to coincide(Adams 1975: Thm. 3.1). For Boolean formulae, moreover, Adams showsthat the three definitions of validity coincide with classicalvalidity. For conditional sentences, however, the resulting logic isweaker, and corresponds in fact to systemP.

As in the semantics considered insections 3 and 4, Monotonicity, Transitivity, and Contraposition failin Adams’ semantics. Consider Transitivity: it is easy toconstruct models in which \(\Pr(B\mkern2mu|\mkern2muA)=1\),\(\Pr(C\mkern2mu|\mkern2muB)>1-\delta\) for positive, butarbitrarily small \(\delta\), and yet \(\Pr(C\mkern2mu|\mkern2muA)=0\)(seeFig. 3).

Figure 3: Counterexample againstTransitivity in an Adams model [Anextended description of Figure 3 is in the supplement.]

This yields a counterexample according to either definition ofprobabilistic validity. Similar examples can be constructed forMonotonicity, Contraposition, and the paradoxes of the materialconditional.

5.2 Lewis’s triviality theorem

Viewed as logics that tell us which conditionals are consequences of agiven set of conditionals, Adams’s logic coincides, over theircommon language, with Stalnaker’s and Lewis’s logics. Thatis, for every argument with a finite set of premises \(\Gamma\) andconclusion \(A\) of \(L_1\), \(A\) is a \(p\)-consequence of\(\Gamma\) iff \(A\) is a consequence of \(\Gamma\) according toStalnaker’s semantics (see Gibbard 1980 for a proof).^[7] However, the main limit of Adams’s logic is that it does notaccount for embeddings of conditionals (negated conditionals,disjunctions of conditionals, right-nested and left-nestedconditionals). Nevertheless, the coincidence of the logics generatedby possible worlds and probabilistic semantics suggests thatAdams’ Thesis could be extended to the language \(L_2\) ofarbitrarily complex conditionals. This hypothesis is known asStalnaker’s Hypothesis, who formulates it in his 1970 article.^[8]

• (Stalnaker’s Hypothesis) For everyprobability function \(\Pr\) and for every conditional \(A>C\),possibly complex: \(\Pr(A>C)=\Pr(C\mkern2mu|\mkern2muA)\), providedthat \(\Pr(A)>0\).

The empirical adequacy of Stalnaker’s Hypothesis depends on thetreatment of nested conditionals. However, Lewis (1976) shows thatunder rather uncontroversial assumptions on the probability of nestedconditionals, the thesis leads to triviality. Lewis assumes thefollowing Factorization Hypothesis (Fitelson 2015):

• (Factorization Hypothesis) For every probabilityfunction \(\Pr\) and for all sentences \(A\) and \(B\) such that\(\Pr(A\wedge B)>0\):
• \[\Pr(B >C\mid A)=\Pr(C\mid A\wedge B).\]

Given Stalnaker’s Hypothesis, the Factorization Hypothesis isequivalent to the Law of Import-Export, namely \(\Pr(A > (B >C))=\Pr((A\wedge B) > C).\)

From Factorization, Lewis derives the following result, whichbasically expresses that the probability of the conditional collapsesto the unconditional probability of its consequent:

• Triviality Theorem (Lewis) If \(A\) isprobabilistically compatible with both \(C\) and \(\neg C\), that is,if \(\Pr(A\wedge C)>0\) and \(\Pr(A\wedge \neg C)>0\), then\(\Pr(A>C)=\Pr(C)\).

The proof goes as follows:

Lewis himself states further variants, and several generalizationshave been proposed since (Hájek & Hall 1994; Bradley 2000;Milne 2003; Fitelson 2015). Bradley (2000) in particular shows thatthe following Preservation Condition inspired by Belief Revisiontheory also leads to triviality.

• Preservation Condition (Bradley): For everyprobability function \(\Pr\) and all sentences \(A\) and \(C\), if\(\Pr(A)>0\) and \(\Pr(C)=0\), then \(\Pr(A>C)=0\).

The condition is weaker than Stalnaker’s Thesis, and thereforeit extends triviality to an even wider class of conditionals.

5.3 Cardinality and imaging

Lewis’s triviality results expose a mismatch between\(\Pr(C\mkern2mu|\mkern2muA)\) and \(\Pr(A>C)\). To clarify thephenomenon, we ask what each quantity represents in light ofLewis’s result, already for simple conditionals.

5.3.1 Cardinality

Let us start with \(\Pr(C\mkern2mu|\mkern2muA)\). Even with a fixedprobability assignment over a finite set of worlds, the conditionalprobability may not correspond to the probability of any set ofworlds. Let \(W=\{a,b,c\}\) and let

Clearly,

However, in the Boolean algebra generated from \(W\), nopossible-worlds proposition gets a probability of ½ under\(\Pr\). Hájek (1989) in fact showed that there are moredistinct conditional probability values than distinct unconditionalprobability values, for any finite set of possible worlds \(W\) thathas at least three worlds with non-zero probability. As pointed out inÉgré and Cozic (2011), this result offers a structuralanalogy with undefinability results in generalized quantifier theory.“Most \(A\)s are \(C\)s” is not definable by applying theunrestricted “most” to first-order definable propositions.Similarly here, we may view the result as showing that in a sentencelike “there is a probability of ½ that if \(A\) then\(C\)”, the embedded conditional does not express anyproposition.

This result, however, is compatible with the idea that if-clausesfundamentally act as restrictors of overt or covert operators (seeLewis 1975; Kratzer 2012; andsection 8.1 below). Égré and Cozic (2011) use the result tovindicate that view, though Charlow (2016) argues that the restrictorview is not in itself immune to triviality. Another take on itconcerns three-valued approaches. In the three-valued case, theassertability of a proposition is naturally viewed as the probabilityof that proposition being true given it has a determinate truth value(McDermott 1996). Assuming De Finetti’s scheme for theconditional, the assertability of \(A>C\) is then equal to theprobability that \(A\wedge C\) is true, given that \(A\) is true, andso to the conditional probability. This derives Adams’ Thesisfor simple conditionals. That fact notwithstanding, note that with\(a, b,c\) as atoms and \(\Pr\) as equiprobable on them, again thereis no three-valued proposition whose probability is ½ for theconditional \(a\vee b > a\), if we define a possible world by anassignment of one of the three values to atomic propositions (andhandle disjunction Strong Kleene style). Rather, if three-valuedconditionals do express propositions, those propositions then need tobe viewed as relations between those worlds where the sentence is trueand those where it is defined, so again as more complex objects thanset of worlds (cf. Dubois & Prade 1994 who represent them aspairs).

5.3.2 Imaging

Consider now \(\Pr(A>C)\). Assume Stalnaker’s semantics for\(A>C\); what does the probability of that conditional correspondto, if not the conditional probability? D. Lewis (1976) identifies arevision rule he callsimaging. Given \(\Pr\), he defines\(\Pr\mkern.1mu'_A\) to be the imaging under the antecedent, where\(\Pr\mkern.1mu'_A(w)=0\) if \(w\) is not an \(A\)-world; and if \(w\)is an \(A\)-world, then \(\Pr\mkern.1mu'_A(w)\) equals \(\Pr(w)\)augmented by the sum of the non-\(A\) worlds to which \(w\) comesclosest according to Stalnaker’s selection function. Lewis showsthat the probability of Stalnaker’s conditional is theprobability of the consequent under imaging by the antecedent. Whereasconditioning upon \(A\) kills off the non-\(A\) worlds and spreadstheir mass uniformly over the \(A\)-worlds, imaging upon \(A\) is adifferent operation, which shifts the weight of non-\(A\) worldsnon-uniformly to the \(A\)-worlds depending on the selection function(seeFig. 4).

(a) Conditionalization on \(A\) results in posterior probabilities\(\Pr'(u)=\frac{p}{p+q}\), \(\Pr'(v)=\frac{q}{p+q}\) and\(\Pr'(w)=0\).

(b) Imaging on \(A\) results in posterior probabilities\(\Pr'(u)=p+r\), \(\Pr'(v)=q\) and \(\Pr'(w)=0\).

Figure 4: Example with \(W=\{u,v,w\}\)and prior probabilities \(\Pr(u)=p\), \(\Pr(v)=q\) and \(\Pr(w)=r\).In the figure a red dashed arrow from \(x\) to \(y\) represents that\(y\) is the closest \(A\)-world to \(x\) [Anextended description of Figure 4 is in the supplement.]

An example from Edgington (2020) can help to figure out thedifference. Assume a stack of 100 straws, out of which 90 are 10 cmlong, 1 is 11 cm long, and 9 are 20 cm long. Consider the conditional“if the straw is longer than 10 cm, then it is shorter than 15cm”. Assume similarity between worlds is set by the closeness inlengths between straws. The probability of the set of worlds in whichStalnaker’s conditional is true is \(\nicefrac{91}{100}\), sincethe 10-cm-straw worlds and the 11-cm-straw world are such that theclosest world in which the straw is longer than 10 cm is indeed aworld where it is less than 15 cm. But the conditional probability forthe conditional is \(\nicefrac{1}{10}\), since only one of the strawsthat are 11 cm or 20 cm long is less than 15 cm.

The upshot of these considerations is that starting fromStalnaker’s conditional, we can see that its probability is notthe conditional probability, but the imaging probability (for more onimaging, we refer to Günther 2018). Conversely, starting from theconditional probability of \(C\) given \(A\), we have seen that it isnot in general equal to the probability of any possible worldproposition. Importantly, Adams’s Thesis remains coherent forsimple conditionalsqua definition, because it stipulatesthat the probability of a conditional sentence is a certainconditional probability. However, as an empirical thesis,Stalnaker’s Hypothesis states a substantive identity betweenquantities defined separately, and we now see better why it shouldfail. Whether Adams’ Thesis holds as a matter of empirical factand even for simple conditionals is still debated. Recent findings, towhich we return insection 7, suggest that relevance considerations also come into play (Douven2016; Skovgaard-Olsen, Singmann, & Klauer 2016).

5.4 Probabilities for compound conditionals

Stalnaker’s Thesis is consistent for simple conditionals, butfails for conditionals of arbitrary complexity. A challenge is toextend Stalnaker’s Thesis to more or less extended classes ofcompound conditionals. Several proposals have been made toward thatgoal. The associated literature is quite technical, however, and wegive only a few landmarks here. Important is that all the resultingsystems mentioned here conservatively extend Adams’s basiclogic, namely systemP (i.e., they preserve the sameaxioms for simple conditionals, though not necessarily for compoundconditionals).

Van Fraassen (1976) extends Stalnaker’s Thesis to right-nestedconditionals of the form \(A> (B>C)\) in which \(A, B,C\) areBoolean, and to left-nested conditionals \((A>B)>C\) with thesame restriction, and he can also deal with specific conjunctions anddisjunctions of conditionals. Van Fraassen’s proposal involves aproduct space construction, relying on so-called Stalnaker-Bernoullimodels, allowing him to assign probabilities to infinite sequences ofworlds based on standard Stalnaker models. The construction is used inStalnaker and Jeffrey (1994) and extended in S. Kaufmann (2009). Onenoteworthy aspect of the construction is that the probability ofright-nested conditionals of the form \(A>(B>C)\) is not in allcases identical to the probability of \((A\wedge B) > C\). McGee(1989) on the other hand proposes a distinct construction that doesnot permit left-nested conditionals, but which is intended to securethe law of import-export for right-nested conditionals (famously, bygiving up Modus Ponens for nested conditionals). Both approaches havesome probabilistic principles in common, however, for instance

holds in both theories. For both approaches, moreover, the associatedlogics are semantically interpretable starting from Stalnaker’spossible world semantics. Bradley (2012) builds on theJeffrey-Stalnaker and McGee approaches to show that for simpleconditionals Adams Thesis can be derived by assigning probabilities topairs of worlds, instead of just worlds. The probability of a pair\((w_0,w_1)\) represents the probability that \(w_0\) is the actualworld, and that \(w_1\) is its selected alternative given theantecedent under discussion (note that one needs such probabilitiesfor all potential antecedents). A generalization of this property tonested conditionals is outlined, with philosophical emphasis on theidea that the probability of a conditional does not depend on a singleworld, but also on the nonfactual relations of counterfactualdependence between worlds.

A related approach to iterated conditionals is based on the so-calledcoherence approach to probability, coming from de Finetti’sbetting view of probability (see Coletti & Scozzafava 1999; Gilio2002; Gilio & Sanfilippo 2014; and Pfeifer 2014 for anintroduction). The idea is that probability degrees can be attached toconditional events so long as they do not violate certain coherenceconstraints on those bets (no sure losses or “Dutchbooks”). In this framework the probability of an event isdefined in terms of the technical notion of prevision. The prevision\(x\) attached to \(A>C\) corresponds to the fact that

for every real number \(s\), you are willing to pay (resp., toreceive) an amount \(sx\) and to receive (resp., to pay) \(s\), or\(0\), or \(sx\), according to whether \(A\wedge C\) is true, or\(A\wedge\neg C\) is true, or \(\neg A\) is true (bet called off),respectively. (Sanfilippo, Gilio, Over, & Pfeifer (2020 notationadapted))

Depending on independence assumptions concerning basic propositions,previsions can be defined not only for conjunctions of conditionals,but also for various nested conditionals. A recent overview of resultsin that domain appears in Sanfilippo et al. (2020).

6. Belief Revision Approaches

While Stalnaker rendered Ramsey’s suggestion about the role of“if \(A\)” in terms of closest antecedent world, and Adamsfocused on the probabilistic interpretation of it, PeterGärdenfors (1978) introduced a semantics of conditionals thatuses the Ramsey idea in a direct and purely qualitative way. Hisapproach is resolutely epistemic—or more precisely,doxastic— and it is based on the notion of belief or acceptance.Precursors of this approach were William Harper (1976) and Isaac Levi(1977).

6.1 Belief revision models and the Ramsey test

Abelief set over a language \(L\) is a set of sentences thatis closed under a given background logic \(\iCn\) (i.e., atheory in the logician’s sense). Abelief revisionmodel (BRM) is a pair \(M = \langle \cK, *\rangle\), where\(\cK\) is a set of belief sets over \(L\) and *: \(\cK \times L \to\cK\) is a belief revision function assigning to every belief set\(K\) and every sentence \(A\) a revised belief set \(K*A\). Theacceptance condition for conditionals at a belief set in a beliefrevision model is a straightforward formalization of Ramsey’stest:

Gärdenfors’s (1978, 1988: sects. 7.1–7.2) initialidea was to take conditionals that are accepted at a belief set asbeliefs, i.e., as elements of this very belief set. Let uscall this Gärdenfors’s Ramsey test:

Principles for the logic of conditionals then fall out directly of thefollowing rationality postulates for belief revision introduced byGärdenfors (1978) and Alchourrón, Gärdenfors, andMakinson (“AGM”, 1985). Using the abbreviation \(K+A\) for\(\iCn(K\cup\{A\})\), the eight AGM-postulates read as follows: Forall belief sets \(K\) (in all BRMs \(M\)) and all sentences \(A\) and\(B\),

The numbering is AGM’s, the names are due to Hansson (1999).Postulates (\(*\)3) and (\(*\)4) taken together say that if the input\(A\) is consistent with the belief set \(K\), then the backgroundlogic alone is sufficient to guide belief revision. When (\(*\)1) and(\(*\)2) are present and \(\iCn\) is a monotonic consequenceoperation, (\(*\)4) is equivalent to thePreservationPrinciple stating that if \(A\) is consistent with \(K\), thenall elements of \(K\) are preserved in \(K*A\):

(Pres)

If \(\neg A\not\in K\), then \(K \subseteq K*A\).(Preservation)

AGM called (\(*\)1)–(\(*\)6) thebasic set ofpostulates. Thesupplementary postulates (\(*\)7) and(\(*\)8) concern composite belief revisions of the form \(K*(A\wedgeB)\). The idea is that, if \(K\) is to be changed minimally so as toinclude two sentences \(A\) and \(B\), such a change should bepossible by first revising \(K\) with respect to \(A\) and thenexpanding \(K*A\) by \(B\)—provided that \(B\) does notcontradict the beliefs in \(K*A\).

The logic of conditionals then mirrors the logic of belief revision:(\(*\)1) corresponds to RCK, (\(*\)2) to ID, (\(*\)3) to SM, aweakened version of (\(*\)4) (viz., if \(A \in K\) and \(K\) isconsistent, then \(K \subseteq K*A\)) to CS, (\(*\)6) corresponds toLLE, (\(*\)7) to OR and (\(*\)8) to RMon. In establishing this logic,Gärdenfors used the following notion of validity: a sentence\(A\) is valid in a BRM \(M\) if there is no consistent \(K\in \cK\)with \(\neg A\in K\). A sentence \(A\) is valid iff it is valid inevery BRM. Arló-Costa and Levi (1996) call this notionnegative validity and contrast it with a notion ofpositive validity that they argue is preferable: \(A\) ispositively valid in a BRM \(M\) if \(A\in K\) for every \(K\in\cK\).

6.2 Gärdenfors’s triviality theorem

Gärdenfors (1986; 1988 sects. 7.4–7.7) soon noticed thatcoupling the Ramsey test with the AGM theory of belief revision leadsto a trivialization, which is in many respects similar toLewis’s trivialization (seesection 5.2).

• Triviality Theorem (Gärdenfors):A beliefrevision model satisfies the AGM postulates(\(*\)4) and(\(*\)5) and Gärdenfors’s Ramsey Test only if it is trivial in thesense that it does not contain any belief set that is logicallycompatible with three sentences \(A\), \(B\) and \(C\) that arepairwise inconsistent with each other.

A quick and simple proof goes as follows: Suppose for reductio that wehave a BRM \(M = \langle \cK, *\rangle\) with a belief set \(K\in\cK\) that is logically compatible with three pairwise inconsistentsentences \(A\), \(B\) and \(C\). First notice that

by (\(*\)4), and similarly \(C\in (K+(A\vee C))*\neg A\). So byGRT,

and

Since \(A\) is logically stronger than both \(A\vee B\) and \(A\veeC\), we have

and

But since \(A\) is compatible with \(K\), \(K+A \subseteq K*A\), by(\(*\)4). So both \(\neg A > B\) and \(\neg A > C\) are in\(K*A\). This implies, by GRT again, that both \(B\) and \(C\) are in\((K*A)*\neg A\). Since \(B\) and \(C\) are inconsistent with eachother, the latter belief set is inconsistent, contradicting (\(*\)5).So the supposition is impossible.

Gärdenfors’s triviality theorem was widely perceived as anunexpected and somewhat shocking result. Postulate(\(*\)5) can hardly been seen as the culprit, so the triviality theorem wasinterpreted by Gärdenfors basically as a conflict between(\(*\)4) or the Preservation Principle on the one hand, and the Ramsey Test onthe other. But the reactions to his result were mixed, and noconsensus has emerged so far.

A first position, taken by Levi and very much in line with thosedenying that conditionals have truth values, was that conditionals donot express propositions, but epistemic appraisals, and thus shouldnot be objects of belief and elements of belief sets, but rathersentences that can only be accepted. This amounts to rejectingGRT.

Another group of researchers suggested to modify or restrict theRamsey test. Solutions of this kind were discussed, amongst others, byRott (1986), Gärdenfors (1987, 1988), Levi (1996: Chapter 2),Lindström and Rabinowicz (1998) and Nute and Cross (2001).Arguably, none of these adaptations of the Ramsey test has retainedthe intuitive appeal of the original idea.

Others think that the Ramsey test is perfectly all right. Some authorsargue that the right reaction is to say that conditionals should notbe analyzed by AGM-stylerevisions, but rather byupdates in the style of Katsuno and Mendelzon (1992). Membersof this party include Ryan and Schobbens (1997), Grahne (1998) andCrocco and Herzig (2002). Updates in this sense are closely related toLewis’s (1976) imaging (seesection 5.3.2 above). These operations violate the AGM-postulate (\(*\)4) and thePrinciple of Preservation, but satisfy the following Monotonicitycondition which is immediately entailed by GRT:

Rott (2011) defends Gärdenfors’s Ramsey test by showingthat it does not create a specific problem for the PreservationPrinciple. Its consequence is rather that both preservation of factualbeliefs and preservation of flat conditionals entail that there is nopreservation of right-nested conditionals of depth 2. GRT allows forright-nested conditionals if a model for iterated belief revision isavailable (other compounds of conditionals are more difficult to makesense of). It can be shown, for instance, that supposing there is anatom \(p\) on which the believer suspends judgment, theconditional

functions exactly like the epistemic modal “might \(\negA\)”: it is believed (or accepted) if and only if \(A\) is notbelieved. Thus, if our language includes right-nested conditionals,there are no two belief sets \(K\) and \(K'\) such that \(K\) is astrict subset of \(K'\). Rott argues that it has simply been a mistaketo take it for granted that Preservation can and should be transferredfrom factual language to languages containing nested conditionals.

Detailed analyses of the dialectics of the situation emerging fromGärdenfors’s triviality theorem can be found inLindström and Rabinowicz (1998), Nute and Cross (2001), andArló-Costa (2007). An up-to-date discussion of related problemsis Boylan & Schultheis (forthcoming).

7. Relevance and Difference-Making

Since MacColl’s and C. I. Lewis’s strictures againstmaterial implication, there has been an increasing number of voicesarguing that some notion of relevant connection between antecedent andconsequent is part of the meaning of conditionals. In the area ofconditionals, the notion of relevance is claimed by distincttraditions, however, and means slightly different things.^[9]

One tradition concerns so-called relevant logics, designed toformulate a more demanding notion of implication than Lewis’sstrict implication (cf. Meyer & Routley 1973). The idea ofrelevant implication is based in large part on the idea that theantecedent and the consequent of an acceptable conditional \(A>B\)must be connectedin terms of topics. For example, aconditional of the form \((p \wedge \neg p) > q\) need not bevalid, despite having an impossible antecedent, since the consequentshares no content with the antecedent. Various constraints can beimposed on the idea of content-sharing, leading to variousnon-classical logics of so-called relevant implication (seeMares’s 2020 SEP entry on relevant logics for an overview, andWeiss 2019 on the use of analytic entailment to deal withSextus’ so-called fourth conditional).

Another approach to the notion of relevance is in terms of the ideathat the antecedent of a conditional should make a difference to thetruth, assertability or probability of the consequent. Consider aconditional like “if London is in England, then Aristotle was aGreek philosopher”. The conditional sounds odd because the truthof the antecedent does not matter to the truth of the consequent.While relevant logics capture the idea of relevance in terms oftopic-sharing, here the idea of relevance is captured in terms of thedifference that the antecedent makes for the consequent.

The idea ofdifference-making itself can be formulated inseveral ways. One way is proposed by Douven, in what he calls the“Evidential support thesis” (“EST”, Douven[2008, 2016: 108]). Basically, the EST says that a conditional\(A>C\) is relevantly assertable provided \(\Pr(C|A)\) exceeds asufficiently high threshold and \(\Pr(C|A)>\Pr(C)\). Douven’smain argument in support of the latter condition—which is arelevance criterion—is that high probability is not enough forassertion. He gives the example of the following minimal pair:

(8)

a.

If there is at least 1 Head in the first 10 tosses of this coin,then there is at least 1 Head in the first 100,000 tosses.

b.

If Chelsea wins the Champion’s league, then there is atleast 1 Head in the first 100,000 tosses.

Both statements are such that the probability of the consequent giventhe antecedent is high. However, in (8)-b the antecedent does notincrease the probability of the consequent, it is irrelevant in thatsense, and the conditional may be called a “non-interferenceconditional”. Douven (2016) defines validity in terms ofacceptability-preservation according to the EST criterion, for allthresholds \(t\in [0.5, 1)\). The resulting logic is weak, inparticular it violates Modus Ponens, RW, AND, CMon, Cut, OR andContraposition. Douven establishes the validity and invalidity ofspecific principles (see Douven 2016: Theorem 5.2.1), but a sound andcomplete axiomatization of his logic is still an open problem.

More recently, an alternative approach was proposed by Crupi andIacona (forthcoming-b), yielding a stronger logic. Crupi and Iaconause the Rips measure of confirmation and associate to a conditional\(A>C\) a score defined as

when \(P(C|A)>P(C)\), and defined to be equal to \(0\) otherwise.This yields a modified score of “uncertainty” for aconditional, which they substitute for the standard notion ofuncertainty in Adams’s definition of \(p\)-validity. The logicthey identify violates RW and Cut, but it validates AND, CMon, OR,Modus Ponens, restricted forms of Aristotle’s andAbelard’s theses (principles of connexive logic), and evenContraposition. Crupi and Iacona (forthcoming-a) present apossible-worlds semantics for what appears to be the same logic, inwhich contraposition plays a central role.

Ahead of this recent wave of work, Rott (1986) pioneered anothermeasure of relevance in a belief revision framework, a measure pickedup by Spohn (2013), and proposed independently in various accounts ofcausality. Arguing against Conjunctive Sufficiency (seesections 2.2), Rott’ suggests that a “difference-making”conditional \(A>C\) is accepted provided \(C\in K*A\) and \(C\notinK*\neg A\). This is a relevantized variant of the Ramsey Test, rulingthat the conditional is accepted provided the revision by \(A\)results in the belief that \(C\), but the revision by \(\neg A\) doesnot result in the belief that \(C\). Using the terminology of“difference-making conditionals”, Rott (forthcoming) callsthe violation of RW the hallmark of conditionals that encode the ideathat the antecedent is positively relevant to the consequent. Thisyields a logic that also invalidates CMon, Cut, Or and Contraposition,but it validates And, Aristotle’s theses, a restricted form ofAbelard’s thesis as well as conditions such as

(\(>\)1)

If \(A >B\wedge C\), then \(A>B\) or \(A>C\).

(\(>\)2a)

\(A>C \tiff (A >A\wedge C \text{ and }A >A\vee C)\).

In general conditionals with a relevance reading tend to behave inrather unusual and unruly ways. Raidl (2021) develops a generaltechnique of transferring completeness results for more“well-behaved” suppositional conditionals \(>\) torelevance-expressing conditionals definable in terms of \(>\). Heis thus able to cover the proposal of Crupi and Iacona as well as thatof Rott in a unified manner.

Rott’s proposal turns out to be related to what is known as the“difference measure” or as the “score ofcontingency”

in the psychology of causality (Shanks 1995; Cheng 1997). Inparticular, (Skovgaard-Olsen et al. 2016) found that depending onwhether this measure is positive, negative, or null, the probabilityof a conditional may or may not equal the conditional probability. Inother words, relevance considerations appear to interact withassignments of probabilities to conditionals, the main finding beingthat Adams’ Thesis appears to hold good only when \(\Delta(p)\)is positive. The equation breaks down, on the other hand, when\(\Delta(p)\) is null, so precisely in the case of non-interferenceconditionals. Furthermore, Rooij and Schulz (2019) argue that an evenbetter predictor of the acceptability of conditionals is given by anormalized measure of contingency, that is, by the measure

Whether in logic or in the psychology of reasoning, the investigationof relevance measures and of the resulting inferences is currently avery active field in the study of conditionals.

8. Modals and Speech Acts

The previous sections have looked at propositional logics ofconditionals, which extend Boolean logic with a distinguishedconditional operator. To wrap up, we look briefly at some richerlanguages, in particular at the interplay of conditionals withmodalities, and at the interplay of conditionals with other speechacts than assertions.

8.1 Modalities

Possible-worlds analyses of conditionals make it possible to definemodalities from conditionals as follows (Stalnaker 1968; Nute1980):

Conversely, we saw how strict conditional analyses treat theconditional as definable from modalities. Those facts may seem tosuggest that modals and conditionals are interdefinable. However, theinteraction between them is actually more complex.

Linguistic facts about conditionals suggest that across a wide rangeof languages, modal operators commute with conditionalantecedents.

(9)

If \(A\) then necessarily \(C \equiv\) Necessarily if \(A\) then\(C\).

(10)

If \(A\) then possibly \(C \equiv\) Possibly if \(A\) then\(C\).

Such facts centrally motivate Kratzer’s account of conditionalsas restrictors of modal operators (Kratzer (2012), inspired by D.Lewis 1975). As Kratzer (1991, pp. 468–469) puts it,“there is a very close relationship betweenif-clausesand operators likemust”, and more specifically“for each world, theif-clause has the function ofrestricting the set of worlds which are accessible from thatworld”.

Kratzer formulates the restrictor view of conditionals by relativizingthe semantics of modals to two premise sets, a so-called modal base(“hard” facts known or believed by the speaker) and anordering source (“soft” facts taken to be revisable).Consider a model \(\langle W, \Gamma, \Delta\rangle\) where \(\Gamma\)is a premise function for the ordering source, and \(\Delta\) thepremise function for the modal base. Let us assume the Limitassumption and let \(\max_{\Gamma(w)}(F)\) be the set of worlds in aproposition \(F\) that satisfy a maximal set of propositions in\(\Gamma(w)\). Kratzer’s semantics for \(\Box\) is:

(Kratzer’sMust)

The conditional “if \(A\) then \(C\)” can be treated as abinary modality \(\Box(A,C)\), with the following truth conditions,where \((\Delta+A)(w)\) equals \(\Delta(w) \cup \{\lvertA\rvert^{M,w,\Gamma, \Delta}\}\):

(RestrictedMust) \[M, w,\Gamma, \Delta\Vdash \Box (A,C) \tiff M, w, \Gamma, (\Delta+A) \Vdash \Box C\]

For bare conditionals, Kratzer postulates that the antecedent alsorestricts a covert modal. This particular assumption is a source ofcontroversy. Several authors point out that whileif-clausesmay restrict operators, modal operators and conditional operators canalso have separate contributions (von Fintel 1994; M. Kaufmann &S. Kaufmann 2015).

An important issue is whether the interaction of modals andconditionals can be accounted for when both types of operators arelogically represented in the language. We highlight two recentaccounts along those lines. Both take inspiration from Yalcin’s(2007) conception of modal operators as expressive devices, and alsofrom Veltman’s (1996) and Gillies’s (2004) emphasis on theidea that utterances involving epistemic modals update informationalstates.

Ciardelli (forthcoming) states truth conditions for a language withmodal operators and a conditional operator. He defines the semanticvalue of sentences relative to two parameters, a state parameter \(s\)(set of worlds), and an attitude parameter \(a\). In Ciardelli’sapproach,if-clauses restrict the state \(s\), as inKratzer’s approach, but modals basically act on the attitudeparameter \(a\), which can express various modal forces (universal,existential, or probabilistic). The framework allows him to derive asequivalences the facts(9) and(10), in the following guise:

Bare conditionals of the form \(A>B\) turn out equivalent toconditionals \(A>\Box B\), but only under the default attitudecoming with universal force. Ciardelli (forthcoming) also derives theincompatibility of the conditional \(A>C\) with themight-conditional \(A>{\Diamond \neg C}\) (If-Might Contradiction,henceforth IMC). The same incompatibility is at the heart ofSantorio’s (forthcoming) so-called path semantics forconditionals. Santorio exposes a contradiction between this principleand two other principles of conditional logic, namely CEM and theDuality principle whereby \(A>\Diamond C\,\equiv\, \neg (A >\negC)\) (endorsed by D. Lewis 1973, though rejected by Stalnaker 1968).His semantics validates IMC and preserves CEM, but drops Duality. Anumber of further predictions follow from Santorio’s semantics,in particular OI becomes valid, as well as restricted versions of IE.Like Ciardelli, Santorio does not have to treat bare conditionals asimplicitly modalized, and like Ciardelli he can derive commutationfacts about modalities.

Beside those proposals, Omori (2019) focuses also on the interplay ofnegation, modals, and conditionals in the framework of connexivelogics. Omori uses Odintsov and Wansing’s (2010) four-valuedframework for modal logic, to show that conditional negations ofvarying strength for \(A>B\) can be derived from the outer negationof a conditional (building on distinctions made in Égré& Politzer 2013), in particular \(A>{\Diamond\neg B}\) insteadof \(A >\neg B\). He gives sound and complete axiomatizations forthe resulting conditional logics. An earlier connexive system isOlkhovikov (2002)’s logic LImp,effectively combining Cooper’s conditional with a trivalentnecessity operator of determinate truth (mapping 1 to 1and ½ and 0 to 0).

8.2 Beyond assertion

Conditionals do not only appear in declarative sentences, but also inquestions and in commands, as in:

(11)

If John visits, will Mary visit too?

(12)

If John visits, make sure to bring cheese!

The semantics of imperatives and interrogatives lies beyond the scopeof this entry. However, we mention such constructions because they toomay be seen as supporting Kratzer’s restrictor analysis. Inparticular, S. Kaufmann and Schwager (2011) treat imperatives asinvolving a covert modal operator, which can have a deontic, orbouletic meaning, but basically acting like “must”. In thecase of questions, Isaacs and Rawlins (2008) treat questions asprefixed with a covert speech act operator, coming with a modal domainrestricted by theif-clause. In a related way, Ciardelli,Groenendijk, and Roelofsen (2018) point out that the semantics of aquestion can be specified by resolution conditions relative to aninformation state. Again, conditional questions can be viewed asappropriately restricting the information state to the antecedentworlds, so as to resolve the question relative to those (seeCiardelli, forthcoming). For a polar question of form \(?M\)(“will Mary visit?”) the information state \(s\) resolvesit if either \(s\) supports \(M\) or \(s\) supports \(\neg M\). In thecase of a conditional polar question like(11), the semantics of Ciardelli, Groenendijk, and Roelofsen (2018)predicts that a state \(s\) will resolve a question of the form \(J>{?M}\) provided either \(s\) supports \(J >M\) or \(s\)supports \(J >\neg M\).

Further connections have been explored between the semantics ofinterrogatives and the very structure of conditional sentences. Starr(2014b) in particular argues that all conditionals have atopic-comment structure, allowing to treat the antecedent as a covertquestion itself. Finally, more aspects of the interaction ofconditional phrases with speech acts deserve to be mentioned,including so-calledbiscuit-conditionals (“if you wantsome biscuits, there are some in the cupboard”, Austin 1956,Lycan 2001), as well as concessives (“even if A, B”sentences, see Lycan 2001, Guerzoni and Lim 2007). For both kinds,permitted inferences differ significantly from those found in regularconditionals (in particular, both entail the truth of the consequent).This is evidence that the diversity found in logics of conditionalsalso depends on the variety of uses and linguistic environments inwhichif-clauses occur.

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• Arlo-Costa, Horacio, “The Logic of Conditionals”,Stanford Encyclopedia of Philosophy (Summer 2021 Edition),Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/sum2021/entries/logic-conditionals/>. [This was the previous entry on this topic in theStanfordEncyclopedia of Philosophy — see theversion history.]

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Acknowledgments

We dedicate our work to the memory of Horacio Arló-Costa. Weare grateful to Vincenzo Crupi, Hitoshi Omori, and Paolo Santorio forhelpful comments, and to participants of the Dagstuhl Seminar 19032“Conditional Logics and Conditional Reasoning: New JointPerspectives” held in Dagstuhl in 2019 for helpful discussions.Thanks to grants ANR-19-CE28-0004-01 (Probasem) and ANR-17-EURE-0017(FrontCog) for support.