This discussion note examines a recent argument for the principle that any counterfactual with true components is itself true. That argument rests upon two widely accepted principles of counterfactual logic to which the paper presents counterexamples. The conclusion speculates briefly upon the wider lessons that philosophers should draw from these examples for the semantics of counterfactuals.

The natural interpretation of counterfactuals with disjunctive antecedents involves selecting from each of the disjuncts the worlds that come closest to the world of evaluation. It has been long noticed that capturing this interpretation poses a problem for a minimal change semantics for counterfactuals, because selecting the closest worlds from each disjunct requires accessing the denotation of the disjuncts from the denotation of the disjunctive antecedent, which the standard boolean analysis of or does not allow (Creary and Hill, Philosophy of (...) Science 43:341–344, 1975; Nute, Journal of Philosophy 72:773–778, 1975; Fine, Mind 84(335):451–458, 1975; Ellis et al. Journal of Philosophical Logic 6:335–357, 1977). This paper argues that the failure to capture the natural interpretation of disjunctive counterfactuals provides no reason to abandon a minimal change semantics. It shows that the natural interpretation of disjunctive counterfactuals is expected once we refine our assumptions about the semantics of or and the logical form of conditionals, and (i) we assume that disjunctions introduce propositional alternatives in the semantic derivation, in line with independently motivated proposals about the semantics of or (Aloni, 2003a; Simons, Natural Language Semantics 13:271–316, 2005; Alonso-Ovalle, Disjunction in Alternative Semantics. PhD thesis, 2006); and (ii) we treat conditionals as correlative constructions, as advocated in von Fintel (1994), Izvorski (Proceedings of NELS 26, 1996), Bhatt and Pancheva (2006), and Schlenker (2004). (shrink)

In this paper I present some of Robert N. McLaughlin's critique of a truth functional approach to conditionals as it appears in his book On the Logic of Ordinary Conditionals. Based on his criticism I argue that the basic principles of logic together amount to epistemological and metaphysical implications that can only be accepted from a logical atomist perspective. Attempts to account for conditional relations within this philosophical framework will necessarily fail. I thus argue that it is not truth functionality (...) as such that is the problem, but the philosophical foundation of modern logic. (shrink)

entry for the Entry for the Stanford Encyclopedia of Philosophy, 2007.

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use.

The notion of probability occupies a central role in contemporary epistemology and cognitive science. Nevertheless, the classical notion of probability is hard to reconcile with the central notions postulated by the epistemological tradition.

An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached with certainty. This goal (...) is achieved by taking Adams" (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams" logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of ¬ C given B are both close to one but the bound on the former"s deviation from 1 is much smaller than the latter"s, then it may be concluded that in all likelihood the conditional probability of C given A B is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor"s rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent to entailment in Goldszmidt and Pearl"s System-Z +. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their random-structures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their random-worlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL. (shrink)

In the paper, the proof of the non-locality of quantum mechanics, given by Bedford and Stapp (1995), and appealing to the GHZ example, is analyzed. The proof does not contain any explicit assumption of realism, but instead it uses formal methods and techniques of the Lewis calculus of counterfactuals. To ascertain the validity of the proof, a formal semantic model for counterfactuals is constructed. With the help of this model it can be shown that the proof is faulty, because it (...) appeals to the unwarranted principle of “elimination of eliminated conditions” (EEC). As an additional way of showing unreasonableness of the assumption (EEC), it is argued that yet another alleged and highly controversial proof of non-locality of QM, using the Hardy example, can be made almost trivial with the help of (EEC). Finally, a general argument is produced to the effect that the locality condition in the form accepted by Stapp and Bedford is consistent with the quantum-mechanical predictions for the GHZ case under the assumption of indeterminism. This result undermines any future attempts of proving the incompatibility between the predictions of quantum theory and the idea of no faster-than-light influence in the GHZ case, quite independently of the negative assessment of the particular derivation proposed by Stapp and Bedford. (shrink)

It is widely agreed that contraposition, strengthening the antecedent and hypothetical syllogism fail for subjunctive conditionals. The following putative counter-examples are frequently cited, respectively.

The paper presents a non-monotonic inference relation on a language containing a conditional that satisfies the Ramsey Test. The logic is a weakening of classical logic and preserves many of the ‘paradoxes of implication’ associated with the material implication. It is argued, however, that once one makes the proper distinction between supposing that something is the case and accepting that it is the case, these ‘paradoxes’ cease to be counterintuitive. A representation theorem is provided where conditionals are given a non-bivalent (...) semantics and epistemic states are represented via preferential models. (shrink)

This paper discusses an important puzzle about the semantics of indicative conditionals and deontic necessity modals ( should , ought , etc.): the Miner Puzzle (Parfit, ms; Kolodny and MacFarlane, J Philos 107:115–143, 2010 ). Rejecting modus ponens for the indicative conditional, as others have proposed, seems to solve a version of the puzzle, but is actually orthogonal to the puzzle itself. In fact, I prove that the puzzle arises for a variety of sophisticated analyses of the truth-conditions of indicative (...) conditionals. A comprehensive solution requires rethinking the relationship between relevant information (what we know) and practical rankings of possibilities and actions (what to do). I argue that (i) relevant information determines whether considerations of value may be treated as reasons for actions that realize them and against actions that don’t, (ii) incorporating this normative fact requires a revision of the standard ordering semantics for weak (but not for strong) deontic necessity modals, and (iii) an off-the-shelf semantics for weak deontic necessity modals, due to von Fintel and Iatridou, which distinguishes “basic” and “higher-order” ordering sources, and interprets weak deontic necessity modals relative to both, is well-suited to this task. The prominence of normative considerations in our proposal suggests a more general methodological lesson: formal semantic analysis of natural language modals expressing normative concepts demands that close attention be paid to the nature of the underlying normative phenomena. (shrink)

The logical properties of the 'if-then' connective of ordinary English differ markedly from the logical properties of the material conditional of classical, two-valued logic. This becomes apparent upon examination of arguments in conversational English which involve (noncounterfactual) usages of if-then'. A nonclassical system of propositional logic is presented, whose conditional connective has logical properties approximating those of 'if-then'. This proposed system reduces, in a sense, to the classical logic. Moreover, because it is equivalent to a certain nonstandard three-valued logic, its (...) decision procedure is almost as efficient as that of the classical logic. It therefore provides a rational and convenient system in which to formalize English arguments. (shrink)

Berit Brogaard and Joe Salerno (Counterfactuals and Context, ANALYSIS 68 (2008): 39-46) argue that the standard Stalnaker-Lewis counterexamples to hypothetical syllogism, strengthening the antecedent, and contraposition trade on a failure to hold fixed the context in which truth values are determined for the premises and conclusion in each counterexample. I argue that no contextual fallacy is committed in the standard counterexamples, and I offer a different view of what it is for a fact to be held fixed by a counterfactual (...) antecedent. On my view, for a fact to be held fixed is an epistemological matter, not a semantic matter. (shrink)

In this essay I renew the case for Conditional Excluded Middle (CXM) in light of recent developments in the semantics of the subjunctive conditional. I argue that Michael Tooley’s recent backward causation counterexample to the Stalnaker-Lewis comparative world similarity semantics undermines the strongest argument against CXM, and I offer a new, principled argument for the validity of CXM that is in no way undermined by Tooley’s counterexample. Finally, I formulate a simple semantics for the subjunctive conditional that is consistent with (...) both CXM and Tooley’s counterexample. (shrink)

In “Backward Causation and the Stalnaker–Lewis Approach to Counterfactuals,” Analysis 62:191–7, (2002), Michael Tooley argues that if a certain kind of backward causation is possible, then a Stalnaker–Lewis comparative world similarity account of the truth conditions of counterfactuals cannot be sound. In “Tooley on Backward Causation,” Analysis 63:157–62, (2003), Paul Noordhof argues that Tooley’s example can be reconciled with a Stalnaker–Lewis account of counterfactuals if the comparative world similarity relation on which the Stalnaker–Lewis account relies is allowed to be antecedent-relative. (...) In this paper I show that taking comparative world similarity to be antecedent-relative results in a formal semantics which is a comparative world similarity semantics in name only. (shrink)

In "Backward causation and the Stalnaker-Lewis approach to counterfactuals," Analysis 62 (2002): 191–97, Michael Tooley argues that if a certain kind of backward causation is possible, then a Stalnaker-Lewis style comparative world similarity account of the truth conditions of counterfactuals cannot be sound. Tooley’s target is one particular type of semantics, but, as I show, the significance of Tooley’s example goes well beyond its consequences for any one semantics for the conditional.

Temporal necessity and the subjunctive conditional appear to be related by the principle of Past Predominance, according to which past similarities and differences take priority over future similarities and differences in determining the comparative similarity of alternative possible histories with respect to the present moment. R. H. Thomason and Anil Gupta have formalized Past Predominance in a semantics that combines selection functions with branching time; in this paper I show that Past Predominance can be formalized and axiomatized using ordinary possible (...) worlds semantics (without branching time). (shrink)

I show that given Jonathan Bennett's theory of 'even if,' the following statement is logically true iff the principle of conditional excluded is valid: (SE) If Q and if P wouldn't rule out Q, then Q even if P. Hence whatever intuitions support the validity of (SE) support the validity of Conditional Excluded Middle, too. Finally I show that Bennett's objection to John Bigelow's theory of the conditional can be turned into a (perhaps) more telling one, viz. that on Bigelow's (...) theory 'if P then Q' and 'if P and Q then R' do not jointly entail 'if P then R'. (shrink)

Berit Brogaard and Joe Salerno (2008) have defended the validity of counterfactual hypothetical syllogism (CHS) within the Stalnaker-Lewis account. Whenever the premisses of an instance of CHS are non-vacuosly true, a shift in context has occurred. Hence the standard counterexamples to CHS suffer from context failure. Charles Cross (2011) rejects this argument as irreconcilable with the Stalnaker-Lewis account. I argue against Cross that the basic Stalnaker-Lewis truth condition may be supplemented in a way that makes (CHS) valid. Yet pace Brogaard (...) and Salerno, there are alternative ways of spelling out the basic truth condition which are standard in most debates; and given these ways, the counterexamples to CHS are successful. (shrink)

In a series of articles, David Barnett (2006, 2009, 2010) has developed a general theory of conditionals. The grand aim is to reconcile the two main rivals: a suppositional and a truth-conditional view (Barnett 2006, 521). While I confine my critical discussion to counterfactuals, I will give some hints how they might spell trouble for his suppositional view in general.

Proponents of the projection strategy take an epistemic rule for the evaluation of English conditionals, the Ramsey test, as clue to the truth-conditional semantics of conditionals. They also construe English conditionals as stronger than the material conditional. Given plausible assumptions, however, the Ramsey test induces the semantics of the material conditional. The alleged link between Ramsey test and truth conditions stronger than those of the material conditional can be saved by construing conditionals as ternary, rather than binary, (...) propositional functions with a hidden contextual parameter. But such a ternary construal raises problems of its own. (shrink)

The commonplaces, all grammatically confused, are that ?conditionals? are ternary in structure, have ?antecedents? and conform to the traditional taxonomy. It is maintained en route that ?The bough will not break? is consistent with ?If the bough breaks ??, that there is no logical difference between ?future indicatives? and ?subjunctives?, and that there is a difference between the logic of propositions (e.g. ?The bough broke?) and that of judgments (?The bough will/might/could/should/must/needn't break?).

I defend a formulation of the Ramsey Test with a condition for accepting negations of conditionals. It is implicit in the assumptions of the triviality theorems of Gärdenfors, Harper, and Lewis; and it allows for a unified proof of those theorems, from weaker assumptions about belief revision. This leads to a proof of McGee’s thesis that iterated conditionals do not obey modus ponens. †To contact the author, please write to: Institute of Philosophy, University of Leuven, Kardinaal Mercierplein 2, B‐3000 Leuven, (...) Belgium; e‐mail: etlin@alum.mit.edu. (shrink)

This paper presents a unified, more-or-less complete, and largely pragmatic theory of indicative conditionals as they occur in natural language, which is entirely truth-functional and does not involve probability. It includes material implication as a special—and the most important—case, but not as the only case. The theory of conditional elements, as we term it, treats if-statements analogously to the more familiar and less controversial other truth-functional compounds, such as conjunction and disjunction.

Nested conditionals are indeed vexatious. This 'counterexample' to Modus Ponens relies for its apparent integrity on implicit premises in the background, which once disclosed show how and that this is not a counterexample at all. Modus Ponens is perhaps the most straightforward rule of all and has no counterexamples that I am aware of at all.

"If you turn left at the next corner, you will see a blue house at the end of the street." That sentence -- a conditional -- might be true even though it is possible that you will not see a blue house at the end of the street when you turn left at the next corner. A moving van may block your view; the house may have been painted pink; a crow might swoop down and peck out your eyes. Still, (...) in some contexts, we might ignore these possibilities and correctly assert the conditional. In this book, Christopher Gauker argues that such context-relativity is the key to understanding the semantics of conditionals. Contexts are defined as objective features of the situation in which a conversation takes place, and the semantic properties of sentences -- conditionals included -- are defined in terms of assertibility in a context. One of the primary goals of a theory of conditionals has to be to distinguish correctly between valid and invalid arguments containing conditionals. According to Gauker, an argument is valid if the conclusion is assertible in every context in which the premises are assertible. This runs counter to what Gauker sees as a systematic misreading of the data by other authors, who judge arguments to be invalid if they can think of a context in which the premises are judged true and some other context in which the conclusion is judged false. Different schools of thought on conditionals reflect fundamentally different approaches to semantics. Gauker offers his theory as a motive and test case for a distinctive kind of semantics that dispenses with reference relations and possible worlds. (shrink)

Claims that necessary and sufficient conditions are not converse relations are discussed, as well as the related claim that If A, then B is not equivalent to A only if B . The analysis of alleged counterexamples has shown, among other things, how necessary and sufficient conditions should be understood, especially in the case of causal conditions, and the importance of distinguishing sufficient-cause conditionals from necessary-cause conditionals. It is concluded that necessary and sufficient conditions, adequately interpreted, are converse relations in (...) all cases. (shrink)

Four different conditionals were known to the Stoics. The so-called ‘first’ (Philonian) conditional has been interpreted fairly uncontroversially as an ancient counterpart to the material conditional of modern logic; the ‘fourth’ conditional is obscure, and seemingly of little historical interest, as it was probably not held widely by any group in antiquity. The ‘second’ (Diodorean) and ‘third’ (Chrysippean) conditionals, on the other hand, pose challenging interpretive questions, raising in the process issues in philosophical logic that are as relevant today as (...) they were then. This paper is a critical survey of some modern answers to four of the most tantalizing of these questions; the issues that I will discuss arise out of interpretations of the Diodorean and Chrysippean conditionals as expressions of natural law, and as strict implications. I will reject these interpretations, concluding with my own proposal for where they should be located on a ‘ladder’ of logical strength. The following passage from Sextus will form the basis of my discussion (from Outlines of Pyrrhonism [Pyrrhoneae Hypotyposes], as presented by Long and Sedley 1987b, 211). He has just introduced Philo’s account of “a sound conditional”—by which I understand a true conditional—with the example “when it is day and I am talking, ‘If it is day, I am talking’”. He then continues. (shrink)

Owing to the problem of inescapable clashes, epistemic accounts of might-counterfactuals have recently gained traction. In a different vein, the might argument against conditional excluded middle has rendered the latter a contentious principle to incorporate into a logic for conditionals. The aim of this paper is to rescue both ontic mightcounterfactuals and conditional excluded middle from these disparate debates and show them to be compatible. I argue that the antecedent of a might-counterfactual is semantically underdetermined with respect to the counterfactual (...) worlds it selects for evaluation. This explains how might-counterfactuals select multiple counterfactual worlds as they apparently do and why their utterance confers a weaker alethic commitment on the speaker than does that of a would-counterfactual, as well as provides an ontic solution to inescapable clashes. I briefly sketch how the semantic underdetermination and truth conditions of mightcounterfactuals are regulated by conversational context. (shrink)

In a previous paper I described a range of nonmonotonic conditionals that behave like conditional probability functions at various levels of probabilistic support. These conditionals were defined as semantic relations on an object language for sentential logic. In this paper I extend the most prominent family of these conditionals to a language for predicate logic. My approach to quantifiers is closely related to Hartry Field''s probabilistic semantics. Along the way I will show how Field''s semantics differs from a substitutional interpretation (...) of quantifiers in crucial ways, and show that Field''s approach is closely related to the usual objectual semantics. One of Field''s quantifier rules, however, must be significantly modified to be adapted to nonmonotonic conditional semantics. And this modification suggests, in turn, an alternative quantifier rule for probabilistic semantics. (shrink)

I will describe the logics of a range of conditionals that behave like conditional probabilities at various levels of probabilistic support. Families of these conditionals will be characterized in terms of the rules that their members obey. I will show that for each conditional, , in a given family, there is a probabilistic support level r and a conditional probability function P such that, for all sentences C and B, C->B holds just in case P[B|C] is greater than or equal (...) to r. Thus, each conditional in a given family behaves like conditional probability above some specific support level. (shrink)

We show how the problem of strengthening the antecedent which is both formally valid and yet often intuitively invalid, concessive conditionals, and conditional rhetorical questions fit into the scheme put forth in Fulda (2010, /Acta Analytica/).

Jonathan Lowe has argued that a particular variation on C.I. Lewis' notion of strict implication avoids the paradoxes of strict implication. We show that Lowe's notion of implication does not achieve this aim, and offer a general argument to demonstrate that no other variation on Lewis' notion of constantly strict implication describes the logical behaviour of natural-language conditionals in a satisfactory way.

The book also includes an appendix of related writings by Lewis.

Firstly, the authors analyzed the properties of primary-onditionals and secondary-conditionals, establish the minimum system C2L m of primary-conditionals and secondary-conditionals, and then prove some of the formal theorems of the system which have important intuitive meanings. Secondly, the authors constructed the neighborhood semantics, prove the soundness of C2L m, introduce a general concept of canonical model by the neighborhood semantics, and then prove the completeness of C2L m by the canonical model. Finally, according to the technical results of the minimum (...) system C2L m, the authors discuss some of the important problems concerning primary-conditionals and secondary-onditionals. (shrink)

Jan Heylen and Leon Horsten object to my proposed analysis of ordinary-language conditionals by appealing to certain putative counter-examples. In this reply, I explain how, by ignoring my reading of the indicative/subjunctive distinction, their objection misses its target. I also criticize their underlying methodology.

This book contends that insufficient attention has been paid to the syntax of conditionals, as investigated by linguists.

We discuss similarities and residual differences, within the general semantic framework of minimality, between defeasible inference, belief revision, counterfactual conditionals, updating — and also conditional obligation in deontic logic. Our purpose is not to establish new results, but to bring together existing material to form a clear overall picture.

Gärdenfors' impossibility theorem draws attention to certain formal difficulties in defining a conditional connective from a notion of theory revision, via the Ramsey test. We show that these difficulties are not avoided by taking the background inference operation to be non-monotonic.

This paper presents ConR (Conditional R), a logic of conditionals based on Anderson and Belnap''s system R. A Routley-Meyer-style semantics for ConR is given for the system (the completeness of ConR over this semantics is proved in E. Mares and A. Fuhrmann, A Relevant Theory of Conditionals (unpublished MS)). Moreover, it is argued that adopting a relevant theory of conditionals will improve certain theories that utilize conditionals, i.e. Lewis'' theory of causation, Lewis'' dyadic deontic logic, and Chellas'' dyadic deontic logic.

In this paper we set out a semantics for relevant (counterfactual) conditionals. We combine the Routley-Meyer semantics for relevant logic with a semantics for conditionals based on selection functions. The resulting models characterize a family of conditional logics free from fallacies of relevance, in particular counternecessities and conditionals with necessary consequents receive a non-trivial treatment.

Boethius' treatise De Hypotheticis Syllogismis provided twelfth-century philosophers with an introduction to the logic of conditional and disjunctive sentences but this work is the only part of the logica vetus which is no longer studied in the twelfth century. In this paper I investigate why interest in Boethius acount of hypothetical syllogisms fell off so quickly. I argue that Boethius' account of compound sentences is not an account of propositions and once a proper notion of propositionality is available the argument (...) forms accepted by Boethius are seen to be incoherent. It was Peter Abaelard who first understood the nature of propositionality and propositional connectives and used this to criticise Boethius' claims in De Hypothetics Syllogismis. In place Boethius' confusion Abaelard offered a simple and correct account of the hypothetical syllogism. (shrink)

In this note I discuss what seems to be a new kind of counterexample to Lewis’s account of counterfactuals. A coin is to be tossed twice. I bet on ‘Two heads’, and I win. Common sense says that (1) is false. But Lewis’s theory says that it is true. (1) If at least one head had come up, I would have won.

This article begins by outlining some of the history—beginning with brief remarks of Quine's—of work on conditional assertions and conditional events. The upshot of the historical narrative is that diverse works from various starting points have circled around a nexus of ideas without convincingly tying them together. Section 3 shows how ideas contained in a neglected article of de Finetti's lead to a unified treatment of the topics based on the identification of conditional events as the objects of conditional bets. (...) The penultimate section explores some of the consequences of the resulting logic of conditional events while the last defends it. (shrink)

The hypothetical syllogism is invalid in standard interpretations of conditional sentences. Many arguments of this sort are quite compelling, though, and you can wonder what makes them so. I shall argue that it is our parsimony in regard to connections among events and states of affairs. All manner of things just might, for all we know, be bound up with one another in all sorts of ways. But ordinarily it is better, being simpler, to assume they are unconnected. In so (...) doing, we jump to the conclusions of some compelling but invalid arguments. (shrink)

David Lewis has recently developed the notion of conversational score-keeping as a way of explaining the acceptability of utterances in various contexts and the manner in which this acceptability changes in a rule-governed manner. I will expand Lewis's discussion by showing how the acceptibility of conditionals is linked to conversational score. In particular, I will argue that at least one controversial issue concerning the logic of conditionals, the interpretation and use of conditionals with disjunctive antecedents, may be resolved by applying (...) Lewis's notion of an accommodation rule for conversational scorekeeping. (shrink)

The properties of belief revision operators are known to have an informal semantics which relates them to the axioms of conditional logic. The purpose of this paper is to make this connection precise via the model theory of conditional logic. A semantics for conditional logic is presented, which is expressed in terms of algebraic models constructed ultimately out of revision operators. In addition, it is shown that each algebraic model determines both a revision operator and a logic, that are related (...) by virtue of the stable Ramsey test. (shrink)

One criticism of David Lewis''s account of counterfactuals is that it sometimes assigns the wrong truth-value to a counterfactual when both antecedent and consequent happen to be true. Lewis has suggested a possible remedy to this situation, but commentators have found this to be unsatisfactory. I suggest an alternative solution which involves a modification of Lewis''s truth conditions, but which confines itself to the resources already present in his account. This modification involves the device of embedding one counterfactual within another. (...) On the revised set of truth conditions, counterfactuals with true components are sometimes true and sometimes false, in a way that is more in keeping with our intuitive judgments about such statements. (shrink)

We report three experimental studies of reasoning with double conditionals, i.e. problems based on premises of the form: If A then B. If B then C. where A, B, and C, describe everyday events. We manipulated both the logical structure of the problems, using all four possible arrangements (or ''figures" of their constituents, A, B, and C, and the believability of the two salient conditional conclusions that might follow from them, i.e. If A then C , or If C then (...) A . The experiments showed that with figures for which there was a valid conclusion, the participants more often, and more rapidly, drew the valid conclusion when it was believable than when it was unbelievable. With figures for which there were no valid conclusions, the participants tended to draw whichever of the two conclusions was believable. These results were predicted by the theory that reasoning depends on constructing mental models of the premises. (shrink)

It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent. Given a bivalent background logic, this is equivalent to asking about the conjunction of Conditional Excluded Middle (CEM, opposite conditionals are not both false) and Weak Boethius' Thesis (WBT, opposite conditionals are not both true). In the system CI.0 of consequential implication, which is intertranslatable with the modal logic KT, WBT is a theorem, so it is natural to ask which instances of CEM (...) are derivable. We also investigate the systems CIw and CI of consequential implication, corresponding to the modal logics K and KD respectively, with occasional remarks about stronger systems. While unrestricted CEM produces modal collapse in all these systems, CEM restricted to contingent formulas yields the Alt2 axiom (semantically, each world can see at most two worlds), which corresponds to the symmetry of consequential implication. It is proved that in all the main systems considered, a given instance of CEM is derivable if and only if the result of replacing consequential implication by the material biconditional in one or other of its disjuncts is provable. Several related results are also proved. The methods of the paper are those of propositional modal logic as applied to a special sort of conditional. (shrink)

p and q, one of “were p true, q would be true” and “were p true, not- q would be true” is true. Therefore, even if Curley is not offered the bribe, either he would take it were he offered it or he would not take it were he offered it.

Consider the reasonable axioms of subjunctive conditionals (1) if p q 1 and p q 2 at some world, then p (q 1 & q 2) at that world, and (2) if p 1 q and p 2 q at some world, then (p 1 ∨ p 2) q at that world, where p q is the subjunctive conditional. I show that a Lewis-style semantics for subjunctive conditionals satisfies these axioms if and only if one makes a certain technical assumption (...) about the closeness relation, an assumption that is probably false. I will then show how Lewisian semantics can be modified so as to assure (1) and (2) even when the technical assumption fails, and in fact in one sense the semantics actually becomes simpler then. (shrink)

Peter Gärdenfors has developed a semantics for conditional logic, based on the operations of expansion and revision applied to states of information. The account amounts to a formalisation of the Ramsey test for conditionals. A conditional A > B is declared accepted in a state of information K if B is accepted in the state of information which is the result of revising K with respect to A. While Gärdenfors's account takes the truth-functional part of the logic as given, the (...) present paper proposes a semantics entirely based on epistemic states and operations on these states. The semantics is accompanied by a syntactic treatment of conditional logic which is formally similar to Gentzen's sequent formulation of natural deduction rules. Three of David Lewis's systems of conditional logic are represented. The formulations are attractive by virtue of their transparency and simplicity. (shrink)

We point out a simple but hitherto ignored link between the theoryof updates, the theory of counterfactuals, and classical modal logic: update is a classicalexistential modality, counterfactual is a classical universalmodality, and the accessibility relations corresponding to these modalities are inverses. The Ramsey Rule (often thought esoteric) is simply an axiomatisation of this inverse relationship. We use this fact to translate between rules for updates andrules for counterfactuals. Thus, Katsuno and Mendelzons postulatesU1--U8 are translated into counterfactual rules C1--C8(Table VII), and (...) many of the familiar counterfactual rulesare translated into rules for updates (Table VIII). Ourconclusions are summarised in Table V. (shrink)

The purpose of this paper is to argue that the hybrid formalism fits naturally in the context of David Lewis’s counterfactual logic and that its introduction into this framework is desirable. This hybridization enables us to regard the inference “The pig is Mary; Mary is pregnant; therefore the pig is pregnant” as a process of updating local information (which depends on the given situation) by using global information (independent of the situation). Our hybridization also has the following technical advantages: (i) (...) it preserves the completeness and decidability of Lewis’s logic; (ii) it allows us to characterize the Limit Assumption as a proof-rule with some side-conditions; and (iii) it enables us to establish a general Kripke completeness result by using the proof-rule corresponding to the Limit Assumption. (shrink)

This paper consists of some lecture notes in which conditional logic is treated as an extension of modal logic. Completeness and filtration theorems are provided for some basis systems.

The aim of this paper is to present a modified version of the notion of strong proof from hypotheses (definition D2), and to give three deduction theorems for the relevant logicsR (theoremsT1, andT2) andE (theoremT3).