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A Course in Philosophical Logic A Course in Philosophical Logic Peter K. Schotch Dalhousie University June 27, 2005 Contents Contents i Preface . iv 1 Classical Logic 1 1.1 What is Philosophical Logic? Some Schools of Thought 1 1.2 Notation . 3 1.3 Classical Logic: A Hydroplane Tour . 5 1.4 Metalogic . 15 1.5 Overview of the Main Result . 18 2 Modal Logic: Introduction 21 2.1 Introduction and Brief History . 21 2.2 An Account of The Necessary . 27 2.3 Proof Theory . 31 2.4 Model Theory . 33 2.5 Metatheory of Modal Logic . 36 3 Modal Logic: General Considerations 43 3.1 Introduction . 43 i ii CONTENTS 3.2 Classical theories of the truth operator . 44 3.3 Inference without truth . 49 4 Many Valued Logic 51 4.1 Introduction . 51 4.2 Three-Valued Logic . 54 4.3 So How Di®erent are ÃL3and SK3? . 86 4.4 Interpretation Issues . 86 4.5 Proof theory and Metalogic of B3 . 92 4.6 Beyond Three Values . 97 5 Epistemic Logic 103 5.1 Introduction . 103 5.2 Some General Considerations on Epistemic Logic . 107 5.3 Constructing a Modal Operator . 109 5.4 The Logic - A Sketch . 111 5.5 Metalogic of Epistemic Logic . 122 6 Ethics as a Formal Science: Deontic Logic 127 6.1 Introduction . 127 6.2 Standard Deontic Logic . 128 6.3 Why SDL Fails . 130 7 Ethics as a Formal Science:Goal Theory 139 7.1 Goal Theory: A Fresh Start . 139 7.2 Structures . 140 7.3 Formalizing Moral Philosophy . 144 iii 8 Ethics as a Formal Science: Action Theory 153 8.1 Introduction . 153 8.2 Elements of the Theory . 154 8.3 Logic and Action . 164 8.4 The Structure of R . 169 Index of Symbols 170 Bibliography 177 iv CONTENTS Preface This book is a monograph in the strict sense, but it is not an introduc- tion, nor is it a survey. It is intended to be a collection of resources, at least some of which would be useful in a course on philosophical logic. I have never actually used the whole book in any one-semester course but that shouldn't be taken as a proof that it cannot or should not be done. In most cases, the chapters are built upon a previously published article or articles though I hasten to assure my prospective readers, that the building in question required more than staples. In some cases the bones of the original work are hardly visible at all, while in every case the flesh has been extensively re-worked. I have made a serious attempt to make the material ¯t together as much as possible and most of it does ¯t in the sense of being motivated by a small collection of ideas which I often regard in my more optimistic moments, as insights. Chapter 1 Classical Logic 1.1 What is Philosophical Logic? Some Schools of Thought The subject of this book is philosophical logic. This term has covered a multitude of sins, and covers at least a bunch, to this very day. At one time the adjective\philosophical" was taken to be a synonym for \in- formal" (or perhaps "not very formal"). For example, P.F. Strawson's Introduction to Logic, was once taken to be the epitome of philosophical logic. The contrast used to be with \mathematical" which meant both formal and pertaining to metamathematics. This usage has declined (but not entirely vanished) and nowadays one is more liable to mean \the sort of thing which is published in the Journal of Philosophical Logic." While there is no doubt that one will typically ¯nd a lot more prose per cc in the JPL than in The Journal of Symbolic Logic, much of the work is as formal as anyone could wish. So what makes logic philosophical is its subject matter (at least in some cases). On the issue of what logic itself is supposed to be, however, we have an unparalleled unanimity. Logic is the science of reasoning. In this science, everyone recognizes a division between deductive logic and the rest (non-deductive, let's call it), which parallels a similar division in our informal notion of reasoning. This informal division can be char- acterized as follows: A piece of reasoning is deductive provided that if 1 2 CHAPTER 1. CLASSICAL LOGIC somebody were to accept the premises, that person would be bound, on grounds of rationality, to accept the conclusion. The science of logic has made great strides in formalizing deduction but has no similar suc- cess to report in the non-deductive arena (no strides at all, some would say). Certainly we shall con¯ne ourselves, by and large, to deduction. Having such a consensus on the foundations does not imply, unfortu- nately, that logicians everywhere are united on the means whereby to prosecute their science. In fact there are two great `logical cultures' which we would do well to canvass before we embark upon our own studies. In the ¯rst culture, a piece of deductive reasoning (a deduc- tion, as we shall say) is correct if and only if the steps by which we progress from the premises to the conclusion are, one and all, in accord with a (properly speci¯ed) set of rules of inference (or rules of proof ). Those who recognize this as the central paradigm are said to be in the proof-theoretic or syntactical camp. In this paradigm, any sequence of steps which is rule-compliant is called a proof, and when the inference from a batch of premises to a conclusion is correct in this sense, the premises are said to prove the conclusion. Using ¡ to represent the premise set and P to represent the conclusion, the notation ¡ ` P represents that gamma proves P. In the other culture, a deduction is correct if and only if it is truth- preserving, which is to say that one could never infer a false conclusion from true premises. This assumes that we have on hand some gloss on \a sentence is true." The provision of such can be problematic, but, in simple cases, these problems can be overcome (for the most part). Anyone who thinks that truth has a central role to play in de¯ning correct (deductive) inference is said to be a model-theorist or to be influenced mainly by the semantic paradigm. When the inference from gamma to P is truth preserving, we say that gamma entails P and write: ¡ ² P . Within both great traditions, there are many subdivisions which stem from disagreements and heresies regarding just what are and will be the correct rules of proof, the number and character of the truth-values, and many other occasions for contention and name-calling. At least one of these divisions is of interest because it cuts across the proof- theory/model-theory divide. The distinction in question is between the followers of Boole who believe that the central items of logical enquiry 1.2. NOTATION 3 are sentences, and the followers of the angels (or of Aristotle anyway) who believe to the contrary that inferences (or arguments as some insist on calling the things, in spite of the unfortunate agonistic overtones of that term) are the heart and soul of logic. It has no agreed-upon name, so we are free to baptize as we will, and we call the former the laws of thought tradition, and the latter, the inferential tradition. This division comes down to a di®erence over what are the \good- guys". In the laws of thought stream, the job of logic is to winnow the theorems (proof-theorists), or the logical-truths (model-theorists), from the cha® of the great mass of other sentences. On the inferen- tial view, we try to demarcate the class of derivations or proofs if we are proof-theorists, and the class of valid inferences if not. A word of warning is in order here: logic is not chemistry. The terminology just introduced conforms, by and large, to current usage, but there are sig- ni¯cant variations. Many logicians have a foot, or at least a toe, in both the proof-theory and model-theory camps, and these non-aligned folk often want some top-level notion to capture these ideas. So one some- times sees the term \logical-truth" used to denote a law of thought, in either semantic or syntactic guise. Similarly, one ¯nds the term \valid," applied in this ambidextrous fashion to inferences. In these cases, since the model theoretic terms have been co-opted, new terminology must be fabricated. The term tautology can be dredged up to serve as the model-theoretic way of referring to a law of thought (for all that it has connotations of triviality), with entailment (or sometimes semantic entailment) doing duty on the inferential side. Nor does this exhaust, by any means, the deviations from normal usage. It sometimes seems to be taken as a measure of a logician's power of thought that new and exotic terminology drops from her pen with ease, if not grace. Needless to say, this sort of display would be better greeted with our earnest displeasure than with gasps of admiration, if logic is to remain comprehensible even to logicians. 1.2 Notation In the object language we use ⊃, ´, ^, _, :, and ? for the conditional, biconditional, conjunction, disjunction, negation, and falsum (or \the false") respectively. Although we have a separate piece of notation for 4 CHAPTER 1. CLASSICAL LOGIC the biconditional, we don't normally regard it as `primitive'; i.e. we usually regard it as an abbreviation for the conjunction of two condi- tionals.
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