Logic as a Tool
Written in a clear, precise and user-friendly style, Logic as a Tool: A Guide to Formal Logical Reasoning is intended for undergraduates in both mathematics and computer science, and will guide them to learn, understand and master the use of classical logic as a tool for doing correct reasoning. It offers a systematic and precise exposition of classical logic with many examples and exercises, and only the necessary minimum of theory.
The book explains the grammar, semantics and use of classical logical languages and teaches the reader how grasp the meaning and translate them to and from natural language. It illustrates with extensive examples the use of the most popular deductive systems -- axiomatic systems, semantic tableaux, natural deduction, and resolution -- for formalising and automating logical reasoning both on propositional and on first-order level, and provides the reader with technical skills needed for practical derivations in them. Systematic guidelines are offered on how to perform logically correct and well-structured reasoning using these deductive systems and the reasoning techniques that they employ.
Concise and systematic exposition, with semi-formal but rigorous treatment of the minimum necessary theory, amply illustrated with examples
Emphasis both on conceptual understanding and on developing practical skills
Solid and balanced coverage of syntactic, semantic, and deductive aspects of logic
Includes extensive sets of exercises, many of them provided with solutions or answers
Supplemented by a website including detailed slides, additional exercises and solutions
Logic as a Tool
Understanding Propositional Logic
Propositional logic is about reasoning with propositions. These are sentences that can be assigned a truth value: true or false. They are built from primitive statements, called atomic propositions, by using propositional logical connectives. The truth values propagate over all propositions through truth tables for the propositional connectives. In this chapter I explain how to understand propositions and compute their truth values, and how to reason using schemes of propositions called propositional formulae. I will formally capture the concept of logically correct propositional reasoning by means of the fundamental notion of propositional logical consequence.1.1 Propositions and logical connectives: truth tables and tautologies
The basic concept of propositional logic is proposition. A proposition is a sentence that can be assigned a unique truth value: true or false.
Some simple examples of propositions include:
- The Sun is hot.
- The Earth is made of cheese.
- 2 plus 2 equals 22.
- The 1000th decimal digit of the number is 9.
(You probably don't know whether the latter is true or false, but it is surely either true or false.)
The following are not propositions (why?):
- Are you bored?
- Please, don't go away!
- She loves me.
- is an integer.
- This sentence is false.
Here is why. The first sentence above is a question, and it does not make sense to declare it true or false. Likewise for the imperative second sentence. The truth of the third sentence depends on who "she" is and who utters the sentence. Likewise, the truth of the fourth sentence is not determined as long as the variable is not assigned a value, integer or not. As for the last sentence, the reason is trickier: assuming that it is true it truly claims that it is false - a contradiction; assuming that it is false, it falsely claims that it is false, hence it is not false - a contradiction again. Therefore, no truth value can be consistently ascribed to it. Such sentences are known as self-referential and are the main source of various logical paradoxes (see the appetizer and Russell's paradox in Section 5.2.1).1.1.2 Propositional logical connectives
The propositions above are very simple. They have no logical structure, so we call them primitive or atomic propositions. From primitive propositions one can construct compound propositions by using special words called logical connectives. The most commonly used connectives are:
- not, called negation, denoted ;
- and, called conjunction, denoted (or sometimes );
- or, called disjunction, denoted ;
- if then , called implication, or conditional, denoted ;
- ...if and only if ..., called biconditional, denoted .
It is often not grammatically correct to read compound propositions by simply inserting the names of the logical connectives in between the atomic components. A typical problem arises with the negation: one does not say "Not the Earth is square." A uniform way to get around that difficulty and negate a proposition is to say "It is not the case that ."
In natural language grammar the binary propositional connectives, plus others like but, because, unless, although, so, yet, etc. are all called