# Logic

## The Laws of Truth

## Nicholas J.J. Smith

- 544 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android

# Logic

## The Laws of Truth

## Nicholas J.J. Smith

## About This Book

Logic is essential to correct reasoning and also has important theoretical applications in philosophy, computer science, linguistics, and mathematics. This book provides an exceptionally clear introduction to classical logic, with a unique approach that emphasizes both the hows and whys of logic. Here Nicholas Smith thoroughly covers the formal tools and techniques of logic while also imparting a deeper understanding of their underlying rationales and broader philosophical significance. In addition, this is the only introduction to logic available today that presents all the major forms of proof--trees, natural deduction in all its major variants, axiomatic proofs, and sequent calculus. The book also features numerous exercises, with solutions available on an accompanying website. Logic is the ideal textbook for undergraduates and graduate students seeking a comprehensive and accessible introduction to the subject.

- Provides an essential introduction to classical logic
- Emphasizes the how and why of logic
- Covers both formal and philosophical issues
- Presents all the major forms of proof--from trees to sequent calculus
- Features numerous exercises, with solutions available at http://njjsmith.com/philosophy/lawsoftruth/
- The ideal textbook for undergraduates and graduate students

## Frequently asked questions

## Information

## PART I

## Propositional

Logic

### 1

**Propositions and Arguments**

**1.1 What Is Logic?**

*reasoning*.

^{1}Knowing basic logic is indeed essential to being able to reason wellāyet it would be misleading to say that human reasoning is the primary subject matter of logic. Rather, logic stands to reasoning as mathematics stands to carpet cutting. Suppose you are looking for your keys, and you know they are either in your pocket, on the table, in the drawer, or in the car. You have checked the first three and the keys arenāt there, so you reason that they must be in the car. This is a good way to reason. Why? Because reasoning this way cannot lead from true premises or starting points to a false conclusion or end point. As Charles Peirce put it in the nineteenth century, when modern logic was being developed:

*propositions*āthings that are true or falseāand their components, and it seeks to discover laws governing the relationships between the truth or falsity of different propositions. One such law is that if a proposition offers a fixed number of alternatives (e.g., the keys are either (i) in your pocket, (ii) on the table, (iii) in the drawer, or (iv) in the car), and all but one of them are false, then the overall proposition cannot be true unless the remaining alternative is true. Such

*general*laws about truth can usefully be applied in reasoning: it is because the general law holds that the particular piece of reasoning we imagined above is a good one. The law tells us that if the keys really are in one of the four spots, and are not in any of the first three, then they must be in the fourth; hence the reasoning cannot lead from a true starting point to a false conclusion.

*truth*. (Note that by āscienceā we mean simply

*systematic study*.)

^{2}As Gottlob Frege, one of the pioneers of modern logic, put it:

**1.2 Propositions**

*propositions*are those things which can be true or false. Now what sort of things are propositions, and what is involved in a propositionās being true or false? The fundamental idea is this: a proposition is a claim about how things areāit represents the world as being some way; it is true if the world is that way, and otherwise it is false. This idea goes back at least as far as Plato and Aristotle: