Indicative Conditionals, Strictly
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Gibbardian Collapse and Trivalent Conditionals
Gibbardian Collapse and Trivalent Conditionals Paul Égré* Lorenzo Rossi† Jan Sprenger‡ Abstract This paper discusses the scope and significance of the so-called triviality result stated by Allan Gibbard for indicative conditionals, showing that if a conditional operator satisfies the Law of Import-Export, is supraclassical, and is stronger than the material conditional, then it must collapse to the material conditional. Gib- bard’s result is taken to pose a dilemma for a truth-functional account of indicative conditionals: give up Import-Export, or embrace the two-valued analysis. We show that this dilemma can be averted in trivalent logics of the conditional based on Reichenbach and de Finetti’s idea that a conditional with a false antecedent is undefined. Import-Export and truth-functionality hold without triviality in such logics. We unravel some implicit assumptions in Gibbard’s proof, and discuss a recent generalization of Gibbard’s result due to Branden Fitelson. Keywords: indicative conditional; material conditional; logics of conditionals; triva- lent logic; Gibbardian collapse; Import-Export 1 Introduction The Law of Import-Export denotes the principle that a right-nested conditional of the form A → (B → C) is logically equivalent to the simple conditional (A ∧ B) → C where both antecedentsare united by conjunction. The Law holds in classical logic for material implication, and if there is a logic for the indicative conditional of ordinary language, it appears Import-Export ought to be a part of it. For instance, to use an example from (Cooper 1968, 300), the sentences “If Smith attends and Jones attends then a quorum *Institut Jean-Nicod (CNRS/ENS/EHESS), Département de philosophie & Département d’études cog- arXiv:2006.08746v1 [math.LO] 15 Jun 2020 nitives, Ecole normale supérieure, PSL University, 29 rue d’Ulm, 75005, Paris, France. -
Frontiers of Conditional Logic
City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 2-2019 Frontiers of Conditional Logic Yale Weiss The Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/2964 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Frontiers of Conditional Logic by Yale Weiss A dissertation submitted to the Graduate Faculty in Philosophy in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York 2019 ii c 2018 Yale Weiss All Rights Reserved iii This manuscript has been read and accepted by the Graduate Faculty in Philosophy in satisfaction of the dissertation requirement for the degree of Doctor of Philosophy. Professor Gary Ostertag Date Chair of Examining Committee Professor Nickolas Pappas Date Executive Officer Professor Graham Priest Professor Melvin Fitting Professor Edwin Mares Professor Gary Ostertag Supervisory Committee The City University of New York iv Abstract Frontiers of Conditional Logic by Yale Weiss Adviser: Professor Graham Priest 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 cat- alyzed the rapid maturation of the field. -
The Presupposition of Subjunctive Conditionals
The Presupposition of Subjunctive Conditionals Kai von Fintel M.I.T. 0. Introduction Why are some conditionals subjunctive? It is often assumed that at least one crucial difference is that subjunctive conditionals presuppose that their antecedent is false, that they are counterfactual (Lakoff 1970). The traditional theory has apparently been refuted. Perhaps the clearest counter-example is one given by Alan Anderson (1951: 37): If Jones had taken arsenic, he would have shown just exactly those symptoms which he does in fact show. A typical place to use such a subjunctive conditional would be in the course of an argument that tries to bolster the hypothesis that Jones did in fact take arsenic. But then it would of course be self-defeating to presuppose that the hypothesis is false. Thus, something else must be going on. 1. Preliminaries Let’s first assume the following “strict” semantics for “bare” conditionals, those in which the if-clause restricts an implicit quantifier over worlds: (1) Schematic Truth-Conditions for Bare Conditionals A bare conditional if p, q has the logical form " D (if p) (q). If defined, it is true in a world w iff all worlds w' in D(w) such that p(w') are such that q(w'): p Ç D(w) Í q. Here D is a function that assigns to any (evaluation) world w a set of accessible worlds. This semantics departs in some ways from the more standard Stalnaker-Lewis analysis and is justified in a recent paper of mine (von Fintel 1997). What it *Thanks to Paul Portner for very helpful comments on a previous draft. -
Section 2.1: Proof Techniques
Section 2.1: Proof Techniques January 25, 2021 Abstract Sometimes we see patterns in nature and wonder if they hold in general: in such situations we are demonstrating the appli- cation of inductive reasoning to propose a conjecture, which may become a theorem which we attempt to prove via deduc- tive reasoning. From our work in Chapter 1, we conceive of a theorem as an argument of the form P → Q, whose validity we seek to demonstrate. Example: A student was doing a proof and suddenly specu- lated “Couldn’t we just say (A → (B → C)) ∧ B → (A → C)?” Can she? It’s a theorem – either we prove it, or we provide a counterexample. This section outlines a variety of proof techniques, including direct proofs, proofs by contraposition, proofs by contradiction, proofs by exhaustion, and proofs by dumb luck or genius! You have already seen each of these in Chapter 1 (with the exception of “dumb luck or genius”, perhaps). 1 Theorems and Informal Proofs The theorem-forming process is one in which we • make observations about nature, about a system under study, etc.; • discover patterns which appear to hold in general; • state the rule; and then • attempt to prove it (or disprove it). This process is formalized in the following definitions: • inductive reasoning - drawing a conclusion based on experi- ence, which one might state as a conjecture or theorem; but al- mostalwaysas If(hypotheses)then(conclusion). • deductive reasoning - application of a logic system to investi- gate a proposed conclusion based on hypotheses (hence proving, disproving, or, failing either, holding in limbo the conclusion). -
Some Remarks on Indicative Conditionals
Some Remarks on Indicative Conditionals Barbara Abbott Michigan State University 1. Introduction This paper concerns indicative conditionals, such as the English examples in (1): (1) a. If Lynn was at the meeting, she knows what's going on. b. If we leave now, we can make the 5:00 show. c. If it rains tomorrow, we won't have a picnic. We wi11look at several theories of indicative conditionals grouped into three categories: those that base its semantics on its logical counterpart (the material conditional); intensional analyses, which bring in alternative possible worlds; and a third subgroup which denies that indicative conditionals express propositions at all. We will also look at some problems for each kind of approach. 2. The Material Conditional Analysis There is a long tradition that associates natural language indicative conditionals with the material conditional of propositional logic. (Indeed, introductory logic texts typically assume this identification.) The semantics of the material conditional is given entirely by its truthtable, shown in (2). (2) p q p� q T T T T F F F T T F F T On this account natural language indicative conditionals express a truth function of their component sentences. 2.1. Problems fo r the Material Conditional Approach There are several problems for this approach. One stressed by Edgington (1995, 2003) is the following: assigning indicative conditionals the material conditional truth function means failing to distinguish between unequally probable conditionals with equally improbable antecedents, like those in (3) (from Jackson 197911991, 115): © 2004 by Barbara Abbott R. Young (ed), SA LT XIV 1-19, Ithaca, NY: Cornell University. -
'The Denial of Bivalence Is Absurd'1
On ‘The Denial of Bivalence is Absurd’1 Francis Jeffry Pelletier Robert J. Stainton University of Alberta Carleton University Edmonton, Alberta, Canada Ottawa, Ontario, Canada [email protected] [email protected] Abstract: Timothy Williamson, in various places, has put forward an argument that is supposed to show that denying bivalence is absurd. This paper is an examination of the logical force of this argument, which is found wanting. I. Introduction Let us being with a word about what our topic is not. There is a familiar kind of argument for an epistemic view of vagueness in which one claims that denying bivalence introduces logical puzzles and complications that are not easily overcome. One then points out that, by ‘going epistemic’, one can preserve bivalence – and thus evade the complications. James Cargile presented an early version of this kind of argument [Cargile 1969], and Tim Williamson seemingly makes a similar point in his paper ‘Vagueness and Ignorance’ [Williamson 1992] when he says that ‘classical logic and semantics are vastly superior to…alternatives in simplicity, power, past success, and integration with theories in other domains’, and contends that this provides some grounds for not treating vagueness in this way.2 Obviously an argument of this kind invites a rejoinder about the puzzles and complications that the epistemic view introduces. Here are two quick examples. First, postulating, as the epistemicist does, linguistic facts no speaker of the language could possibly know, and which have no causal link to actual or possible speech behavior, is accompanied by a litany of disadvantages – as the reader can imagine. -
Three Ways of Being Non-Material
Three Ways of Being Non-Material Vincenzo Crupi, Andrea Iacona May 2019 This paper presents a novel unified account of three distinct non-material inter- pretations of `if then': the suppositional interpretation, the evidential interpre- tation, and the strict interpretation. We will spell out and compare these three interpretations within a single formal framework which rests on fairly uncontro- versial assumptions, in that it requires nothing but propositional logic and the probability calculus. As we will show, each of the three intrerpretations exhibits specific logical features that deserve separate consideration. In particular, the evidential interpretation as we understand it | a precise and well defined ver- sion of it which has never been explored before | significantly differs both from the suppositional interpretation and from the strict interpretation. 1 Preliminaries Although it is widely taken for granted that indicative conditionals as they are used in ordinary language do not behave as material conditionals, there is little agreement on the nature and the extent of such deviation. Different theories tend to privilege different intuitions about conditionals, and there is no obvious answer to the question of which of them is the correct theory. In this paper, we will compare three interpretations of `if then': the suppositional interpretation, the evidential interpretation, and the strict interpretation. These interpretations may be regarded either as three distinct meanings that ordinary speakers attach to `if then', or as three ways of explicating a single indeterminate meaning by replacing it with a precise and well defined counterpart. Here is a rough and informal characterization of the three interpretations. According to the suppositional interpretation, a conditional is acceptable when its consequent is credible enough given its antecedent. -
The Incorrect Usage of Propositional Logic in Game Theory
The Incorrect Usage of Propositional Logic in Game Theory: The Case of Disproving Oneself Holger I. MEINHARDT ∗ August 13, 2018 Recently, we had to realize that more and more game theoretical articles have been pub- lished in peer-reviewed journals with severe logical deficiencies. In particular, we observed that the indirect proof was not applied correctly. These authors confuse between statements of propositional logic. They apply an indirect proof while assuming a prerequisite in order to get a contradiction. For instance, to find out that “if A then B” is valid, they suppose that the assumptions “A and not B” are valid to derive a contradiction in order to deduce “if A then B”. Hence, they want to establish the equivalent proposition “A∧ not B implies A ∧ notA” to conclude that “if A then B”is valid. In fact, they prove that a truth implies a falsehood, which is a wrong statement. As a consequence, “if A then B” is invalid, disproving their own results. We present and discuss some selected cases from the literature with severe logical flaws, invalidating the articles. Keywords: Transferable Utility Game, Solution Concepts, Axiomatization, Propositional Logic, Material Implication, Circular Reasoning (circulus in probando), Indirect Proof, Proof by Contradiction, Proof by Contraposition, Cooperative Oligopoly Games 2010 Mathematics Subject Classifications: 03B05, 91A12, 91B24 JEL Classifications: C71 arXiv:1509.05883v1 [cs.GT] 19 Sep 2015 ∗Holger I. Meinhardt, Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Englerstr. 11, Building: 11.40, D-76128 Karlsruhe. E-mail: [email protected] The Incorrect Usage of Propositional Logic in Game Theory 1 INTRODUCTION During the last decades, game theory has encountered a great success while becoming the major analysis tool for studying conflicts and cooperation among rational decision makers. -
Conundrums of Conditionals in Contraposition
Dale Jacquette CONUNDRUMS OF CONDITIONALS IN CONTRAPOSITION A previously unnoticed metalogical paradox about contrapo- sition is formulated in the informal metalanguage of propositional logic, where it exploits a reflexive self-non-application of the truth table definition of the material conditional to achieve semantic di- agonalization. Three versions of the paradox are considered. The main modal formulation takes as its assumption a conditional that articulates the truth table conditions of conditional propo- sitions in stating that if the antecedent of a true conditional is false, then it is possible for its consequent to be true. When this true conditional is contraposed in the conclusion of the inference, it produces the false conditional conclusion that if it is not the case that the consequent of a true conditional can be true, then it is not the case that the antecedent of the conditional is false. 1. The Logic of Conditionals A conditional sentence is the literal contrapositive of another con- ditional if and only if the antecedent of one is the negation of the consequent of the other. The sentence q p is thus ordinarily un- derstood as the literal contrapositive of: p⊃ :q. But the requirement presupposes that the unnegated antecedents⊃ of the conditionals are identical in meaning to the unnegated consequents of their contrapos- itives. The univocity of ‘p’ and ‘q’ in p q and q p can usually be taken for granted within a single context⊃ of: application⊃ : in sym- bolic logic, but in ordinary language the situation is more complicated. To appreciate the difficulties, consider the use of potentially equivo- cal terms in the following conditionals whose reference is specified in particular speech act contexts: (1.1) If the money is in the bank, then the money is safe. -
Immediate Inference
Immediate Inference Dr Desh Raj Sirswal, Assistant Professor (Philosophy) P.G. Govt. College for Girls, Sector-11, Chandigarh http://drsirswal.webs.com . Inference Inference is the act or process of deriving a conclusion based solely on what one already knows. Inference has two types: Deductive Inference and Inductive Inference. They are deductive, when we move from the general to the particular and inductive where the conclusion is wider in extent than the premises. Immediate Inference Deductive inference may be further classified as (i) Immediate Inference (ii) Mediate Inference. In immediate inference there is one and only one premise and from this sole premise conclusion is drawn. Immediate inference has two types mentioned below: Square of Opposition Eduction Here we will know about Eduction in details. Eduction The second form of Immediate Inference is Eduction. It has three types – Conversion, Obversion and Contraposition. These are not part of the square of opposition. They involve certain changes in their subject and predicate terms. The main concern is to converse logical equivalence. Details are given below: Conversion An inference formed by interchanging the subject and predicate terms of a categorical proposition. Not all conversions are valid. Conversion grounds an immediate inference for both E and I propositions That is, the converse of any E or I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples either both propositions are true or both are false. Steps for Conversion Reversing the subject and the predicate terms in the premise. Valid Conversions Convertend Converse A: All S is P. -
Indicative Conditionals: Probabilities and Relevance
Philos Stud https://doi.org/10.1007/s11098-021-01622-3 Indicative conditionals: probabilities and relevance 1,2 2 Francesco Berto • Aybu¨ke O¨ zgu¨n Accepted: 21 February 2021 Ó The Author(s) 2021 Abstract 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 u ! w to the extent that (i) the conditional probability pðwjuÞ is high, provided that (ii) u is relevant for w. 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. Keywords Conditionals Á Conditional probabilities Á Relevance Á Adams’ thesis Á Subject matter People do not consider the conditional probability to be the only basis for asserting a conditional, even if they judge the probability of the conditional to be the conditional probability. Evans & Over, If. & Francesco Berto [email protected] Aybu¨ke O¨ zgu¨n [email protected] 1 Arche´, University of St Andrews, St Andrews, Scotland 2 ILLC, University of Amsterdam, Amsterdam, The Netherlands 123 F. -
Chasing Hook: Quantified Indicative Conditionals
1 Chasing Hook: Quantified Indicative Conditionals Angelika Kratzer University of Massachusetts at Amherst May 2015 Abstract This chapter looks at indicative conditionals embedded under quantifiers, with a special emphasis on ‘one-case’ conditionals as in No query was answered if it came from a doubtful address. It agrees with earlier assessments that a complete conditional (with antecedent and consequent) is embedded under a quantifier in those constructions, but then proceeds to create a dilemma by showing that we can’t always find the right interpretation for that conditional. Contrary to earlier assessments, Stalnaker’s conditional won’t always do. The paper concludes that the embedded conditional in the sentence above is a material implication, but the if- clause also plays a pragmatic role in restricting the domain of the embedding quantifier. That an appeal to pragmatics should be necessary at all goes with Edgington’s verdict that “we do not have a satisfactory general account of sentences with conditional constituents.” 2 0. Introduction I should say this upfront. The Hook from Edgington’s Conditionals is a man with opinions. He thinks that if is a truth-functional connective and corresponds to material implication. My Hook is not a ‘he’ or a ‘she’, but an ‘it’. It is material implication itself. It is É. Hook is elusive. We know it has a connection with if, but we don’t quite know what the connection is. My project is to hunt Hook down in the back alleys of English. It’s not that I think Hook is that special. I am interested in Hook because it makes a good probe for exploring the properties of embedded conditionals.