Logic in Computer Science

The first language we consider is the language of propositional logic. It is based on propositions (or declarative sentences) which can either be true or false. Some examples are: 1. Grass is green. 2. The sky is yellow. 3. Every natural number n > 2 is the sum of two prime numbers. 4. If I do not study then I get an F in this course. Sentence (1) is obviously true, whereas sentences (2) is false (at least on earth). Sentence (3) is the so-called Goldbach conjecture. Nobody actually knows whether that sentence is true or false (it is an open problem in mathematics). Nevertheless, it is either one-true or false. Sentence (4) is remarkable since it built from smaller sentences by using certain constructions. Two two propositions 'I do not study' and 'I get an F in this course' are combined using an 'If ... then ...' construction. Even the proposition 'I do not study' can be considered to be the result of applying a 'not ...' construction to the proposition 'I do study'. Some examples for sentences that are not propositions are: • May the force be with you. • Go, clean up your room! • How are you today? 3 4 CHAPTER 1. PROPOSITIONAL LOGIC The sentences above are formulated in plain English. Such a representation of propositions is not very suitable for a computer. Therefore, we are going to introduce a formal language of propositions. This language can easily be manipulated by a program. We are going to consider certain declarative sentences as being atomic. For example, the sentences (1)-(3) are atomic. They cannot be decomposed into smaller propositions. We will use the symbol ⊥ to denote falsehood, and distinct symbols p, q, r,. .. for arbitrary atomic sentences. Definition 1 Let P be a set of propositional variables. The set Prop of propositional formulas are those which can be obtained by using the following rules finitely many times: (a) φ 1 ∧ φ 2 ∈ Prop and (b) φ 1 ∨ φ 2 ∈ Prop and (c) φ 1 → φ 2 ∈ Prop. The previous definition defines the set of formulas by giving a set of rules which may be applied to a base set finitely many times. We also say that the set Prop is defined recursively by those rules. A set defined in such a way always provides a principle of induction. …