Running Head The World is all that is the case http//www.humanities-ebooks.co.uk Philosophy Insights General Editor: Mark ddis Formal Logic Mark Jago What makes an argument valid? For advice on use of this ebook please scroll to page 2
Publication Data Mark Jago, 2007 The uthor has asserted his right to be identified as the author of this Work in accordance with the Copyright, Designs and Patents ct 1988. Published by Humanities-Ebooks.co.uk Tirril Hall, Tirril, Penrith C10 2JE Reading Options * Before continuing, please use the command View > fit to page and then progress by using the next page arrows at the top or bottom right of the Viewer screen. * To navigate through the contents use the Bookmarks at the left of the screen. * To search, click on the search symbol in the toolbar and select show all results. * For ease of reading, use <CTRL+L> to enlarge the page to full screen * Use <CTRL+L> to return to the full menu, with its bookmarks and search tool. * Hyperlinks (if any) appear in Blue Underlined Text. Licence and permissions This book is licensed for a particular computer or computers. The file itself may be copied, but the copy will not open until the new user obtains a licence from the Humanities-Ebooks website in the usual manner. The original purchaser may license the same work for a second computer by applying to support@humanities-ebooks.co.uk with proof of purchase. Permissions: it is permissible to print one (watermarked) copy of the book for your own use, but not to copy and paste text. ISBN 978-1-84760-041-7
Formal Logic Mark Jago Bibliographical Entry: Jago, Mark. Formal Logic. Philosophy Insights. Tirril: Humanities-Ebooks, 2007
Note on the uthor Mark Jago is a lecturer in the Department of Philosophy at the University of Nottingham, UK and a Junior Research ssociate in the Research Group on the Philosophy of Information at the University of Oxford. He wrote the Wittgenstein guide in the Philosophy Insights series and has published articles on truth, belief, logic, fiction and information. Personal website: http://www.nottingham.ac.uk/philosophy/staff/mark-jago.htm.
Philosophy Insights: Formal Logic 5 Contents Introduction 7 1 Logical Reasoning 9 1.1 Preliminaries.............................. 9 1.2 Valid rguments............................ 10 1.3 Valid Forms of Inference........................ 12 Exercises................................... 15 2 Propositional Logic 16 2.1 Introduction............................... 16 2.2 Logical Connectives.......................... 17 2.3 The Logical Language......................... 20 2.4 Construction Trees........................... 23 2.5 Truth Tables............................... 24 2.6 Valuations................................ 27 Exercises................................... 30 3 Entailment and Equivalence 32 3.1 Logical Entailment........................... 32 3.2 Equivalence............................... 34 3.3 Equivalence Schemes.......................... 36 Exercises................................... 39 4 Proof Trees 41 4.1 Proofs in Logic............................. 41 4.2 The Proof Tree Method......................... 41 4.3 Examples of Proof Trees........................ 44 4.4 Decidability............................... 45 4.5 Valuations From Open Finished Trees................. 46 4.6 Soundness and Completeness...................... 48 Exercises................................... 49
Philosophy Insights: Formal Logic 6 5 First Order Logic 51 5.1 More Valid rguments......................... 51 5.2 Constants, Predicates and Relations.................. 52 5.3 Existence and Generality........................ 54 5.4 Formation Rules............................. 57 5.5 Binary Relations............................ 58 5.6 Semantics for First-Order Logic.................... 62 5.7 Satisfaction............................... 65 Exercises................................... 68 6 Identity 70 6.1 The Puzzle of Identity.......................... 70 6.2 Identity in First-Order Logic...................... 70 6.3 Expressing t Least, t Most and Exactly............... 71 6.4 Definite Descriptions.......................... 74 6.5 Leibniz s Law and Second Order Logic................ 75 Exercises................................... 76 7 Proof Trees for First Order Logic 78 7.1 Rules for Quantifiers.......................... 78 7.2 Rules for Identity............................ 83 7.3 Undecidability.............................. 84 7.4 Constructing Models from Open Branches............... 85 7.5 Soundness and Completeness...................... 88 Exercises................................... 89 ppendix. Basic Set Theory 91 ppendix B. Infinity 92 References and Further Reading 94 nswers to Exercises 95
Introduction Logical reasoning is vital to philosophy. Descartes for one recognized this in his Rules for the Direction of the Mind (1628), where he writes: RULE 4: There is need of a method for investigating the truth about things. RULE 5:... we shall be observing this method exactly if we reduce complex and obscure propositions step by step to simpler ones, and then, by retracing our steps, try to rise from intuition of all of the simplest ones to knowledge of all the rest. Descartes aim was first to find principles whose truth he could be certain of and then to deduce further truths from these. This raises the question, just what counts as reasoning correctly from one proposition to another? This is what we hope to understand through studying logic. There are different views as to what studying logic should achieve, including the following: We should aim to discover logical truths, i.e. sentences that could not possibly be false and which we can discover to be true a priori. We should aim to discover valid forms of reasoning to use in our arguments. We should aim to discover the principles of logical entailment, so that we can ascertain the facts that are entailed by what we know to be the case. Fortunately for us, these approaches to logic all turn out to be interchangeable, at least in the form of logic that we will study here, known as classical logic. Strange as it sounds at first, there is not one body of doctrine or method that can be labelled logic. There are disagreements over what principles apply to the notion of logical entailment and over what counts as a valid argument. These disagreements constitute the philosophy of logic, which I will not go into in this book. s a rule of thumb, whenever someone speaks of logic, unqualified as this or that style of logic, they will mean classical first order logic. This is certainly true in most philosophy classes (at least, those not dealing with technical subjects such as the philosophy of logic or mathematics). Classical logic is also adopted as the logic of choice in mathematics and electronics, although not always in computer science. 1 1 Classical logic is focused on truth, whereas computer scientists are often focused on the kinds of tasks that computers can do and in particular, what computers can prove. Focusing on proof rather than truth is the province of constructive logics.
7. Proof Trees for First Order Logic Recap: Rules for the Logical Connectives The basics of proof trees in first order logic are just the same as the basics of proof trees for propositional logic (section 4.2). ll of the rules that we encountered there are still usable. B B B B B B B B B ( B) ( B) ( B) ( B) B B B B B Here, and B can be any first order sentences, atomic or complex. These rules do not allow us to deal with quantified sentences or identity, which we will look at below. 7.1 Rules for Quantifiers Rules for x and x The rules for negated quantified sentences x and x are obvious, given how x and x are interrelated: x x x x
Philosophy Insights: Formal Logic 91 ppendix. Basic Set Theory set is just a collection of objects. The collection is abstract: objects do not have to be arranged in space or time in any particular way form a set. For example, there is a set consisting of Tony Blair, the Emperor ugustus and the number 17. If a, b and c are objects, we write the set that contains them (and no further objects) as {a,b,c} Here, a, b and c are called the elements or members of the set. Sets themselves are considered to be objects, so that a set can be an element of another set, for example, the set {a,b,{a,c}} What makes a set the set it is, rather than some other set, is just its members, so that there cannot be two distinct sets that have exactly the same members. Thus, {a,c,b}, {a,b,c} and {a,b,b,c} are different ways of writing the very same set. But note that {a,b,{c}} is not the same set as {a,b,c}, for the former contains the set {c} whereas the latter does not. We count the collection of no objects at all as a set. This is called the empty set and is written (some authors use {} ). 1 I will use capital roman letters X, Y, Z to name sets, the symbol to abbreviate is a member of and / to abbreviate is not a member of. The union of two sets X and Y, X Y, is the set that contains all the members of X and all the members of Y (and no more). n object a X Y, therefore, if and only if either a X or a Y. For example, if X = {a,b,c} and Y = {b,c,d} then X Y = {a,b,c,d}. The intersection of X and Y, X Y, is the set that contains all of the objects that are both members of X and members of Y. n object a X Y if and only if a X and a Y. If X and Y are as above, then their intersection X Y = {b,c}. X is a subset of Y, written X Y, if and only if every member of X is also a member of Y. {a} is a subset of {a,b}, as is {b}. Note that X X and X for every X. The set of all subsets of X, including and X itself, is the power set of X, written (X). 2 1 There can only be the one empty set. If there were two distinct empty sets, then at least one object must belong to one but not the other. But since no objects belong to the empty set, this cannot be the case. 2 The power set of of X can also be written 2 X.