By Mark V. Lawson
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Additional info for An introduction to logic [Lecture notes]
4. Every wff is logically equivalent to one in DNF. 36 CHAPTER 1. PROPOSITIONAL LOGIC Proof. Let A be a wff. Construct the truth table for A. 6. The wff that results is in DNF and logically equivalent to A. The method of proof used above can be used as a method for constructing DNF though it is a little laborious. Another method is to use logical equivalences. Let A be a wff. First convert A to NNF and then if necessary use the distributive laws to convert to a wff which is in DNF. 5. We show how to convert ¬(p → (p ∧ q)) into DNF using a sequence of logical equivalences.
We show that p → (q → r) ≡ (p ∧ q) → r. 1(1). 4. We show that p → (q → r) ≡ q → (p → r). 1(1). 4. LOGICAL EQUIVALENCE 21 5. We show that (p → q) ∧ (p → r) ≡ p → (q ∧ r). 1(1). The next example is a little different. 13. We shall prove that equivalences. 1(1) ≡ (¬p ∨ p) ∨ ¬q by associativity and commutativity ≡ T since ¬p ∨ p. Finally, here is an attempt to explain the rationale behind the definition of →. 14. I shall try to show how the truth table of → is forced upon us if we make some reasonable assumptions.
B) (p ∧ q) → r ≡ (p → r) ∨ (q → r). (c) p → (q ∨ r) ≡ (p → q) ∨ (p → r). 24 CHAPTER 1. PROPOSITIONAL LOGIC 7. We defined only 5 binary connectives, but there are in fact 16 possible ones. The tables below show all of them. p T T F F p T T F F q T F T F q T F T F ◦9 F T T T ◦1 T T T T ◦2 T T T F ◦10 F T T F ◦3 T T F T ◦11 F T F T ◦4 T T F F ◦5 T F T T ◦6 T F T F ◦12 F T F F ◦13 F F T T ◦14 F F T F ◦7 T F F T ◦15 F F F T ◦8 T F F F ◦16 F F F F (a) Express each of the connectives from 1 to 8 in terms of ¬, →, p, q and brackets only.
An introduction to logic [Lecture notes] by Mark V. Lawson