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University at Albany, State University of New York Scholars Archive Philosophy Faculty Books Philosophy 12-27-2014 Forall x: An introduction to formal logic 1.30 P.D. Magnus University at Albany, State University of New York, [email protected] Follow this and additional works at: https://scholarsarchive.library.albany.edu/ cas_philosophy_scholar_books Part of the Logic and Foundations of Mathematics Commons Recommended Citation Magnus, P.D., "Forall x: An introduction to formal logic 1.30" (2014). Philosophy Faculty Books. 3. https://scholarsarchive.library.albany.edu/cas_philosophy_scholar_books/3 This Book is brought to you for free and open access by the Philosophy at Scholars Archive. It has been accepted for inclusion in Philosophy Faculty Books by an authorized administrator of Scholars Archive. For more information, please contact [email protected]. forallx An Introduction to Formal Logic P.D. Magnus University at Albany, State University of New York fecundity.com/logic, version 1.30 [141227] This book is offered under a Creative Commons license. (Attribution-ShareAlike 3.0) The author would like to thank the people who made this project possible. Notable among these are Cristyn Magnus, who read many early drafts; Aaron Schiller, who was an early adopter and provided considerable, helpful feedback; and Bin Kang, Craig Erb, Nathan Carter, Wes McMichael, Selva Samuel, Dave Krueger, Brandon Lee, Toan Tran, and the students of Introduction to Logic, who detected various errors in previous versions of the book. c 2005{2014 by P.D. Magnus. Some rights reserved. You are free to copy this book, to distribute it, to display it, and to make derivative works, under the following conditions: (a) Attribution. You must give the original author credit. (b) Share Alike. If you alter, transform, or build upon this work, you may distribute the resulting work only under a license identical to this one. | For any reuse or distribution, you must make clear to others the license terms of this work. Any of these conditions can be waived if you get permission from the copyright holder. Your fair use and other rights are in no way affected by the above. | This is a human-readable summary of the full license, which is available on-line at http://creativecommons.org/licenses/by-sa/3.0/ Typesetting was carried out entirely in LATEX2". The style for typesetting proofs is based on fitch.sty (v0.4) by Peter Selinger, University of Ottawa. This copy of forallx is current as of December 27, 2014. The most recent version is available on-line at http://www.fecundity.com/logic Contents 1 What is logic? 5 1.1 Arguments . 6 1.2 Sentences . 6 1.3 Two ways that arguments can go wrong . 7 1.4 Deductive validity . 8 1.5 Other logical notions . 10 1.6 Formal languages . 12 Practice Exercises . 15 2 Sentential logic 17 2.1 Sentence letters . 17 2.2 Connectives . 19 2.3 Other symbolization . 28 2.4 Sentences of SL . 29 Practice Exercises . 33 3 Truth tables 37 3.1 Truth-functional connectives . 37 3.2 Complete truth tables . 38 3.3 Using truth tables . 41 3.4 Partial truth tables . 43 Practice Exercises . 45 4 Quantified logic 48 4.1 From sentences to predicates . 48 4.2 Building blocks of QL . 50 4.3 Quantifiers . 54 4.4 Translating to QL . 57 4.5 Sentences of QL . 68 4.6 Identity . 71 Practice Exercises . 76 5 Formal semantics 83 5.1 Semantics for SL . 84 3 4 CONTENTS 5.2 Interpretations and models in QL . 88 5.3 Semantics for identity . 92 5.4 Working with models . 94 5.5 Truth in QL . 98 Practice Exercises . 103 6 Proofs 107 6.1 Basic rules for SL . 108 6.2 Derived rules . 117 6.3 Rules of replacement . 119 6.4 Rules for quantifiers . 121 6.5 Rules for identity . 127 6.6 Proof strategy . 128 6.7 Proof-theoretic concepts . 130 6.8 Proofs and models . 131 6.9 Soundness and completeness . 132 Practice Exercises . 134 A Other symbolic notation 140 B Solutions to selected exercises 143 C Quick Reference 156 Chapter 1 What is logic? Logic is the business of evaluating arguments, sorting good ones from bad ones. In everyday language, we sometimes use the word `argument' to refer to bel- ligerent shouting matches. If you and a friend have an argument in this sense, things are not going well between the two of you. In logic, we are not interested in the teeth-gnashing, hair-pulling kind of ar- gument. A logical argument is structured to give someone a reason to believe some conclusion. Here is one such argument: (1) It is raining heavily. (2) If you do not take an umbrella, you will get soaked. :_: You should take an umbrella. The three dots on the third line of the argument mean `Therefore' and they indicate that the final sentence is the conclusion of the argument. The other sentences are premises of the argument. If you believe the premises, then the argument provides you with a reason to believe the conclusion. This chapter discusses some basic logical notions that apply to arguments in a natural language like English. It is important to begin with a clear understand- ing of what arguments are and of what it means for an argument to be valid. Later we will translate arguments from English into a formal language. We want formal validity, as defined in the formal language, to have at least some of the important features of natural-language validity. 5 6 forallx 1.1 Arguments When people mean to give arguments, they typically often use words like `there- fore' and `because.' When analyzing an argument, the first thing to do is to separate the premises from the conclusion. Words like these are a clue to what the argument is supposed to be, especially if| in the argument as given| the conclusion comes at the beginning or in the middle of the argument. premise indicators: since, because, given that conclusion indicators: therefore, hence, thus, then, so To be perfectly general, we can define an argument as a series of sentences. The sentences at the beginning of the series are premises. The final sentence in the series is the conclusion. If the premises are true and the argument is a good one, then you have a reason to accept the conclusion. Notice that this definition is quite general. Consider this example: There is coffee in the coffee pot. There is a dragon playing bassoon on the armoire. :_: Salvador Dali was a poker player. It may seem odd to call this an argument, but that is because it would be a terrible argument. The two premises have nothing at all to do with the conclusion. Nevertheless, given our definition, it still counts as an argument| albeit a bad one. 1.2 Sentences In logic, we are only interested in sentences that can figure as a premise or conclusion of an argument. So we will say that a sentence is something that can be true or false. You should not confuse the idea of a sentence that can be true or false with the difference between fact and opinion. Often, sentences in logic will express things that would count as facts| such as `Kierkegaard was a hunchback' or `Kierkegaard liked almonds.' They can also express things that you might think of as matters of opinion| such as, `Almonds are yummy.' Also, there are things that would count as `sentences' in a linguistics or grammar course that we will not count as sentences in logic. ch. 1 what is logic? 7 Questions In a grammar class, `Are you sleepy yet?' would count as an interrogative sentence. Although you might be sleepy or you might be alert, the question itself is neither true nor false. For this reason, questions will not count as sentences in logic. Suppose you answer the question: `I am not sleepy.' This is either true or false, and so it is a sentence in the logical sense. Generally, questions will not count as sentences, but answers will. `What is this course about?' is not a sentence. `No one knows what this course is about' is a sentence. Imperatives Commands are often phrased as imperatives like `Wake up!', `Sit up straight', and so on. In a grammar class, these would count as imperative sentences. Although it might be good for you to sit up straight or it might not, the command is neither true nor false. Note, however, that commands are not always phrased as imperatives. `You will respect my authority' is either true or false| either you will or you will not| and so it counts as a sentence in the logical sense. Exclamations `Ouch!' is sometimes called an exclamatory sentence, but it is neither true nor false. We will treat `Ouch, I hurt my toe!' as meaning the same thing as `I hurt my toe.' The `ouch' does not add anything that could be true or false. 1.3 Two ways that arguments can go wrong Consider the argument that you should take an umbrella (on p. 5, above). If premise (1) is false| if it is sunny outside| then the argument gives you no reason to carry an umbrella. Even if it is raining outside, you might not need an umbrella. You might wear a rain pancho or keep to covered walkways. In these cases, premise (2) would be false, since you could go out without an umbrella and still avoid getting soaked. Suppose for a moment that both the premises are true. You do not own a rain pancho.
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