Logic of Conditionals

The conditional interpretation of general categorical statements like ‘All men are animals’ as universally quantified material conditionals ‘For all x, if x is F, then x is G’ suggests that the logical structure of law statements is conditional rather than categorical. Disregarding the problem that the universally quantified material conditional is trivially true whenever there are no xs that are F, there are some reasons to be sceptical of Frege’s equivalence between categorical and conditional expressions. -/- Now many philosophers will (...) claim that the material conditional interpretation of laws statements, dispositions ascriptions, or any causal claim is generally accepted as wrong and outdated. Still, there seem to be some basic logical assumptions that are shared by most of the participants in the debate on causal matters which at least stems from the traditional truth functional interpretation of conditionals. This is indicated by the vocabulary in the philosophical debate on causation, where one often speaks of ‘counterfactuals’, ‘possible worlds’ and ‘necessity’ without being explicit on whether or to what extent one accepts the logical-technical definition of these notions. To guarantee a non-Humean and non-extensional approach to causal relations, it is therefore important to be aware of the logical and metaphysical implications of the technical vocabulary. -/- In this paper we want to show why extensional logic cannot deal with causal relations. Via a logical analysis of law-like statements ‘All Fs are Gs’ we hope to throw some new light on interrelated notions like causation, laws, induction, hypotheticality and modality. If successful, our analysis should be of relevance for a deeper understanding of any type of causal relations, whether we understand them to be laws, dispositions, singulars or categoricals. (shrink)

This paper presents a new system of conditional logic B2, which is strictly intermediate in strength between the existing systems B1 and B3 from John Burgess (1981) and David Lewis (1973a). After presenting and motivating the new system, we will show that it is characterized by a natural class of frames. These frames correspond to the idea that conditionals are about which worlds are nearly closest, rather than which worlds are closest. Along the way, we will also give new characterization (...) results for B1 and B3, along with two other new systems B1.1 and B1.2. (shrink)

Counterfactuals are somewhat tolerant. Had Socrates been at least six feet tall, he need not have been exactly six feet tall. He might have been a little taller—he might have been six one or six two. But while he might have been a little taller, there are limits to how tall he would have been. Had he been at least six feet tall, he would not have been more than a hundred feet tall, for example. Counterfactuals are not just tolerant, (...) then, but bounded. This paper presents a surprising paradox: If counterfactuals are tolerant and bounded, then we can prove a flat contradiction using natural rules of inference. Something has to go then. But what? (shrink)

This paper defends a theory of fictional truth. According to this theory, there is a fact of the matter concerning the number of hairs on Sherlock Holmes' head, and likewise for any other meaningful question one could ask about what's true in a work of fiction. We argue that a theory of this form is needed to account for the patterns in our judgments about attitude reports that embed fictional claims. We contrast our view with one of the dominant approaches (...) to fictional truth, which originates with David Lewis. Along the way we explore the relationship between fiction, counterfactuals, and vagueness. (shrink)

The main goal of this paper is to investigate the relation between the meaning of a sentence and its truth conditions. We report on a comprehension experiment on counterfactual conditionals, based on a context in which a light is controlled by two switches. Our main finding is that the truth-conditionally equivalent clauses (i) "switch A or switch B is down" and (ii) "switch A and switch B are not both up" make different semantic contributions when embedded in a conditional antecedent. (...) Assuming compositionality, this means that (i) and (ii) differ in meaning, which implies that the meaning of a sentential clause cannot be identified with its truth conditions. We show that our data have a clear explanation in inquisitive semantics: in a conditional antecedent, (i) introduces two distinct assumptions, while (ii) introduces only one. Independently of the complications stemming from disjunctive antecedents, our results also challenge analyses of counterfactuals in terms of minimal change from the actual state of affairs: we show that such analyses cannot account for our findings, regardless of what changes are considered minimal. (shrink)

This paper develops a trivalent semantics for indicative conditionals and extends it to a probabilistic theory of valid inference and inductive learning with conditionals. On this account, (i) all complex conditionals can be rephrased as simple conditionals, connecting our account to Adams's theory of p-valid inference; (ii) we obtain Stalnaker's Thesis as a theorem while avoiding the well-known triviality results; (iii) we generalize Bayesian conditionalization to an updating principle for conditional sentences. The final result is a unified semantic and probabilistic (...) theory of conditionals with attractive results and predictions. (shrink)

The Equation (TE) states that the probability of A → B is the probability of B given A. Lewis (1976) has shown that the acceptance of TE implies that the probability of A → B is the probability of B, which is implausible: the probability of a conditional cannot plausibly be the same as the probability of its consequent, e.g., the probability that the match will light given that is struck is not intuitively the same as the probability that it (...) will light. Here I want to counter Lewis’ claim. My aim is to argue that: (1) TE express the coherence requirements implicit in the probability distributions of a modus ponens inference (MP); (2) the triviality result is not implausible because it is a result from these requirements; (3) these coherence requirements measure MP employability, so TE significance is tied to it; (4) MP employability doesn’t provide either the acceptability or the truth conditions of conditionals, since MP employability depends on previous independent reasons to accept the conditional and some acceptable conditionals are not MP friendly. Consequently, TE doesn’t have the logical significance that is usually attributed to it. (shrink)

Russell’s role in the controversy about the paradoxes of material implication is usually presented as a tale of how even the greatest minds can fall prey to basic conceptual confusions. Quine accused him of making a silly mistake in Principia Mathematica. He interpreted “if- then” as a version of “implies” and called it material implication. Quine’s accusation is that this decision involved a use-mention fallacy because the antecedent and consequent of “if-then” are used instead of being mentioned as the premise (...) and the conclusion of an implication relation. It was his opinion that the criticisms and alternatives to the material implication presented by C. I. Lewis and others would never be made in the first place if Russell simply called the Philonian construction “material conditional” instead of “material implication”. Quine’s interpretation of the topic became hugely influential if not universally accepted. This paper will present the following criticisms against this interpretation: (1) the notion of material implication does not involve a use-mention fallacy, since the components of “if-then” are mentioned and not used; (2) Quine’s belief that the components of “if-then” are used was motivated by a conditional-assertion view of conditionals that is widely controversial and faces numerous difficulties; (3) if anything, it was Quine who could be accused of fallacious reasoning: he ignored that in the assertion of a conditional is the whole proposition that is asserted and not its constituents; (4) the Philonian construction remains counter-intuitive even if it is called “material conditional”; (5) the Philonian construction is more plausible when it is interpreted as a material implication. (shrink)

The material account proposes that indicative conditionals are material, but it is widely believed that this account cannot be applied to subjunctive conditionals. There are three reasons for this consensus: (1) the concern that most subjunctive conditionals would be vacuously true if they were material, which seems implausible; (2) the inconsistency with Adams pair, which suggests that indicative and subjunctive conditionals have different truth conditions; and (3) the belief that the possible world theories are a superior alternative to the material (...) account. In this paper, I will argue against (1) by showing that the counterintuitive aspects of vacuously true conditionals can be explained away in a consistent manner, regardless of whether they are indicative or subjunctive. I will support this argument by demonstrating that the positive arguments for the material account of indicatives also apply to subjunctives. I will counter (2) by explaining the Adams pair as logically equivalent conditionals that may be appropriate at different times, depending on the speaker’s epistemic situation. Finally, I will challenge (3) by arguing that the possible world account faces insurmountable issues and that a comprehensive material account of both indicatives and subjunctives is ultimately a more elegant solution. (shrink)

There is a profound, but frequently ignored relationship between logical consequence (formal implication) and material implication. The first repeats the patterns of the latter, but with a wider modal reach. It is argued that this kinship between formal and material implication simply means that they express the same kind of implication, but differ in scope. Formal implication is unrestricted material implication. This apparently innocuous observation has some significant corollaries: (1) conditionals are not connectives, but arguments; (2) the traditional examples of (...) valid argumentative forms are metalogical principles that express the properties of logical consequence; (3) formal logic is not a useful guide to detect valid arguments in the real world; (4) it is incoherent to propose alternatives to the material implication while accepting the classical properties of formal implication; (5) some of the counter-examples to classical argumentative forms and known conditional puzzles are unsound. (shrink)

It is commonly accepted that unconditional statements are clearer and less problematic than conditional ones. This article challenges this belief by proposing that all unconditional statements can be reduced to conditional ones since epistemic justification is inherently conditional in nature. The distinction between unconditional and conditional statements is similar to the distinction between assumptions and premises, which is an idealization that results from our attempts to limit epistemic complexity. This has perplexing consequences: (1) since any ordinary statement can be viewed (...) as a disguised conditional, this sets off a chain of epistemic regress that results in a larger conditional that encompasses our entire belief systems; (2) this monster conditional can’t never be justified since it always expands in the process of justification; (3) since there is no such thing as unconditional statements, conditionals themselves become meaningless, which implies that arguments and even entire belief systems become meaningless as well; (4) the only way to avoid these disastrous results is to admit that the meaning of conditionals and, ultimately, of our belief systems is a matter of convention. (shrink)

Different logic systems are motivated by attempts to fix the counter-intuitive instances of classical argumentative forms, e.g., strengthening of the antecedent, contraposition and conditional negation. These counter-examples are regarded as evidence that classical logic should be rejected in favour of a new logic system in which these argumentative forms are considered invalid. It is argued that these logical revisions are ad hoc, because those controversial argumentative forms are implied by other argumentative forms we want to keep. It is impossible to (...) remove an argumentative form from a logical system without getting entangled in an intricate logical web, since these revisions imply the removal of other parts of a system we want to maintain. Consequently, these revisions are incoherent and unwarranted. At the very least, the usual approach in the analysis of counter-examples of argumentative forms must be seriously reconsidered. (shrink)

There is almost a consensus among conditional experts that indicative conditionals are not material. Their thought hinges on the idea that if indicative conditionals were material, A → B could be vacuously true when A is false, even if B would be false in a context where A is true. But since this consequence is implausible, the material account is usually regarded as false. It is argued that this point of view is motivated by the grammatical form of conditional sentences (...) and the symbols used to represent their logical form, which misleadingly suggest a one-way inferential direction from A to B. That conditional sentences mislead us into a directionality bias is a phenomenon that is well-documented in the literature about conditional reasoning. It is argued that this directional appearance is deceptive and does not reflect the underlying truth conditions of conditional sentences. This directional bias is responsible for both the unpopularity of the material account of conditionals and some of the main alternative principles and themes in conditional theory, including the Ramsey’s test, the Equation, Adams’ thesis, conditional-assertion and possible world theories. The directional mindset forgets a hard- earned lesson that made classical logic possible in the first place, namely, that grammatical form of sentences can mislead us about its truth conditions. There is a case to be made for a material account of indicative conditionals when we break the domination of words over the human mind. (shrink)

Adam Rieger (2013) has carried out a survey of arguments in favour of the material account of indicative conditionals. These arguments involve simple and direct demonstrations of the material account. I extend the survey with new arguments and clarify the logical connections among them. I also show that the main counter-examples against these arguments are not successful either because their premises are just as counter-intuitive as the conclusions, or because they depend on contextual fallacies. The conclusion is that the unpopularity (...) of the material account is unjustified and that a more systematic approach in the analysis of arguments is long overdue in our attempts to understand the nature of conditionals. (shrink)

In this article I define a strong conditional for classical sentential logic, and then extend it to three non-classical sentential logics. It is stronger than the material conditional and is not subject to the standard paradoxes of material implication, nor is it subject to some of the standard paradoxes of C. I. Lewis’s strict implication. My conditional has some counterintuitive consequences of its own, but I think its pros outweigh its cons. In any case, one can always augment one’s language (...) with more than one conditional, and it may be that no single conditional will satisfy all of our intuitions about how a conditional should behave. Finally, I suspect the strong conditional will be of more use for logic rather than the philosophy of language, and I will make no claim that the strong conditional is a good model for any particular use of the indicative conditional in English or other natural languages. Still, it would certainly be a nice bonus if some modified version of the strong conditional could serve as one. -/- I begin by exploring some of the disadvantages of the material conditional, the strict conditional, and some relevant conditionals. I proceed to define a strong conditional for classical sentential logic. I go on to adapt this account to Graham Priest’s Logic of Paradox, to S. C. Kleene’s logic K3, and then to J. Łukasiewicz’s logic Ł, a standard version of fuzzy logic. (shrink)

Conditional perfection is the phenomenon in which conditionals are strengthened to biconditionals. In some contexts, “If A, B” is understood as if it meant “A if and only if B.” We present and discuss a series of experiments designed to test one of the most promising pragmatic accounts of conditional perfection. This is the idea that conditional perfection is a form of exhaustification—that is a strengthening to an exhaustive reading, triggered by a question that the conditional answers. If a speaker (...) is asked how B comes about, then the answer “If A, B” is interpreted exhaustively to meaning that A is the only way to bring about B. Hence, “A if and only if B.” We uncover evidence that conditional perfection is a form of exhaustification, but not that it is triggered by a relationship to a salient question. (shrink)

This paper develops a trivalent semantics for the truth conditions and the probability of the natural language indicative conditional. Our framework rests on trivalent truth conditions first proposed by Cooper (1968) and Belnap (1973) and it yields two logics of conditional reasoning: (i) a logic C of certainty-preserving inference; and (ii) a logic U for uncertain reasoning that preserves the probability of the premises. We show systematic correspondences between trivalent and probabilistic representations of inferences in either framework, and we use (...) the distinction between the two systems to cast light on the validity of inferences such as Modus Ponens, Or-To-If, and Conditional Excluded Middle. Specifically, the conditional behaves monotonically in C, but non-monotonically in U; Modus Ponens is valid in C, but valid in U only for non-nested conditionals. The result is a unified account of the semantics and epistemology of indicative conditionals that can be fruitfully applied to analyzing the validity of conditional inferences. (shrink)

The Consequence Argument for incompatibilism is beset by two rival interpretations: the counterfactual sufficiency interpretation and the counterfactual might interpretation. Waldrop recently argued that the counterfactual sufficiency interpretation conflicts with certain principles governing the logic of counterfactuals. In this paper, I show that Waldrop’s argument can be adapted to prove that the counterfactual might interpretation also conflicts with the same principles. So the problem Waldrop pointed out is not peculiar to the counterfactual sufficiency interpretation.

This is the second part of a two-part series on the logic of hyperlogic, a formal system for regimenting metalogical claims in the object language (even within embedded environments). Part A provided a minimal logic for hyperlogic that is sound and complete over the class of all models. In this part, we extend these completeness results to stronger logics that are sound and complete over restricted classes of models. We also investigate the logic of hyperlogic when the language is enriched (...) with hyperintensional operators such as counterfactual conditionals and belief operators. (shrink)

A part of Stalnaker (1968)’s influential theory of conditionals has been neglected, namely the role for an accessibility relation between worlds. I argue that the accessibility relation does not play the role intended for it in the theory as stated, and propose a minimal revision which solves the problem, and brings the theory in line with the formulation in Stalnaker & Thomason 1970.

By and large, the conditional connective in three-valued logic has two different functions. First, by means of a deduction theorem, it can express a specific relation of logical consequence in the logical language itself. Second, it can represent natural language structures such as "if/then'' or "implies''. This chapter surveys both approaches, shows why none of them will typically end up with a three-valued material conditional, and elaborates on connections to probabilistic reasoning.

Counterfactual skepticism says that most ordinary counterfactuals are false. While few endorse counterfactual skepticism, the precise costs of the view are disputed and not generally well-understood. I have two aims in this paper. My first and primary aim is to establish, on grounds acceptable to all parties, that counterfactual skepticism is not benign. I argue it leads to significant skepticism about the future: if counterfactual skepticism is true, then we can have only very limited knowledge about the future. I give (...) three arguments to this conclusion, taking as premises principles connecting knowledge of indicative conditionals to knowledge of disjunctions; and principles linking knowledge of indicative conditionals to knowledge of counterfactual conditionals.My second aim is to examine the consequences for non-skeptical theories. My arguments indicate that not only are many ordinary counterfactuals true, but that we also know them. How can this be reconciled with the very real tension between counterfactuals and chance? I consider two solutions, one purely epistemic and one contextualist. Together they suggest something like contextualism about knowledge is required to fully resolve the skeptical threat. (shrink)

Indicative and subjunctive conditionals are in non-complimentary distribution: there are conversational contexts at which both are licensed (Stalnaker 1975; Karttunen and Peters 1979; von Fintel 1998). This means we can ask an important, but under-explored, question: in contexts which license both, what relations hold between the two? In this paper, I’ll argue for an initially surprising conclusion: when attention is restricted to the relevant contexts, indicatives and subjunctives are co-entailing. §1 introduces the indicative/subjunctive distinction, along with a discussion of the (...) relevant notion of entailment; §2 presents the main argument of the paper, and §3 considers some of the philosophical implications of the argument in §2. Finally, §4 argues that we can reconcile the equivalence of indicatives and subjunctives with apparently conflicting judgements. (shrink)

If modus ponens is valid, then you should take up smoking.

This paper is about two controversial inference-patterns involving counterfactual or subjunctive conditionals. Given a plausible assumption about the truth-conditions of counterfactuals, it is shown that one can't go wrong in applying hypothetical syllogism (i.e., transitivity) so long as the set of worlds relevant for the conclusion is a subset of the sets of worlds relevant for the premises. It is also shown that one can't go wrong in applying antecedent strengthening so long as the set of worlds relevant for the (...) conclusion is a subset of that for the premise. These results are then adapted to Lewis's theory of counterfactuals. (shrink)

This paper explores an idea of Stoic descent that is largely neglected nowadays, the idea that an argument is valid when the conditional formed by the conjunction of its premises as antecedent and its conclusion as consequent is true. As it will be argued, once some basic features of our naıve understanding of validity are properly spelled out, and a suitable account of conditionals is adopted, the equivalence between valid arguments and true conditionals makes perfect sense. The account of validity (...) outlined here, which displays one coherent way to articulate the Stoic intuition, accords with standard formal treatments of deductive validity and encompasses an independently grounded characterization of inductive validity. (shrink)

This paper discusses some key connexive principles construed as principles about reasons, that is, as principles that express logical properties of sentences of the form ‘_p_ is a reason for _q_’. Its main goal is to show how the theory of reasons outlined by Crupi and Iacona, which is based on their evidential account of conditionals, yields a formal treatment of such sentences that validates a restricted version of the principles discussed, overcoming some limitations that affect most extant accounts of (...) conditionals. (shrink)

Deontic necessity modals (e.g. 'have to', 'ought to', 'must', 'need to', 'should', etc.) seem to vary in how they interact with negation. According to some accounts, what forces modals like 'ought' and 'should' to outscope negation is their polarity sensitivity -- modals that scope over negation do so because they are positive polarity items. But there is a conflict between this account and a widely assumed theory of if-clauses, namely the restrictor analysis. In particular, the conflict arises for constructions containing (...) a bound pronoun in the if-clause. This note spells out the core conflict. (shrink)

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

The Qualitative Thesis says that if you leave open P, then you are sure of if P, then Q just in case you are sure of the corresponding material conditional. We argue the Qualitative Thesis provides compelling reasons to accept a thesis that we call Conditional Locality, which says, roughly, the interpretation of an indicative conditional depends, in part, on the conditional’s local embedding environment. In the first part of the paper, we present an argument—due to Ben Holguín—showing that, without (...) Conditional Locality, the Qualitative Thesis is in tension with a margin for error principle on rational sureness. We show Conditional Locality reconciles the Qualitative Thesis with the margin for error principle. In the second part, we argue the full range of data supports what we call the Strong Qualitative Thesis. Without Conditional Locality, the Strong Qualitative Thesis has unacceptable consequences. But with Conditional Locality, it is tenable. (shrink)

This paper calls for a re-appraisal of McGee's analysis of the semantics, logic and probabilities of indicative conditionals presented in his 1989 paper Conditional probabilities and compounds of conditionals. The probabilistic measures introduced by McGee are given a new axiomatisation built on the principle that the antecedent of a conditional is probabilistically independent of the conditional and a more transparent method of constructing such measures is provided. McGee's Dutch book argument is restructured to more clearly reveal that it introduces a (...) novel contribution to the epistemology of semantic indeterminacy, and shows that its more controversial implications are unavoidable if we want to maintain the Ramsey Test along with the standard laws of probability. Importantly, it is shown that the counterexamples that have been levelled at McGee's analysis|generating a rather wide consensus that it yields `unintuitive' or `wrong' probabilities for compounds |fail to strike at their intended target; for to honour the intuitions of the counterexamples one must either give up the Ramsey Test or the standard laws of probability. It will be argued that we need to give up neither if we take the counterexamples as further evidence that the indicative conditional sometimes allows for a non-epistemic `causal' interpretation alongside its usual epistemic interpretation. (shrink)

This paper outlines an account of concessive conditionals that rests on two main ideas. One is that the logical form of a sentence as used in a given context is determined by the content expressed by the sentence in that context. The other is that a coherent distinction can be drawn between a reading of ‘if’ according to which a conditional is true when its consequent holds on the supposition that its antecedent holds, and a stronger reading according to which (...) a conditional is true when its antecedent supports its consequent. As we will suggest, the logical form of concessive conditionals can be elucidated by relying on this distinction. (shrink)

This paper outlines an account of conditionals, the evidential account, which rests on the idea that a conditional is true just in case its antecedent supports its consequent. As we will show, the evidential account exhibits some distinctive logical features that deserve careful consideration. On the one hand, it departs from the material reading of ‘if then’ exactly in the way we would like it to depart from that reading. On the other, it significantly differs from the non-material accounts which (...) hinge on the Ramsey Test, advocated by Adams, Stalnaker, Lewis, and others. (shrink)

This paper develops a probabilistic analysis of conditionals which hinges on a quantitative measure of evidential support. In order to spell out the interpreta- tion of ‘if’ suggested, we will compare it with two more familiar interpretations, the suppositional interpretation and the strict interpretation, within a formal framework which rests on fairly uncontroversial assumptions. As it will emerge, each of the three interpretations considered exhibits specific logical features that deserve separate consideration.

Standard accounts of counterfactuals with metaphysically impossible antecedents take them to by trivially true. But recent work shows that nontrivial countermetaphysicals are frequently appealed to in scientific modeling and are indispensable for a number of metaphysical projects. I focus on three recent discussions of counterpossible counterfactuals, which apply counterpossibles in both scientific and metaphysical modeling. I show that a sufficiently developed modal counterpart theory can provide a semantics for a wide range of counterpossibles without any inconsistent possibilities or other forms (...) of impossible worlds. But such a view faces problems: in order for the metaphysical views I discuss to bear weight, there must be a significant difference between the metaphysical possibilities and impossibilities. I will show how the counterpart-theoretic view delineates the possible from impossible, while still making room for the impossible. (shrink)

On the standard analysis, a counterfactual conditional such as “If P had been the case, then Q would have been the case” is true in the actual world if, in all nearest possible worlds in which its antecedent (P) is true, its consequent (Q) is also true. Despite its elegance, this analysis faces a difficulty if the laws of nature are deterministic. Then the antecedent could not have been true, given prior conditions. So, it is unclear what the relevant “nearest (...) possible worlds” are. David Lewis suggested that they are ones in which a local breach of the laws occurred: a “small miracle”. Others have suggested that they are ones in which the initial conditions were different (“backtracking”). I propose another response. It builds on the idea that the special sciences, where counterfactual reasoning is most common, operate at a higher level of description from fundamental physics, and that the world may behave indeterministically at higher levels even if it behaves deterministically at the fundamental physical one. The challenge from determinism can then be bypassed for many special-science counterfactuals. (shrink)

In this paper I would like to offer a new framework for hypothetical reasoning, with the goal of predicting what scenarios we consider when we interpret a conditional or causal claim (such as a sentence containing the word ‘because‘). The idea is that when we interpret a conditional or causal claim, we identify a part of the world to change and imagine changing that. Sentences are about parts of the world: when we interpret a conditional antecedent or because clause, we (...) allow the part of the world it is about to vary. -/- My main evidence for this approach is that it gives just the right range of hypothetical scenarios to account for how we interpret both conditionals and causal claims. Some approaches (such as Stalnaker and Lewis’s semantics of conditionals based on similarity, and Kratzer’s premise semantics) consider too few scenarios, while others (such as Fine’s truthmaker semantics of conditionals) consider too many. The present approach inhabits a Goldilocks zone between these extremes: not too restrictive, not too permissive, but just right. (shrink)

It is commonly believed that the role of context cannot be ignored in the analysis of conditionals, and counterfactuals in particular. On truth conditional accounts involving possible worlds semantics, conditionals have been analysed as expressions of relative necessity: “If A, then B” is true at some world w if B is true at all the A-worlds deemed relevant to the evaluation of the conditional at w. A drawback of this approach is that for the evaluation of conditionals with the same (...) antecedents at some world, the same worlds are deemed as relevant for all occasions of utterance. But surely this is inadequate, if shifts of contexts between occasions are to be accounted for. Both the linguistic and logical implications of this defect are discussed, and in order to overcome it a modification of David Lewis’ ordering semantics for counterfactuals is developed for a modified language. I follow Lewis by letting contexts determine comparative similarity assignments, and show that the addition of syntactic context parameters (context indices) to the language gives the freedom required to switch between sets of relevant antecedent worlds from occasion to occasion by choosing the corresponding similarity assignment accordingly. Thus an account that extends Lewis’ analysis of a language containing a single counterfactual connective > to a language containing infinitely many counterfactual connectives >_c, each indexed by a different context name c, overcomes the limitations of traditional analyses. Finally it is also shown that these traditional accounts can be recovered from the modified account if certain contextual restrictions are in place. -/- . (shrink)

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) is high, provided that (ii) (...) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. (shrink)

Suppose that some contradictions are true – for example, that as I walk through the door, I’m inside and I’m not inside. Then we argue 'if I'm walking through the door, I'm inside; I'm not inside; therefore, I'm not walking through the door' is an invalid instance of modus tollens.

The literature on counterfactuals is dominated by strict accounts and variably strict accounts. Counterexamples to the principle of Antecedent Strengthening were thought to be fatal to SA; but it has been shown that by adding dynamic resources to the view, such examples can be accounted for. We broaden the debate between VSA and SA by focusing on a new strengthening principle, Strengthening with a Possibility. We show dynamic SA classically validates this principle. We give a counterexample to it and show (...) that extra dynamic resources cannot help SA. We then show VSA accounts for the counterexample if it allows for orderings on worlds that are not almost-connected, and that such an ordering naturally falls out of a Kratzerian ordering source semantics. We conclude that the failure of Strengthening with a Possibility tells strongly against Dynamic SA and in favor of an ordering source-based version of VSA. (shrink)

Suppositional theories of conditionals take apparent similarities between supposition and conditionals as a starting point, appealing to features of the former to provide an account of the latter. This paper develops a novel form of suppositional theory, one which characterizes the relationship at the level of semantics rather than at the level of speech acts. In the course of doing so, it considers a range of novel data which shed additional light on how conditionals and supposition interact.

Once upon a time, some thought that indicative conditionals could be effectively analyzed as material conditionals. Later on, an alternative theoretical construct has prevailed and received wide acceptance, namely, the conditional probability of the consequent given the antecedent. Partly following critical remarks recently ap- peared in the literature, we suggest that evidential support—rather than conditional probability alone—is key to understand indicative conditionals. There have been motivated concerns that a theory of evidential conditionals (unlike their more tra- ditional counterparts) cannot generate (...) a sufficiently interesting logical system. Here, we will describe results dispelling these worries. Happily, and perhaps surprisingly, appropriate technical variations of Ernst Adams’s classical approach allow for the construction of a logic of evidential conditionals with distinctive fea- tures, which is also well-behaved and reasonably strong. (shrink)

This essay calls attention to a set of linguistic interactions between counterfactual conditionals, on one hand, and possibility modals like could have and might have, on the other. These data present a challenge to the popular variably strict semantics for counterfactual conditionals. Instead, they support a version of the strict conditional semantics in which counterfactuals and possibility modals share a unified quantificational domain. I’ll argue that pragmatic explanations of this evidence are not available to the variable analysis. And putative counterexamples (...) to the unified strict analysis, on careful inspection, in fact support it. Ultimately, the semantics of conditionals and modals must be linked together more closely than has sometimes been recognized, and a unified strict semantics for conditionals and modals is the only way to fully achieve this. (shrink)

.This paper discusses Aristotle’s thesis and Boethius’ thesis, the most distinctive theorems of connexive logic. Its aim is to show that, although there is something plausible in Aristotle’s thesis and Boethius’ thesis, the intuitions that may be invoked to motivate them are consistent with any account of indicative conditionals that validates a suitably restricted version of them. In particular, these intuitions are consistent with the view that indicative conditionals are adequately formalized as strict conditionals.

We develop and defend a new approach to counterlogicals. Non-vacuous counterlogicals, we argue, fall within a broader class of counterfactuals known as counterconventionals. Existing semantics for counterconventionals, 459–482 ) and, 1–27 ) allow counterfactuals to shift the interpretation of predicates and relations. We extend these theories to counterlogicals by allowing counterfactuals to shift the interpretation of logical vocabulary. This yields an elegant semantics for counterlogicals that avoids problems with the usual impossible worlds semantics. We conclude by showing how this approach (...) can be extended to counterpossibles more generally. (shrink)

The Identity principle says that conditionals with the form 'If p, then p' are logical truths. Identity is overwhelmingly plausible, and has rarely been explicitly challenged. But a wide range of conditionals nonetheless invalidate it. I explain the problem, and argue that the culprit is the principle known as Import-Export, which we must thus reject. I then explore how we can reject Import-Export in a way that still makes sense of the intuitions that support it, arguing that the differences between (...) indicative and subjunctive conditionals play a key role in solving this puzzle. (shrink)

We show that each of three natural sets of assumptions about the conditional entails necessitarianism: that anything possible is necessary.

We deepen the study of conjoined and disjoined conditional events in the setting of coherence. These objects, differently from other approaches, are defined in the framework of conditional random quantities. We show that some well known properties, valid in the case of unconditional events, still hold in our approach to logical operations among conditional events. In particular we prove a decomposition formula and a related additive property. Then, we introduce the set of conditional constituents generated by $n$ conditional events and (...) we show that they satisfy the basic properties valid in the case of unconditional events. We obtain a generalized inclusion-exclusion formula, which can be interpreted by introducing a suitable distributive property. Moreover, under logical independence of basic unconditional events, we give two necessary and sufficient coherence conditions. The first condition gives a geometrical characterization for the coherence of prevision assessments on a family F constituted by n conditional events and all possible conjunctions among them. The second condition characterizes the coherence of prevision assessments defined on $F\cup K$, where $K$ is the set of conditional constituents associated with the conditional events in $F$. Then, we give some further theoretical results and we examine some examples and counterexamples. Finally, we make a comparison with other approaches and we illustrate some theoretical aspects and applications. (shrink)

This paper investigates whether 'even if A, B' is pragmatically polysemic, so that a nonconcessive conditional may have 'even if', and whether concessive conditionals, pragmatically defined, can fail to have 'even if' or a non-temporal 'still'. Different paraphrases are used to help elucidate pragmatic meanings. A theory of the pragmatic meanings of concessive and implicative conditionals is presented. The semantic meaning of 'even if' and the question of whether concessive conditionals imply the truth of their consequents are also discussed.