Truth Functional Logic

1. Truth Functional Logic Compound Statements

2. What is a Truth Table? • A truth table is a way of representing a statement’s meaning symbolically. • Each compound statement has a single identifying characteristic.

3. Negation • Every claim has a negation or contradictory claim. • “Ginger is a dog” has as its negation statement, “Ginger is not a dog.” • “Ginger is a dog” = P • “Ginger is not a dog” = ~P

4. Truth Table for Negation • Every statement has two possible truth values, T & F. (True or false) • P ~P Negation means T F opposite truth value F T If “P” is true, then ~P is false; “P” is false, then ~P is true.

5. Conjunction Statement • A conjunction is a compound claim asserting both the simpler claims contained in it. • Thus, a conjunction is true only if both of the claims are also true. • A conjunction = P & Q

6. Truth Table-Conjunction • A conjunction has two claims; each have two possible truth values and thus the compound statement has four possible truth values. P Q P & Q Since a conjunction is T T T true only when both first T F F are true, the first case is F T F the key case. F F F

7. Disjunction • A disjunction is a compound claim asserting either or both claims contained in it. Thus, a disjunction is false only if both simpler claims are false. • P Q P v Q T T T T F T F T T F F F ↔ This is the key case.

8. Conditional • A conditional asserts the second claim on the condition that the first is true. A conditional thus is false if and only if the first claim is true and the second is false. • P Q P →Q T T T T F F ↔ This is key case. F T T F F T

9. Truth Functional Logic-2 Arguments and Truth Tables

10. Validity of Arguments • The validity of an argument guarantees that if its premises are true, then its conclusion must be true. • Thus, an argument is invalid if there is any case where its premises are true and its conclusion false.= Key case.

11. Truth Table for Arguments • If there are two variables, S and P, then you need four lines; if three variables you need eight lines. • You will need a column for each premise and the conclusion and for each variable, e.g. S and ~S.

12. Truth Table-Argument-2 “If building this requires a small Philips screwdriver, then I will not be able to build it. It does require a small Philips screwdriver. Thus, I will not be able to build it. Symbolize: S → ~B, S, Thus, ~B

13. Truth Table Argument-3 • Build the truth table as follows: • S B S → ~B S ~B • T T F F T F * T F T T T T F T T F F F * F F T T F T There is no line where the premises are true and the conclusion is false and thus the argument is valid.

14. Truth Table-Argument-4 • Martin is not buying a new car {since} he said he would buy a new car or take a Hawaiian vacation. He is now in Maui. • Symbolize: C v H H ~ C

15. Truth Table-Argument- 5 • Build a truth table as follows: C H C v H H ~ C T T TTF ↔ Invalid T F T F F F T T T T F F F F T

16. Truth Table Argument-6 • “If you want to over-clock your processor you must make both hardware and software changes. But you either can’t do hardware or can’t do software. So you won’t be over-clocking your processor. • Symbolize: O→ (H & S) ~H v ~ S ~ O

17. Build the Truth Table • Because you have three variables you will need eight lines. • First column alternates four true with four false • Second column- alternates pairs of true and false. • Third column- alternate one true and one false all the way down.

18. The Truth Table O H S 0 →( H & S) (~H v ~S) ~0 T T T T F F T T F F T F T F T F T F T F F F T F F T T T F T F T F F T T F F T F T T F F F F T T There is no case where the conclusion is false and the premises are all true- so it is a valid argument.