Logic and Proofs

Abstract

• A proposition is a declarative statement (sentence) that is true or false. For example, “Mars is a planet” is a proposition that is true, and “Jupiter is a star” is a proposition that is false. Questions, commands and exclamations are examples of sentences that are not propositions. • Truth values: They are the values “true” (denoted T) and “false” (denoted F). • A proposition that conveys a single fact is called a simple proposition. • Simple propositions can be combined and/or negated, using so-called logical connectives, to form more elaborate propositions called compound propositions. • Logical connectives: They are the logical operations “and”, “or”, and “not”. The first two are binary operations, that is, they take two operands where each operand is a proposition. The “not” is a unary operation in that it takes a single operation. • Examples of compound propositions: o Mars is a planet and Jupiter is a star o Mars is a planet or Jupiter is a star o not (Jupiter is a star). In plain English, this statement means “Jupiter is not a star”. o Mars is a planet and (Jupiter is a star or not(Pluto is a comet)) • Notations: o The operation “and” is denoted ∧. (In Boolean algebra, it is denoted by the dot “.” ). o The operation “or” is denoted ∨. (In Boolean algebra, it is denoted by + ). o The operation “not” is denoted ¬, ~, ̅(overbar), or ′. • Logic is concerned much more about the truth values of propositions than about their meanings. • Therefore, and for convenience and efficiency, propositions are often denoted with single-letter symbols such as p, q, r, etc. And the focus is on whether a proposition p is true or false, rather than on what p means/designates. For example, we can denote “Mars is a planet” by p, “Jupiter is a star” by q, and “Pluto is a comet” by r. Then the above examples of compound statements can be succinctly represented as: o “Mars is a planet and Jupiter is a star”: pp ∧ qq. This (i.e., pp ∧ qq) is called a conjunction. o “Mars is a planet or Jupiter is a star”: pp ∨ qq. This (i.e., pp ∨ qq) is called a disjunction. o “not (Jupiter is a star)”: ¬pp, ~pp, ?̅?p, or pp′.