Garrett - Elementary Logic

First published 2012 by Acumen

Published 2014 by Routledge

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Brian Garrett, 2012

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Isbn: 978-1-84465-517-5 (hardcover)

Isbn: 978-1-84465-518-2 (paperback)

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The book offers a clear and concise introduction to propositional logic. It can be used in a three-month course, but also could be expanded for use in a longer course. The last two chapters contain philosophical material, which should be accessible even to beginning students.

My aim has been to produce a logic text that offers a concise, ground-up introduction to elementary logic, with an emphasis on the ideas underlying logical principles and rules of inference. I am writing for the student who wants to understand the basic ideas of elementary logic, but may have no intention of doing more advanced work in logic. My concern is to get students to see why a given proof is valid or what the rationale is for a particular rule of inference. Such insights are of more value than being able to zip through proofs in record time. To that end, are given up to answering some of the questions at the end of preceding chapters with, quite deliberately, much in the way of authorial intervention.

Since the target audience is philosophy students, I thought it might be useful to look at various meta-logical or philosophical issues that arise from propositional logic, and outline more advanced logics that some students might like to investigate further. It is well for students to know, for example, that the classical logic that underpins elementary logic has counter-intuitive aspects, and that there are rival systems of logic. Such reflections make it apparent that logic and the philosophy of logic cannot be cleanly separated. These issues are discussed in .

In . Since paradoxes matter, logic matters too.

Three further features of the book are worth noting: the concept boxes, the Glossary and the Further Reading section. In many of the chapters I have interspersed the text with concept boxes. These boxes either summarize some concept or technique central to a given chapter or else further elaborate on some claim made therein. In the Glossary I give clear and explicit definitions of many of the key words and phrases used in the book. Key words or phrases in the text are indicated using bold toning on first appearance. A firm grasp of the terms that appear in the Glossary is crucial to understanding the nature of logic and logical truth. As well as the articles and books referred to in the text and footnotes, I have selected a number of useful and interesting books and articles for the Further Reading. These readings are themselves divided into various subsections, making it easy for the student to follow up any particular topic.

Some are sceptical of the value of teaching students elementary logic. I am not. Many arguments in philosophy and elsewhere are fallacious, and we need logic to expose them. Even if many philosophical debates turn on the truth or falsity of premises, rather than the validity of arguments, it is still important to be able to identify the form of argument in question. In addition, it is undeniable that competency in logic, like any intellectual skill, helps to sharpen and perfect the mind. A student trained in logic will be better placed to assess and evaluate any piece of reasoning they come across, in philosophy or in real life.

This book has grown out of a short Introduction to Logic course that I teach at the Australian National University (ANU). Thanks to various students in my logic classes over the years for their helpful comments. I am also grateful to three anonymous Acumen referees, and to Peter Eldridge-Smith, Katrina Hutchison, J. J. Joaquin, Thomas Mautner, Peter Roeper and Ryan Young for useful feedback.

The aim of this book is to introduce students to the ideas and techniques of symbolic logic. Logic is the study of arguments. After working through this book the reader should be in a position to identify and evaluate a wide range of arguments.

Once an argument has been identified, we need to determine whether it is a good argument or a bad one. By good argument we mean a valid argument; by bad argument we mean an invalid argument. Our primary method for determining validity will be , but we also use the (simpler but more cumbersome) method of truth-trees. In addition, we briefly show how truth-tables can also be used to test for validity.

Elementary logic studies arguments, and, in doing so, it studies the logical or inferential properties of the so-called logical connectives: and, if then , or, not and if and only if. We use these logical words much of the time, even if we might find it hard to say what they mean. In logic, however, these key words have a clear and explicit meaning.

In elementary logic, the premises and conclusion of an argument are all declarative sentences; that is, they are sentences that are either true or false. There are only two truth-values and each declarative sentence has one and only one of them. The cat is on the mat, no one loves Raymond and all bachelors are bald are examples of declarative sentences.

Is the cat on the mat?, Put the cat on the mat!, Dont park there and I pronounce you man and wife are examples of non-declarative sentences. It would be odd to respond to utterances of any of these four sentences with Thats true! (or Thats false!). In these latter cases, the utterer is not stating or asserting things to be a certain way (so that things would either be that way or not).

As noted, we are concerned with five logical connectives: and, if then , or, not and if and only if. These connectives are used to form new sentences. Thus, from the sentences Bill is bald and Fred is fat we can use and to form the sentence Bill is bald and Fred is fat. We can use if then to form the sentence if Bill is bald then Fred is fat. And so on with the other connectives. (Note that all the connectives except not take two sentences in order to form a new sentence. Not takes only one.)

The logical connectives are thus ).

The meaning of the logical connectives is captured not only by their truth-tables, but also by the rules of inference associated with each connective. Thus, for example, and is associated with the rules: from A and B infer the conjunction A and B; from A and B infer A (or infer B).