Logic

Chapter

2
Introduction to Logic
Chapter Outline
• • • • • Logic and Propositions Logical Connectives Truth Tables Conditional Statements Tautology, Contradiction, and Contingency • Logical Equivalences and Implications

Introduction to Logic

Logic and Propositions
Logic is the discipline that deals with the study of reasoning. It provides rules and techniques for determining whether a given argument is valid or not. The rules of logic are used in many areas, particularly in mathematics. Logical methods are used to prove theorems or to distinguish between valid and invalid mathematical arguments. It is also essential in computer science in verifying the correctness of programs. A proposition is any declarative sentence that is either true or false, but not both. The truth value of a proposition is true, denoted by T (or 1), if it is a true proposition and false, denoted by F (or 0), if it is a false proposition.

Example 1 Which of the following are propositions? a. Today is Monday. b. Did you buy the textbook? c. 13 + 17 = 40 d. Take your medicine. e. There are nine planets in the universe. f. 23 – x = 11 Solution

Logical Connectives
A proposition is said to be atomic if it cannot be further subdivided. Atomic propositions may be combined to form compound propositions. Atomic propositions are usually denoted by lowercase letters p, q, r etc. Compound propositions are formed by combining one or more atomic propositions using logical connectives. There are five basic logical connectives or logical operators used in logic. These are the negation, conjunction, disjunction, conditional, and biconditional.

2
Lecture Notes in Mathematical Logic

Introduction to Logic

Negation

The negation of a proposition p, denoted by ¬p (read as "not p"), is the statement "It is not the case that p." The truth value of ¬p is the opposite of the truth value of p. Example 2 Find the negation of each proposition. a. Today is Wednesday. b. There is no... [continues]

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