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  1. Review essay, Justin Khoo's The Meaning of 'If'[REVIEW]Nate Charlow - forthcoming - Mind.
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  2. An even simpler defense of material implication.Matheus Silva - manuscript
    Lee Archie argued that if any truth-values are consistently assigned to a natural language conditional, where modus ponens and modus tollens are valid argument forms, and affirming the consequent is invalid, this conditional will have the same truth-conditions as a material implication. This argument is simple and requires few and relatively uncontroversial assumptions. We show that it is possible to extend Archie’s argument to three- and five-valued logics and vindicate a slightly weaker conclusion, but one that is still important: Even (...)
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  3. Progressive Specificity.Ginger Schultheis & Nathaniel Baron-Schmitt - manuscript
    We defend a new constraint on the progressive that says that what you are doing is always specific in an important sense. This principle is Progressive Specificity: if you are Ving and to V is to X or to Y, then you are Xing or you are Ying. We offer three arguments for Progressive Specificity. We then extend those arguments to an analogous principle governing the futurative progressive. Finally, we explore the relationship between Progressive Specificity and the well-known principle of (...)
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  4. Molinism's kryptonite: Counterfactuals and circumstantial luck.Andre Leo Rusavuk - forthcoming - Philosophical Quarterly.
    According to Molinism, logically prior to his creative decree, God knows via middle knowledge the truth value of the counterfactuals or conditionals of creaturely freedom (CFs) and thus what any possible person would do in any given circumstance. Critics of Molinism have pointed out that the Molinist God gets lucky that the CFs allow him to actualize either a world of his liking or even a good-enough world at all. In this paper, I advance and strengthen the popular critique in (...)
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  5. Reasonable Inferences for Counterfactuals.Ginger Schultheis - manuscript
    This paper is about four inferences patterns governing conditionals: Transitivity, Simplification, Contraposition, and Antecedent Strengthening. Transitivity, Simplification, and Contraposition are intuitively compelling. Although Antecedent Strengthening may seem less attractive at first, close attention to the full range of data reveals that it too has considerable appeal. An adequate theory of conditionals should account for these facts. The strict theory does so by validating them. But the variably strict theory invalidates them. So the variably strict theorist faces a question: why do (...)
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  6. Vagueness, conditionals, and context-sensitivity.Tom Beevers - forthcoming - Pacific Philosophical Quarterly.
    Abstract: I argue that practically all vague language is context-sensitive in a covert and unfamiliar way. I first outline a novel puzzle concerning the interaction of conditionals and vagueness. I then argue that the best way of resolving the puzzle is through positing context-sensitive penumbral connections between sundry parts of language. I argue that these penumbral connections shift through a distinct form of Lewisian accommodation. The upshot is that meaning is a far shiftier thing than has typically been thought.
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  7. Naïve Truth and the Evidential Conditional.Andrea Iacona & Lorenzo Rossi - 2024 - Journal of Philosophical Logic 53 (2):559-584.
    This paper develops the idea that valid arguments are equivalent to true conditionals by combining Kripke’s theory of truth with the evidential account of conditionals offered by Crupi and Iacona. As will be shown, in a first-order language that contains a naïve truth predicate and a suitable conditional, one can define a validity predicate in accordance with the thesis that the inference from a conjunction of premises to a conclusion is valid when the corresponding conditional is true. The validity predicate (...)
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  8. 'Might' Counterfactuals.Ginger Schultheis - forthcoming - Linguistics and Philosophy.
    The epistemic thesis is the thesis that a 'might' counterfactual like 'If Matt had gone to the parade, David might have gone to the parade' has the same meaning as 'Maybe, if Matt had gone to the parade, David would have gone to the parade.' I offer a new theory of the counterfactual interpretation of the modal 'might' on which 'might' has the same meaning as 'maybe would'. And I show that, when coupled with a plausible semantics for 'if' clauses, (...)
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  9. A Puzzle about Weak Belief.Joshua Edward Pearson - forthcoming - Analysis.
    I present an intractable puzzle for the currently popular view that belief is weak—the view that expressions like ‘S believes p’ ascribe to S a doxastic attitude towards p that is rationally compatible with low credence that p. The puzzle concerns issues that arise on considering beliefs in conditionals. I show that proponents of weak belief either cannot consistently apply their preferred methodology when accommodating beliefs in conditionals, or they must deny that beliefs in conditionals can be used in reasoning.
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  10. Logical Form, Conditionals, Pseudo-Conditionals.Andrea Iacona - forthcoming - Logic and Logical Philosophy:1-18.
    This paper raises some questions about the formalization of sentences containing ‘if’ or similar expressions. In particular, it focuses on three kinds of sentences that resemble conditionals in some respects but exhibit distinctive logical features that deserve separate consideration: whether-or-not sentences, biscuit conditionals, and concessive conditionals. As will be suggested, the examples discussed show in different ways that an adequate formalization of a sentence must take into account the content expressed by the sentence. This upshot is arguably what one should (...)
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  11. Disambiguating will-conditionals.Ivano Ciardelli & Adrian Ommundsen - 2022 - Proceedings of the 23Rd Amsterdam Colloquium.
    We bring together two ideas that are widely accepted in the literature: (i) conditionals with modal consequents are systematically ambiguous between a reading on which the if-clause restricts the modal, and a reading on which the if-clause introduces an epistemic supposition and the modal is unrestricted; (ii) ‘will’ is a modal. Together, (i) and (ii) imply that ‘will’-conditionals are systematically ambiguous between a reading as restricted historical claims, and one as unrestricted historical claims made under a supposition. We argue that (...)
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  12. Counterfactual Probability.Ginger Schultheis - 2023 - Journal of Philosophy 120 (11):581-614.
    Stalnaker's Thesis about indicative conditionals is, roughly, that the probability one ought to assign to an indicative conditional equals the probability that one ought to assign to its consequent conditional on its antecedent. The thesis seems right. If you draw a card from a standard 52-card deck, how confident are you that the card is a diamond if it's a red card? To answer this, you calculate the proportion of red cards that are diamonds -- that is, you calculate the (...)
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  13. Arguments, Suppositions, and Conditionals.Pavese Carlotta - forthcoming - Semantics and Linguistic Theory.
    Arguments and conditionals are powerful means language provides us to reason about possibilities and to reach conclusions from premises. These two kinds of constructions exhibit several affinities—e.g., they both come in different varieties depending on the mood; they share some of the same connectives (i.e., ‘then’); they allow for similar patterns of modal subordination. In the light of these affinities, it is not surprising that prominent theories of conditionals—old and new suppositionalisms as well as dynamic theories of conditionals—as well as (...)
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  14. Possible worlds truth table task.Niels Skovgaard-Olsen, Peter Collins & Karl Christoph Klauer - 2023 - Cognition 238 (105507):1-24.
    In this paper, a novel experimental task is developed for testing the highly influential, but experimentally underexplored, possible worlds account of conditionals (Stalnaker, 1968; Lewis, 1973). In Experiment 1, this new task is used to test both indicative and subjunctive conditionals. For indicative conditionals, five competing truth tables are compared, including the previously untested, multi-dimensional possible worlds semantics of Bradley (2012). In Experiment 2, these results are replicated and it is shown that they cannot be accounted for by an alternative (...)
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  15. Conditionals, Support and Connexivity.Hans Rott - manuscript
    In natural language, conditionals are frequently used for giving explanations. Thus the antecedent of a conditional is typically understood as being connected to, being relevant for, or providing evidential support for the conditional's consequent. This aspect has not been adequately mirrored by the logics that are usually offered for the reasoning with conditionals: neither in the logic of the material conditional or the strict conditional, nor in the plethora of logics for suppositional conditionals that have been produced over the past (...)
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  16. Reasoning Studies. From Single Norms to Individual Differences.Niels Skovgaard-Olsen - 2022 - Dissertation, University of Freiburg
    Habilitation thesis in psychology. The book consists of a collection of reasoning studies. The experimental investigations will take us from people’s reasoning about probabilities, entailments, pragmatic factors, argumentation, and causality to morality. An overarching theme of the book is norm pluralism and individual differences in rationality research.
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  17. What a Clause Does: Raising Its Question and Answering It Too.Da Fan - 2021 - Dissertation, University of California, Davis
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  18. Causal and Evidential Conditionals.Mario Günther - 2022 - Minds and Machines 32 (4):613-626.
    We put forth an account for when to believe causal and evidential conditionals. The basic idea is to embed a causal model in an agent’s belief state. For the evaluation of conditionals seems to be relative to beliefs about both particular facts and causal relations. Unlike other attempts using causal models, we show that ours can account rather well not only for various causal but also evidential conditionals.
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  19. Logical connectives.Varol Akman - 2006 - In A. C. Grayling, Andrew Pyle & Naomi Goulder (eds.), The Continuum encyclopedia of British philosophy. Bristol: Thoemmes Continuum. pp. 1939-1940.
    Logical connectives (otherwise known as 'logical constants' or 'logical particles') have seemed challenging to philosophers of language. This article gives a concise account of logical connectives.
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  20. Challenges for independence-driven and context-repair responses to the proviso problem.Alex Silk - 2022 - Natural Language Semantics 30 (3):363-377.
    This note presents challenge cases for prominent pragmatic responses to the proviso problem. I offer examples of uses of conditionals _if_ \(\psi,\,\phi_{P}\) that seem to commit the speaker unconditionally to the presupposition _P_ of the consequent clause _ϕ_, even though the sentence’s predicted semantic presupposition _ψ_⊃_P_ is antecedently satisfied (contrary to context-repair accounts), and independence between _ψ_ and _P_ isn’t antecedently assumed (contrary to independence-driven accounts). The examples provided avoid problems with other examples from the literature used against pragmatic accounts. (...)
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  21. Considering the exceptions: on the failure of cumulative transitivity for indicative conditionals.Ryan Simonelli - 2022 - Synthese 200 (5):1-21.
    According to existing accounts of indicative conditionals, any argument of the following form is valid: ϕ → ψ, ( ϕ ∧ ψ ) → χ ∴ ϕ → χ. Here, I present a set of counterexamples to show that there exist invalid arguments of this form. I argue that this data poses serious problems to variably strict accounts of conditionals, as such accounts are structurally unable to accommodate it. Dynamic strict accounts, however, are a different story. While existing dynamic strict (...)
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  22. Outline of a Theory of Reasons.Vincenzo Crupi & Andrea Iacona - 2023 - Philosophical Quarterly 73 (1):117-142.
    This paper investigates the logic of reasons. Its aim is to provide an analysis of the sentences of the form ‘p is a reason for q’ that yields a coherent account of their logical properties. The idea that we will develop is that ‘p is a reason for q’ is acceptable just in case a suitably defined relation of incompatibility obtains between p and ¬q. As we will suggest, a theory of reasons based on this idea can solve three challenging (...)
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  23. Conditionals, Causal Claims and Objectivity.Michał Sikorski - 2020 - Dissertation, Università di Torino
    In my thesis, I develop two distinct themes. The first part of my thesis is devoted to indicative conditionals and approaching them from an empirically informed perspective. In the second part, I am developing classical topics of philosophy of science, specifically, scientific objectivity and the role of values in science, in connection to recent methodological developments, revolving around the Replication Crisis.
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  24. Path Semantics for Indicative Conditionals.Paolo Santorio - 2022 - Mind 131 (521):59-98.
    The literature on indicative conditionals contains two appealing views. The first is the selectional view: on this view, conditionals operate by selecting a single possibility, which is used to evaluate the consequent. The second is the informational view: on this view, conditionals don’t express propositions, but rather impose constraints on information states of speakers. Both views are supported by strong arguments, but they are incompatible on their standard formulations. Hence it appears that we have to choose between mutually exclusive options. (...)
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  25. The Use and Misuse of Counterfactuals in Ethical Machine Learning.Atoosa Kasirzadeh & Andrew Smart - 2021 - In Atoosa Kasirzadeh & Andrew Smart (eds.), ACM Conference on Fairness, Accountability, and Transparency (FAccT 21).
    The use of counterfactuals for considerations of algorithmic fairness and explainability is gaining prominence within the machine learning community and industry. This paper argues for more caution with the use of counterfactuals when the facts to be considered are social categories such as race or gender. We review a broad body of papers from philosophy and social sciences on social ontology and the semantics of counterfactuals, and we conclude that the counterfactual approach in machine learning fairness and social explainability can (...)
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  26. Sly Pete in Dynamic Semantics.Simon Goldstein - 2022 - Journal of Philosophical Logic 51 (5):1103-1117.
    In ‘Sly Pete’ or ‘standoff’ cases, reasonable speakers accept incompatible conditionals, and communicate them successfully to a trusting hearer. This paper uses the framework of dynamic semantics to offer a new model of the conversational dynamics at play in standoffs, and to articulate several puzzles posed by such cases. The paper resolves these puzzles by embracing a dynamic semantics for conditionals, according to which indicative conditionals require that their antecedents are possible in their local context, and update this body of (...)
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  27. Counterfactuals for causal responsibility in legal contexts.Holger Andreas, Matthias Armgardt & Mario Gunther - 2023 - Artificial Intelligence and Law 31 (1):115-132.
    We define a formal semantics of conditionals based on _normatively ideal worlds_. Such worlds are described informally by Armgardt (Gabbay D, Magnani L, Park W, Pietarinen A-V (eds) Natural arguments: a tribute to john woods, College Publications, London, pp 699–708, 2018) to address well-known problems of the counterfactual approach to causation. Drawing on Armgardt’s proposal, we use iterated conditionals in order to analyse causal relations in scenarios of multi-agent interaction. This results in a refined counterfactual approach to causal responsibility in (...)
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  28. Conditional Logics Accommodating Stalnaker's Thesis.Andrew Bacon - manuscript
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  29. Morgenbesser’s Coin.Yael Loewenstein - 2021 - Canadian Journal of Philosophy 51 (5):317-328.
    Before a fair, indeterministic coin is tossed, Lucky, who is causally isolated from the coin-tossing mechanism, declines to bet on heads. The coin lands heads. The consensus is that the following counterfactual is true: (M:) If Lucky had bet heads, he would have won the bet. It is also widely believed that to rule (M) true, any plausible semantics for counterfactuals must invoke causal independence. But if that’s so, the hope of giving a reductive analysis of causation in terms of (...)
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  30. 'Deduction' versus 'inference' and the denotation of conditional sentences.Carsten Breul - manuscript
    The paper defends a variant of the material implication approach to the meaning of conditional sentences against some arguments that are considered to be widely subscribed to and/or important in the philosophical, psychological and linguistic literature. These arguments are shown to be wrong, debatable, or to miss their aim if the truth conditions defining material implication are viewed as determining nothing but the denotation of conditional sentences and if the function of conditional sentences in deduction (logic) is focused on rather (...)
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  31. G. Priest's An Introduction to Non-Classical Logic (2001). [REVIEW]Hans-Peter Leeb - 2003 - History and Philosophy of Logic 24:65-66.
    The review gives a short description of the content of the book and discusses the treatment of conditionals in it.
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  32. If a man buys a horse, … you have no argument against material implication: On a flaw in the foundations of the restrictor approach to conditionals.Carsten Breul - 2022 - Linguistische Berichte 269:43-54.
    The paper discusses a prominent one of Kratzer's (1986, 1991, 2012) arguments against material implication analyses of the denotation of (indicative) conditional sentences. This is the argument based on the sentence _Most of the time, if a man buys a horse, he pays cash for it_. It is shown that material implication makes a prediction that does conform to speakers' intuitions, contrary to Kratzer's claim. The paper also argues that Lewis's (1975) attack on material implication analyses of conditional sentences based (...)
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  33. Gender in conditionals.Fabio Del Prete & Alessandro Zucchi - 2021 - Linguistics and Philosophy 44 (4):953–980.
    The 3sg pronouns “he” and “she” impose descriptive gender conditions on their referents. These conditions are standardly analysed as presuppositions. Cooper argues that, when 3sg pronouns occur free, they have indexical presuppositions: the gender condition must be satisfied by the pronoun’s referent in the actual world. In this paper, we consider the behaviour of free 3sg pronouns in conditionals and focus on cases in which the pronouns’ gender presuppositions no longer seem to be indexical and project locally instead. We compare (...)
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  34. "If Oswald had not killed Kennedy" – Spohn on Counterfactuals.Hans Rott - 2016 - In Wolfgang Freitag, Hans Rott, Holger Sturm & Alexandra Zinke (eds.), Von Rang und Namen. Philosophical Essays in Honour of Wolfgang Spohn (edited book). Münster, Germany: Mentis. pp. 379–399.
    Wolfgang Spohn's theory of ranking functions is an elegant and powerful theory of the structure and dynamics of doxastic states. In two recent papers, Spohn has applied it to the analysis of conditionals, claiming to have presented a unified account of indicative and subjunctive (counterfactual) conditionals. I argue that his analysis fails to account for counterfactuals that refer to indirect causes. The strategy of taking the transitive closure that Spohn employs in the theory of causation is not available for counterfactuals. (...)
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  35. Anankastic conditionals are still a mystery.Milo Phillips-Brown - 2019 - Semantics and Pragmatics 12 (13):1-17.
    A compositional semantics for anankastic conditionals (‘If you want p, you must φ’) has been elusive. Condoravdi and Lauer (2016) decisively object to all semantics that precede their own. CL's view rests on a response to *the problem of conflicting goals*; CL use an interpretation of 'want' on which an agent's desires don't conflict with her beliefs. But a proper response requires lack of conflict with the facts. CL's view fails. Anankastic conditionals are still a mystery.
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  36. Pensamento e linguagem nas afirmações condicionais.Gilberto Gomes - 2013 - D.E.L.T.A 1 (29):121-134.
    Conditional assertions are a peculiar language structure that manifests a specific cognitive operation. In order to express it, different languages have found different ways of using verb forms. Primary conditionals are here defined as those that presuppose the possibility of the falsity of both the antecedent and the consequent. In them, the truth of the antecedent appears as a sufficient condition for the truth of the consequent. The truth condition of primary conditionals is defined as the impossibility of the conjunction (...)
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  37. Meaning-preserving contraposition of conditionals.Gilberto Gomes - 2019 - Journal of Pragmatics 1 (152):46-60.
    It is argued that contraposition is valid for a class of natural language conditionals, if some modifications are allowed to preserve the meaning of the original conditional. In many cases, implicit temporal indices must be considered, making a change in verb tense necessary. A suitable contrapositive for implicative counterfactual conditionals can also usually be found. In some cases, the addition of certain words is necessary to preserve meaning that is present in the original sentence and would be lost or changed (...)
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  38. The Dialogical Entailment Task.Niels Skovgaard-Olsen - 2019 - Cognition (C):104010.
    In this paper, a critical discussion is made of the role of entailments in the so-called New Paradigm of psychology of reasoning based on Bayesian models of rationality (Elqayam & Over, 2013). It is argued that assessments of probabilistic coherence cannot stand on their own, but that they need to be integrated with empirical studies of intuitive entailment judgments. This need is motivated not just by the requirements of probability theory itself, but also by a need to enhance the interdisciplinary (...)
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  39. Norm Conflicts and Conditionals.Niels Skovgaard-Olsen, David Kellen, Ulrike Hahn & Karl Christoph Klauer - 2019 - Psychological Review 126 (5):611-633.
    Suppose that two competing norms, N1 and N2, can be identified such that a given person’s response can be interpreted as correct according to N1 but incorrect according to N2. Which of these two norms, if any, should one use to interpret such a response? In this paper we seek to address this fundamental problem by studying individual variation in the interpretation of conditionals by establishing individual profiles of the participants based on their case judgments and reflective attitudes. To investigate (...)
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  40. Disjunctive antecedent conditionals.Justin Khoo - 2018 - Synthese 198 (8):7401-7430.
    Disjunctive antecedent conditionals —conditionals of the form if A or B, C—sometimes seem to entail both of their simplifications and sometimes seem not to. I argue that this behavior reveals a genuine ambiguity in DACs. Along the way, I discuss a new observation about the role of focal stress in distinguishing the two interpretations of DACs. I propose a new theory, according to which the surface form of a DAC underdetermines its logical form: on one possible logical form, if A (...)
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  41. Counteridenticals.Alexander W. Kocurek - 2018 - The Philosophical Review 127 (3):323-369.
    A counteridentical is a counterfactual with an identity statement in the antecedent. While counteridenticals generally seem non-trivial, most semantic theories for counterfactuals, when combined with the necessity of identity and distinctness, attribute vacuous truth conditions to such counterfactuals. In light of this, one could try to save the orthodox theories either by appealing to pragmatics or by denying that the antecedents of alleged counteridenticals really contain identity claims. Or one could reject the orthodox theory of counterfactuals in favor of a (...)
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  42. Non-Epicurean Desires.Fabien Schang - 2016 - Filosofia Unisinos 17 (1):63-68.
    In this paper, it is argued that there can be necessary and non-natural desires. After a discussion about what seems wrong with such desires, Epicurus’ classification of desires is treated similarly to Kripke’s treatment of the Kantian table of judgments. A sample of three cases is suggested to make this point.
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  43. Relevance differently affects the truth, acceptability, and probability evaluations of “and”, “but”, “therefore”, and “if–then”.Niels Skovgaard-Olsen, David Kellen, Hannes Krahl & Karl Christoph Klauer - 2017 - Thinking and Reasoning 23 (4):449-482.
    In this study we investigate the influence of reason-relation readings of indicative conditionals and ‘and’/‘but’/‘therefore’ sentences on various cognitive assessments. According to the Frege-Grice tradition, a dissociation is expected. Specifically, differences in the reason-relation reading of these sentences should affect participants’ evaluations of their acceptability but not of their truth value. In two experiments we tested this assumption by introducing a relevance manipulation into the truth-table task as well as in other tasks assessing the participants’ acceptability and probability evaluations. Across (...)
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  44. Motivating the Relevance Approach to Conditionals.Niels Skovgaard-Olsen - 2016 - Mind and Language 31 (5):555-579.
    The aim is to motivate theoretically a relevance approach to conditionals in a comparative discussion of the main alternatives. In particular, it will be argued that a relevance approach to conditionals is better motivated than the suppositional theory currently enjoying wide endorsement. In the course of this discussion, an argument will be presented for why failures of the epistemic relevance of the antecedent for the consequent should be counted as genuine semantic defects. Furthermore, strategies for dealing with compositionality and the (...)
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  45. Ranking Theory and Conditional Reasoning.Niels Skovgaard-Olsen - 2016 - Cognitive Science 40 (4):848-880.
    Ranking theory is a formal epistemology that has been developed in over 600 pages in Spohn's recent book The Laws of Belief, which aims to provide a normative account of the dynamics of beliefs that presents an alternative to current probabilistic approaches. It has long been received in the AI community, but it has not yet found application in experimental psychology. The purpose of this paper is to derive clear, quantitative predictions by exploiting a parallel between ranking theory and a (...)
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  46. If, since and because: a study in conditional connection.Storrs McCall - 1983 - Logique Et Analyse 26 (3):309.
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  47. Reflective and Evaluative Modes of Mental Simulation.Keith D. Markman & Matthew N. McMullen - 2005 - In David R. Mandel, Denis J. Hilton & Patrizia Catellani (eds.), The psychology of counterfactual thinking. New York: Routledge. pp. 77--93.
    A number of researchers have focused on the distinction between upward counterfactuals that simulate a better reality and downward counterfactuals that simulate a worse reality. In this chapter the authors will discuss the important aspects of a model (Markman and McMullen 2003) that attempts to explain how the very same counterfactual can engender dramatically different affective reactions. According to the model, the consequences of simulation direction are moderated by what we have termed simulation mode--relatively stronger tendencies to engage in reflective (...)
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  48. Decision theory, intelligent planning and counterfactuals.Michael John Shaffer - 2008 - Minds and Machines 19 (1):61-92.
    The ontology of decision theory has been subject to considerable debate in the past, and discussion of just how we ought to view decision problems has revealed more than one interesting problem, as well as suggested some novel modifications of classical decision theory. In this paper it will be argued that Bayesian, or evidential, decision-theoretic characterizations of decision situations fail to adequately account for knowledge concerning the causal connections between acts, states, and outcomes in decision situations, and so they are (...)
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Truth-Conditional Accounts of Indicative Conditionals
  1. The material and the suppositional conditional.Jan Sprenger - manuscript
    The material conditional and the suppositional analysis of the indicative conditional are based on different philosophical foundations and they leave important successes of their competitor unexplained. This paper unifies both accounts within a truth-functional, trivalent model of the suppositional analysis. In this model, we observe that the material and the suppositional conditional exhibit the same logical behavior while they have different truth conditions and different probabilities. The result is a unified semantic analysis that closes an important gap in the suppositional (...)
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  2. Conditionals: Inferentialism Explicated.Vincenzo Crupi & Andrea Iacona - 2024 - Erkenntnis.
    According to the view of conditionals named 'inferentialism', a conditional holds when its consequent can be inferred from its antecedent. This paper identifies some major challenges that inferentialism has to face, and uses them to assess three accounts of conditionals: one is the classical strict account, the other two have recently been proposed by Douven and Rott. As will be shown, none of the three proposals meets all challenges in a fully satisfactory way. We argue through novel formal results that (...)
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