I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as 'If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable,' which is true, and 'If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable,' which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What’s more, I (...) argue that these counterpossibles don’t just appear in the periphery of relative computability theory but instead they play an ineliminable role in the development of the theory. Finally, I present and discuss a model theory for these counterfactuals that is a straightforward extension of the familiar comparative similarity models. (shrink)

The standard semantics for counterfactuals ensures that any counterfactual with a true antecedent and true consequent is itself true. There have been many recent attempts to amend the standard semantics to avoid this result. I show that these proposals invalidate a number of further principles of the standard logic of counterfactuals. The case against the automatic truth of counterfactuals with true components does not extend to these further principles, however, so it is not clear that rejecting the latter should be (...) a consequence of rejecting the former. Instead I consider how one might defuse putative counterexamples to the truth of true-true counterfactuals. (shrink)

In contemporary natural languages semantics one will often see the use of special brackets to enclose a linguistic expression, e.g. ⟦carrot⟧. These brackets---so-called denotation brackets or semantic evaluation brackets---stand for a function that maps a linguistic expression to its "denotation" or semantic value (perhaps relative to a model or other parameters). Even though this notation has been used in one form or another since the early development of natural language semantics in the 1960s and 1970s, Montague himself didn't make use (...) of this notation in his series of groundbreaking papers on semantics. That raises the question: When was the ⟦.⟧-notation introduced to semantics? This note answers that question. (shrink)

The aim of this paper is to identify what I take to be the main conceptual problem in Lewis’ semantics for counterfactuals when the Limit Assumption is not satisfied, what I call the Discrimination Problem , and to present and discuss a modification of Lewis’ semantics that aims at solving DP. First, I outline Lewis’ semantics, highlighting the aspects that will be relevant for our discussion. Second, I present DP and discuss it with a heuristic example. Third, I present the (...) new proposal and comment some formal consequences and the objections they give rise to. Fourth, I present an allegedly equivalent version of the proposal and discuss the difference between counterfactuals and strict conditionals. Finally, I defend that the two proposals are different and argue for the first one as the best way of capturing two intuitions that seem in conflict when the LA is not satisfied. I conclude with a dilemma: either the proposal here defended works or LA is inescapable. Thus, if the proposal doe.. (shrink)

Are counterfactuals with true antecedents and consequents automatically true? That is, is Conjunction Conditionalization: if (X & Y), then (X > Y) valid? Stalnaker and Lewis think so, but many others disagree. We note here that the extant arguments for Conjunction Conditionalization are unpersuasive, before presenting a family of more compelling arguments. These arguments rely on some standard theorems of the logic of counterfactuals as well as a plausible and popular semantic claim about certain semifactuals. Denying Conjunction Conditionalization, then, requires (...) rejecting other aspects of the standard logic of counterfactuals, or else our intuitive picture of semifactuals. (shrink)

Is A & C sufficient for the truth of ‘if A were the case, C would be the case’? Jonathan Bennett thinks not, although the counterexample he gives is inconsistent with his own account of counterfactuals. In any case, I argue that anyone who accepts the case of Morgenbesser's coin, as Bennett does, should reject Bennett’s counterexample. Moreover, I show that the principle underlying his counterexample is unmotivated and indeed false. More generally, I argue that Morgenbesser’s coin commits us to (...) the sufficiency of A & C for the truth of the corresponding counterfactual. (shrink)

According to the analysis of concessive conditionals suggested by Crupi and Iacona, a concessive conditional $$p{{\,\mathrm{\hookrightarrow }\,}}q$$ p ↪ 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.

This paper investigates the implicative conditional, a connective intended to describe the logical behavior of an empirically defined class of natural language conditionals, also named implicative conditionals, which excludes concessive and some other conditionals. The implicative conditional strengthens the strict conditional with the possibility of the antecedent and of the contradictory of the consequent. $${p\Rightarrow q}$$ p ⇒ q is thus defined as $${\lnot } \Diamond {(p \wedge \lnot q) \wedge } \Diamond {p \wedge } \Diamond {\lnot q}$$ ¬ ◊ (...) ( p ∧ ¬ q ) ∧ ◊ p ∧ ◊ ¬ q. We explore the logical properties of this conditional in a reflexive normal Kripke semantics, provide an axiomatic system and prove it to be sound and complete for our semantics. The implicative conditional validates transitivity and contraposition, which we take to be integral parts of reasoning and communication. But it only validates restricted versions of strengthening the antecedent, right weakening, simplification, and rational monotonicity. Apparent counterexamples to some of these properties are explained as due to contextual factors. Finally, the implicative conditional avoids the paradoxes of material and strict implication, and validates some connexive principles such as Aristotle’s theses and weak Boethius’ thesis, as well as some highly entrenched principles of conditionals, such as conjunction of consequents, disjunction of antecedents, modus ponens, cautious monotonicity and cut. (shrink)

Motivated by supplying a new strategy for connexive logic and a better semantics for conditionals so that negating a conditional amounts to negating its consequent under the condition, we propose a new semantics for connexive conditional logic, by combining Kleene’s three-valued logic and a slight modification of Stalnaker’s semantics for conditionals. In the new semantics, selection functions for selecting closest worlds for evaluating conditionals can be undefined. Truth and falsity conditions for conditionals are then supplemented with a precondition that the (...) selection function is defined. Based on the new semantics, we define six conditional logics by different classes of models, using two notions of validity, one preserving truth and the other preserving tolerant truth. We prove soundness and completeness of these logics and compare them with some conditional logics in the literature. Our partial function plus precondition strategy may be used in other semantic frameworks to generate more connexive logics. (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 offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

There are many things—call them ‘experts’—that you should defer to in forming your opinions. The trouble is, many experts are modest: they’re less than certain that they are worthy of deference. When this happens, the standard theories of deference break down: the most popular (“Reflection”-style) principles collapse to inconsistency, while their most popular (“New-Reflection”-style) variants allow you to defer to someone while regarding them as an anti-expert. We propose a middle way: deferring to someone involves preferring to make any decision (...) using their opinions instead of your own. In a slogan, deferring opinions is deferring decisions. Generalizing the proposal of Dorst (2020a), we first formulate a new principle that shows exactly how your opinions must relate to an expert’s for this to be so. We then build off the results of Levinstein (2019) and Campbell-Moore (2020) to show that this principle is also equivalent to the constraint that you must always expect the expert’s estimates to be more accurate than your own. Finally, we characterize the conditions an expert’s opinions must meet to be worthy of deference in this sense, showing how they sit naturally between the too-strong constraints of Reflection and the too-weak constraints of New Reflection. (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)

In some recent works, Crupi and Iacona have outlined an analysis of ‘if’ based on Chrysippus’ idea that a conditional holds whenever the negation of its consequent is incompatible with its antecedent. This paper presents a sound and complete system of conditional logic that accommodates their analysis. The soundness and completeness proofs that will be provided rely on a general method elaborated by Raidl, which applies to a wide range of systems of conditional logic.

The two main features of this thesis are (i) an account of contextualized (context indexed) counterfactuals, and (ii) a non-vacuist account of counterpossibles. Experience tells us that the truth of the counterfactual is contingent on what is meant by the antecedent, which in turn rests on what context is assumed to underlie its reading (intended meaning). On most conditional analyses, only the world of evaluation and the antecedent determine which worlds are relevant to determining the truth of a conditional, and (...) consequently what its truth value is. But that results in the underlying context being fixed, when evaluating distinct counterfactuals with the same antecedent on any single occasion, even when the context underlying the evaluation of each counterfactual may vary. Alternative approaches go some of the way toward resolving this inadequacy by appealing to a difference in the consequents associated with counterfactuals with the same antecedent. That is, in addition to the world of evaluation and the antecedent, the consequent contributes to the counterfactual’s evaluation. But these alternative approaches nevertheless give a single, determinate truth value to any single conditional (same antecedent and consequent), despite the possibility that this value may vary with context. My reply to these shortcomings (chapter 4) takes the form of an analysis of a language that makes appropriate explicit access to the intended context available. That is, I give an account of a contextualized counterfactual of the form ‘In context C: If it were the case that … , then it would be the case that …’. Although my proposal is largely based on Lewis’ (1973, 1981) analyses of counterfactuals (the logic VW and its ordering semantics), it does not require that any particular logic of counterfactuals should serve as its basis – rather, it is a general prescription for contextualizing a conditional language. The advantage of working with ordering semantics stems from existing results (which I apply and develop) concerning the properties of ordering frames that facilitate fashioning and implementing a notion of contextual information preservation. Analyses of counterfactuals, such as Lewis’ (1973), that cash out the truth of counterfactuals in terms of the corresponding material conditional’s truth at possible worlds result in all counterpossibles being evaluated as vacuously true. This is because antecedents of counterpossibles are not true at any possible world, by definition. Such vacuist analyses have already been identified and challenged by a number of authors. I join this critical front, and drawing on existing proposals, I develop an impossible world semantics for a non-vacuist account of counterpossibles (chapter 5), by modifying the same system and semantics that serve the basis of the contextualized account offered in chapter 4, i.e. Lewis’ (1986) ordering semantics for the logic VW. I critically evaluate the advantages and disadvantages of key conditions on the ordering of worlds on the extended domain and show that there is a sense in which all of Lewis’ analysis of mere counterfactuals can be preserved, whilst offering an analysis of counterpossibles that meets our intuitions. The first part of chapter 1 consists of an outline of the usefulness of impossible worlds across philosophical analyses and logic. That outline in conjunction with a critical evaluation of Lewis’ logical arguments in favour of vacuism in chapter 2, and his marvellous mountain argument against impossible worlds in chapter 3, serves to motivate and justify the impossible world semantics for counterpossibles proposed in chapter 5. The second part of chapter 1 discusses the limitations that various conditional logics face when tasked to give an adequate treatment of the influence of context. That introductory discussion in conjunction with an overview of conditional logics and their various semantics in chapter 2 – which includes an in-depth exposition of Stalnaker-Lewis similarity semantics for counterfactuals – serves as the motivation and conceptual basis for the contextualized account of counterfactuals proposed in chapter 4. (shrink)

Theoria , the international Swedish philosophy journal, was founded in 1935. Its contributors in the first 75 years include the major Swedish philosophers from this period and in addition a long list of international philosophers, including A. J. Ayer, C. D. Broad, Ernst Cassirer, Hector Neri Castañeda, Arthur C. Danto, Donald Davidson, Nelson Goodman, R. M. Hare, Carl G. Hempel, Jaakko Hintikka, Saul Kripke, Henry E. Kyburg, Keith Lehrer, Isaac Levi, David Lewis, Gerald MacCallum, Richard Montague, Otto Neurath, Arthur N. (...) Prior, W. V. Quine, Nicholas Rescher, Ernest Sosa, Robert C. Stalnaker, P. F. Strawson, Patrick Suppes, Johan van Benthem, Georg Henrik von Wright and many others. Hempel's confirmation paradoxes, Ross's deontic paradox, Montague's universal grammar and Lindström's theorem are among the contributions to philosophy that were first published in Theoria. (shrink)

The intuitive notion behind the usual semantics of most systems of modal logic is that of ?possible worlds?. Loosely speaking, an expression is necessary if and only if it holds in all possible worlds; it is possible if and only if it holds in some possible world. Of course, contradictory expressions turn out to hold in no possible worlds, and logically true expressions turn out to hold in every possible world. A method is presented for transforming standard modal systems into (...) systems of modal logic for impossible worlds. To each possible world there corresponds an impossible world such that an expression holds in the impossible world if and only if it does not hold in the possible world. One can then talk about such worlds quite consistently, and there seems to be no logical reason for excluding them from consideration. (shrink)

In this essay I renew the case for Conditional Excluded Middle 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 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)

According to the Relevant Alternatives (RA) Theory of knowledge, knowing that something is the case involves ruling out (only) the relevant alternatives. The conception of knowledge in epistemic logic also involves the elimination of possibilities, but without an explicit distinction, among the possibilities consistent with an agent’s information, between those relevant possibilities that an agent must rule out in order to know and those remote, far-fetched or otherwise irrelevant possibilities. In this article, I propose formalizations of two versions of the (...) RA theory. Doing so clarifies a famous debate in epistemology, pitting Fred Dretske against David Lewis, about whether the RA theorist should accept the principle that knowledge is closed under known implication, familiar as the K axiom in epistemic logic. Dretske’s case against closure under known implication leads to a study of other closure principles, while Lewis’s defense of closure by appeal to the claimed context sensitivity of knowledge attributions leads to a study of the dynamics of context. Having followed the first lead at length in other work, here I focus more on the second, especially on logical issues associated with developing a dynamic epistemic logic of context change over models for the RA theory. (shrink)

The article offers a rigorous truth condition for subjunctively conditional statements. The theory is framed in the system of transparent intensional logic and takes connections (especially the cause-Effect relation) as basic. Counterexamples are given to rival theories based on the notion of world similarity.

This paper uses Popper's treatment of probability and an epistemic constraint on probability assignments to conditionals to extend the Bayesian representation of rational belief so that revision of previously accepted evidence is allowed for. Results of this extension include an epistemic semantics for Lewis' theory of counterfactual conditionals and a representation for one kind of conceptual change.

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)

Hans Rott has argued, most recently in his book Change, Choice and Inference, that certain formal correspondences between belief revision and rational choice have important philosophical implications, claiming that the former strongly indicate the unity of practical and theoretical reason as well as the primacy of practical reason. In this paper, I confront Rott's argument with three serious challenges. My conclusion is that, while Rott's work is indisputable as a formal achievement, the philosophical consequences he wants to draw are not (...) forthcoming. (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.

Galles & Pearl (l998) claimed that s [possible-worlds] framework.s framework. Recursive models are shown to correspond precisely to a subclass of (possible-world) counterfactual structures. On the other hand, a slight generalization of recursive models, models where all equations have unique solutions, is shown to be incomparable in expressive power to counterfactual structures, despite the fact that the Galles and Pearl arguments should apply to them as well. The problem with the Galles and Pearl argument is identified: an axiom that they (...) viewed as irrelevant, because it involved disjunction (which was not in their language), is not irrelevant at all. (shrink)

Epistemic closure has been a central issue in epistemology over the last forty years. According to versions of the relevant alternatives and subjunctivist theories of knowledge, epistemic closure can fail: an agent who knows some propositions can fail to know a logical consequence of those propositions, even if the agent explicitly believes the consequence (having “competently deduced” it from the known propositions). In this sense, the claim that epistemic closure can fail must be distinguished from the fact that agents do (...) not always believe, let alone know, the consequences of what they know—a fact that raises the “problem of logical omniscience” that has been central in epistemic logic. This paper, part I of II, is a study of epistemic closure from the perspective of epistemic logic. First, I introduce models for epistemic logic, based on Lewis’s models for counterfactuals, that correspond closely to the pictures of the relevant alternatives and subjunctivist theories of knowledge in epistemology. Second, I give an exact characterization of the closure properties of knowledge according to these theories, as formalized. Finally, I consider the relation between closure and higher-order knowledge. The philosophical repercussions of these results and results from part II, which prompt a reassessment of the issue of closure in epistemology, are discussed further in companion papers. As a contribution to modal logic, this paper demonstrates an alternative approach to proving modal completeness theorems, without the standard canonical model construction. By “modal decomposition” I obtain completeness and other results for two non-normal modal logics with respect to new semantics. One of these logics, dubbed the logic of ranked relevant alternatives, appears not to have been previously identified in the modal logic literature. More broadly, the paper presents epistemology as a rich area for logical study. (shrink)

The variably strict analysis of conditionals does not only largely dominate the philosophical literature, since its invention by Stalnaker and Lewis, it also found its way into linguistics and psychology. Yet, the shortcomings of Lewis–Stalnaker’s account initiated a plethora of modifications, such as non-vacuist conditionals, presuppositional indicatives, perfect conditionals, or other conditional constructions, for example: reason relations, difference-making conditionals, counterfactual dependency, or probabilistic relevance. Many of these new connectives can be treated as strengthened or weakened conditionals. They are definable conditionals. (...) This article develops a technique to infer the logic for such definable conditionals from the known logic of the underlying defining conditional. The technique is applied to central examples. The results show that a large part of the zoo of conditionals arises from a basic conditional—a constant nucleus of the different contextual and conceptual variations of variably strict conditionals. (shrink)

According to Williamson, our knowledge of metaphysical necessities and possibilities is just a “special case” of our knowledge of counterfactual conditionals. This subsumption of modal under counterfactual thinking mainly serves a methodological role: to sign the end of “philosophical exceptionalism” in modal epistemology, namely the view that our knowledge of metaphysical modalities is obtained by means of a special, dedicated, possibly a priori, capacity. In this paper, I show that a counterfactual approach to modal epistemology is structurally similar to more (...) traditional “conceivability-based” approaches. On this basis, I then show that the counterfactual approach suffers some of the same problems and I conclude that it is still based on a quite exceptional capacity to determine the truth of modal metaphysical claims. Given that, for Williamson, the epistemology of thought experiments should also be subsumed under the counterfactual approach, the problem I raise in this paper has consequences for his approach to thought experiments. (shrink)

I present and discuss two logical results. The first shows that a non-trivial counterfactual analysis exists for any contingent proposition that is false in at least two possible worlds. The second result identifies a set of conditions that are individually necessary and jointly sufficient for the success of a counterfactual analysis. I use these results to shed light on the question whether disposition ascribing propositions can be analyzed as Stalnaker-Lewis conditional propositions. The answer is that they can, but, in order (...) for a counterfactual analysis to work, the antecedent and consequent must be related in a particular way, and David Lewis’s Time’s Arrow constraints on comparative world similarity must be relaxed. The upshot is that counterfactual analyses are easy to come by, in principle, even if not in practice. In that sense, it’s easy to be iffy. (shrink)

Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional—especially counterfactual—expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that catalyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. counterfactuals) to the exotic (e.g. conditional (...) obligation). Despite the close connections between conditional and modal logic, both the technical development and philosophical exploitation of the latter has outstripped that of the former, with the result that noticeable lacunae exist in the literature on conditional logic. My dissertation addresses a number of these underdeveloped frontiers, producing new technical insights and philosophical applications. -/- I contribute to the solution of a problem posed by Priest of finding sound and complete labeled tableaux for systems of conditional logic from Lewis' V-family. To develop these tableaux, I draw on previous work on labeled tableaux for modal and conditional logic; errors and shortcomings in recent work on this problem are identified and corrected. While modal logic has by now been thoroughly studied in non-classical contexts, e.g. intuitionistic and relevant logic, the literature on conditional logic is still overwhelmingly classical. Another contribution of my dissertation is a thorough analysis of intuitionistic conditional logic, in which I utilize both algebraic and worlds semantics, and investigate how several novel embedding results might shed light on the philosophical interpretation of both intuitionistic logic and conditional logic extensions thereof. -/- My dissertation examines deontic and connexive conditional logic as well as the underappreciated history of connexive notions in the analysis of conditional obligation. The possibility of interpreting deontic modal logics in such systems (via embedding results) serves as an important theoretical guide. A philosophically motivated proscription on impossible obligations is shown to correspond to, and justify, certain (weak) connexive theses. Finally, I contribute to the intensifying debate over counterpossibles, counterfactuals with impossible antecedents, and take—in contrast to Lewis and Williamson—a non-vacuous line. Thus, in my view, a counterpossible like "If there had been a counterexample to the law of the excluded middle, Brouwer would not have been vindicated" is false, not (vacuously) true, although it has an impossible antecedent. I exploit impossible (non-normal) worlds—originally developed to model non-normal modal logics—to provide non-vacuous semantics for counterpossibles. I buttress the case for non-vacuous semantics by making recourse to both novel technical results and theoretical considerations. (shrink)

Conditional logics have traditionally been intended to formalize various intuitively correct modes of reasoning involving conditional expressions in natural language. Although conditional logics have by now been thoroughly studied in a classical context, they have yet to be systematically examined in an intuitionistic context, despite compelling philosophical and technical reasons to do so. This paper addresses this gap by thoroughly examining the basic intuitionistic conditional logic ICK, the intuitionistic counterpart of Chellas’ important classical system CK. I give ICK both worlds (...) semantics and algebraic semantics, and prove that these are equivalent. I give a Gödel-type embedding of ICK into CK and a Glivenko-type embedding of CK into ICK. I axiomatize ICK and prove soundness, completeness, and decidability results. Finally, I discuss extending ICK. (shrink)

We prove some embedding theorems for classical conditional logic, covering 'finitely cumulative' logics, 'preferential' logics and what we call 'semi-monotonic' logics. Technical tools called 'partial frames' and 'frame morphisms' in the context of neighborhood semantics are used in the proof.

In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in (...) this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update. (shrink)

Proves Completeness for the Evidential Conditional.

Standard conditionals $\varphi > \psi$, by which I roughly mean variably strict conditionals à la Stalnaker and Lewis, are trivially true for impossible antecedents. This article investigates three modifications in a doxastic setting. For the neutral conditional, all impossible-antecedent conditionals are false, for the doxastic conditional they are only true if the consequent is absolutely necessary, and for the metaphysical conditional only if the consequent is ‘model-implied’ by the antecedent. I motivate these conditionals logically, and also doxastically by properties of (...) conditional belief and belief revision. For this I show that the Lewisian hierarchy of conditional logics can be reproduced within ranking semantics, provided we slightly stretch the notion of a ranking function. Given this, acceptance of a conditional can be interpreted as a conditional belief. The epistemic and the neutral conditional deviate from Lewis’ weakest system $V$, in that ID or even CN are dropped, and new axioms appear. The logic of the metaphysical conditional is completely axiomatised by $V$ to which we add the known Kripke axioms T5 for the outer modality. Related completeness results for variations of the ranking semantics are obtained as corollaries. (shrink)