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Counterfactuals Counterfactuals Notes by R.J. Buehler Based on Counterfactuals by David Lewis May 6, 2014 ii Contents Preface v 1 An Analysis of Counterfactuals1 1.1 Conditionals..........................................1 1.1.1 Overview.......................................1 1.1.2 The Necessity of Being Vague............................3 1.2 Counterfactual Conditional Operators...........................4 1.2.1 The New Operators.................................4 1.2.2 Modal Logic.....................................4 1.3 Fixed Strict Conditionals..................................5 1.3.1 Strictness Orderings.................................5 1.3.2 Counterfactuals as Strict Conditionals.......................5 1.4 Variably Strict Conditionals.................................6 1.4.1 Truth Conditions...................................7 1.4.2 The Limit Assumption................................7 1.5 ‘Might’ Counterfactuals and Outer Modalities.......................8 1.5.1 Truth Conditions...................................8 1.5.2 Generating a Modal Logic..............................8 1.5.3 A New System....................................9 1.6 Impossible Antecedents................................... 10 1.7 True Antecedents and the Inner Modalities........................ 11 1.7.1 True Antecedents................................... 11 1.7.2 Weakly Centered Systems and the Inner Modalities................ 12 1.8 Counterfactual Fallacies................................... 13 1.8.1 Strengthening the Antecedent............................ 13 1.8.2 Transitivity...................................... 13 1.8.3 Contraposition.................................... 14 1.9 Quantifiers and Counterparts................................ 15 1.9.1 Potentialities..................................... 15 1.9.2 Counterparts..................................... 15 2 Reformulations 17 2.1 Multiple Modalities...................................... 17 2.2 Propositional Quantification................................. 17 2.3 Comparative Similarity................................... 18 2.4 Comparative Possibility................................... 18 2.5 Cotenability.......................................... 18 2.6 Selection Functions and Operators............................. 19 iii iv CONTENTS 3 Comparisons 21 3.1 Metalinguistic Theories................................... 21 3.1.1 Implicit Premises................................... 21 3.1.2 Factual Premises................................... 22 3.1.3 Laws of Nature.................................... 22 3.2 Stalnaker’s Theory...................................... 22 3.2.1 The Counterfactual Excluded Middle........................ 23 4 Foundations 25 4.1 Possible Worlds........................................ 25 4.2 Similarity........................................... 26 5 Analogies 27 5.1 Conditional Obligation.................................... 27 5.2 ‘When Next’ and ‘When Last’................................ 29 5.3 Contextually Definite Descriptions............................. 30 6 Logics 31 6.1 Completeness and Soundness Results............................ 31 6.1.1 Model Theory..................................... 31 6.1.2 Proof Theory..................................... 33 6.1.3 Completeness and Soundness............................ 34 6.1.4 An Alternative Axiomatization of VC....................... 34 6.2 Decidability Results..................................... 35 6.3 Derived Modal Logics.................................... 35 7 Criticism and Reception 37 7.1 Fine 1975........................................... 37 Preface What follows are my personal notes on David Lewis’ Counterfactuals. Most of the ideas presented in this document are not my own, but rather Lewis’ and should be treated accordingly. This text is not meant for reproduction or as a replacement for Lewis’ book, but rather as a convenient reference and summary, suitable for use as lecture notes or a review and little more. For a complete presentation of the thoughts and arguments presented, please see the full text of Counterfactuals. v vi PREFACE Chapter 1 An Analysis of Counterfactuals 1.1 Conditionals 1.1.1 Overview Conditionals–that is, statements of the form ‘If..., Then...’–have long been a sore point for those who desire a logic that corresponds closely to natural language; to make sense of the difficulties which have plagued this effort, we consider five classes of conditionals, ways that–traditionally–philosophers and logicians have thought it helpful to delineate the conditional construction. These classes are, by no means, definitive or mutually exclusive. Definition: Material Conditional The conditional of classical logic (→); the material conditional is false only if the antecedent is true and the consequent is false. Otherwise, it is considered true. Note that the truth or falsity of the antecedent and consequent are judged independently of one another. Example 1. 3 is a prime number → the sky is blue Definition: Indicative Conditional The standard conditional of natural language; indicative conditionals state that their consequent is true (in this world) if their antecedent is. Example 2. If Socrates is a man, then Socrates is mortal. The material conditional of classical logic is, of course, intended to capture the usage of the indicative conditional of natural language–and, quite famously, seems to do a rather poor job. In particular, the so-called paradoxes of the material implication (e.g., ¬p → (p → q)) appear to provide good reason to suppose that the material conditional is not adequate (although Grice has argued against this conclusion with some success). C.I. Lewis, in an attempt to rectify the perceived deficiency of the material conditional, proposed the strict conditional as the proper logical representation for the indicative conditional: 1 2 CHAPTER 1. AN ANALYSIS OF COUNTERFACTUALS Definition: Strict Conditional Using the necessity operator of modal logic (), a strict conditional is defined as (φ → ψ) where φ and ψ are well-formed formulas of the logic and → is the material conditional. In terms of its intended interpretation, the strict conditional asserts that it is necessarily the case that φ → ψ. Example 3. (Socrates is immortal → 2+2 =4 ) As the example given shows, the strict conditional–while escaping the paradoxes of material implication–still appears to struggle with proposition that are necessarily true or necessarily false. Definition: Subjunctive Conditional At its most basic, a subjunctive conditional is simply a natural language conditional using the subjunctive verb mood; the subjunctive mood is usually associated with states of unreality, with sentences expressing something which runs counter to what is actually the case. Example 4. If Socrates were immortal, he would not be human. If I were to study all day tomorrow, I would get a perfect score on the quiz. Definition: Counterfactual Conditional A counterfactual conditional is a natural language conditional which asserts that its consequent would obtain if its antecedent were an accurate description of reality. Note that, following Lewis, this allows for counterfactual conditionals which are not, actually, counterfactual. Example 5. If Socrates were immortal, he would not be human. If I were to study all day tomorrow, I would get a perfect score on the quiz. A natural and important question is whether the distinctions given above are meaningful, especially those describing natural language. It is commonly accepted that there is a meaningful difference between indicative and subjunctive/counterfactual conditionals. The standard argument for this claim is owed to Ernest Adams (3): Consider the sentences below: If Oswald did not kill Kennedy, someone else did. If Oswald hadn’t killed Kennedy, someone else would have. The first is an indicative conditional while the second is a subjunctive/counterfactual conditional. If there were no meaningful difference between the classes, two conditionals with only tense and mood differences as here should always receive the same truth value. Nevertheless, it is easy to imagine the first sentence being true and the second false. It follows, then, that there is a logical difference I don’t know if I’m between indicative and subjunctive/counterfactual conditionals. entirely on board with even the counterfactual- Lewis’ distinction between counterfactual and subjunctive conditionals has not, however, been as indicative distinction sketched so far; as well received with most philosophers using the terms interchangeably. Lewis argues for the distinction Lewis points out, there seem to be conditionals by pointing out that future subjunctive conditionals seem to behave as if they were indicative condi- of each type which behave more like the tionals and that a subjunctive verb mood isn’t necessary to express a counterfactual conditional, e.g. other (e.g., future subjunctives). Could ‘If no Hitler, then no A-bomb’. indicative conditionals just be a special case of counterfactual condi- tionals (e.g., when the actual world matches the antecedent)? 1.1. CONDITIONALS 3 1.1.2 The Necessity of Being Vague This is a very important point and bears think- It’s important to note that counterfactuals have, perhaps necessarily so, a certain vagueness; Lewis ing about; at the mo- ment, I tend to agree accepts this and argues that any explication of counterfactuals must therefore either be stated in though. vague terms or be made relative to some parameter which is fixed only within some rough limits. 4 CHAPTER 1. AN ANALYSIS OF COUNTERFACTUALS 1.2 Counterfactual
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