By Mark V. Lawson
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Additional resources for An introduction to logic [Lecture notes]
An we write A1 , . . , An B and we say this is a valid argument. This definition encapsulates many examples of logical reasoning. It is the foundation of mathematics and the basis of trying to prove that programs do what we claim they do. We shall see later that there are examples of logical reasoning that cannot be captured by PL and this will lead us to the generalization of PL called first-order logic or FOL. 1. Here are some examples of valid arguments. 1. p, p → q q. We show that this is a valid argument.
PROPOSITIONAL LOGIC 3. p ∨ q, ¬p q. We show this is a valid argument. p T T F F q T F T F ¬p p ∨ q F T F T T T T F We are only interested in the cases where both p ∨ q and ¬p are true. p F q T ¬p p ∨ q T T We see that if p ∨ q and ¬p are true then q must be true. 4. p → q, q → r p → r. We show this is a valid argument. p T T T T F F F F q T T F F T T F F r T F T F T F T F p→q q→r p→r T T T T F F F T T F T F T T T T F T T T T T T T We are only interested in the cases where both p → q and q → r are true.
The satisfiability problem (SAT) Given a wff decide whether there is some truth assignment to the atoms that makes the wff take the value true. I shall discuss this problem in more detail later and explain why it is so important. The following examples illustrate an idea that we shall develop in the next section. 4. Compare the true tables of p → q and ¬p ∨ q. p T T F F p→q T F T T q T F T F p T T F F q T F T F ¬p F F T T ¬p ∨ q T F T T They are clearly the same. 5. Compare the true tables of p ↔ q and (p → q) ∧ (q → p).