By Schaubel D.E., Cai J.

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LOGICAL VALIDITY AND RELATED NOTIONS ¬ f t t t (P t t f f ∧ t f f f Q) t f t f (¬ f f t t P t t f f ∨ f t t t 43 ¬ f t f t Q) t f t f The outcome of the calculation shows that the formulas are equivalent: note that the column under the ¬ of ¬(P ∧ Q) coincides with that under the ∨ of ¬P ∨ ¬Q. Notation: Φ ≡ Ψ indicates that Φ and Ψ are equivalent3. 6) shows that ¬(P ∧ Q) ≡ (¬P ∨ ¬Q). 7 (De Morgan Again) The following truth table shows that ¬(P ∧ Q) ⇔ (¬P ∨ ¬Q) is a logical validity, which establishes that ¬(P ∧ Q) ≡ (¬P ∨ ¬Q).

2 Converse and Contraposition. The converse of an implication P ⇒ Q is Q ⇒ P ; its contraposition is ¬Q ⇒ ¬P . The converse of a true implication does not need to be true, but its contraposition is true iff the implication is. Cf. 10, p. 45. Necessary and Sufficient Conditions. The statement P is called a sufficient condition for Q and Q a necessary condition for P if the implication P ⇒ Q holds. An implication P ⇒ Q can be expressed in a mathematical text in a number of ways: 1. if P , then Q, 2.

21 Answer as many of the following questions as you can: 1. Construct a formula Φ involving the letters P and Q that has the following truth table. P t t f f Q t f t f Φ t t f t 2. How many truth tables are there for 2-letter formulas altogether? 3. Can you find formulas for all of them? 4. Is there a general method for finding these formulas? 5. And what about 3-letter formulas and more? 50 CHAPTER 2. 3 Making Symbolic Form Explicit In a sense, propositional reasoning is not immediately relevant for mathematics.