Truth Functional Logic Compound Statements

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.

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

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.

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

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

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.

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

Truth Functional Logic-2 Arguments and Truth Tables

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.

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.

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

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.

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

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

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

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.

The Truth Table T T T T F F T T F F T F T F T F T F T F F F T F 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.