Propositional Logic A symbolic representation of deductive inference. Use upper case letters to represent simple propositions. E.g. Friday class rocks. = F Linda loves logic lectures. = L Use upper case letters and logical operators (~, ,&, v) to represent compound propositions ( e.g. If it’s Tuesday, Raja is in class) T  R)

Logical Operators (Connectives) ~ Negation, ~P “not P” “P is false”  Implication, P  Q “ if P, then Q” “P implies Q” & Conjunction, P & Q “P and Q” v Disjunction, P v Q “P or Q”

Translating Compound Statements Gonzaga will not win the tournament. ~G If UCLA wins, Minnesota will not advance. U  ~M Either Louisville or Kansas will win. L v K The Lobos will win and the Bruins won’t. L & ~B

Negation Heather is not happy. ~H The show is not over. ~S P________~P__ T F F T Negate falsehoods. Affirm truths. Suspend judgment on propositions with unknown truth values Get a life. Negating a true statement produces a false one, negating a false statement creates a truth.

Conjunction Your wife is gone and so is your car. W & C P QP&Q T T T T F F F T F F F F A conjunction is true only when both conjuncts are true

Implication (Hypotheticals) If Pat is busy, Mari will help you. P  M If the death penalty is unfairly administered, then the ABA will oppose it. D  A P___Q___P  _Q T T T T F F F T T F F T If the candle is lit, the wax will melt. C  W If P, then Q is true whenever either P is false or Q is true.

Disjunction Either your crankcase or your transmission is leaking oil. C v T It’s either Lupus or Arthritis. L v A P__Q____P_v_Q T T T T F T F T T F F F P or Q is true when either P or Q or both are true.

Modus Ponens If Tiny drinks Tequila, he gets sick. Tiny drank Tequila. Therefore, Tiny is sick. T  S T____  S P  Q P___  Q A valid deductive argument form.

The fallacy of affirming the consequent. If Tiny drinks Tequila, he gets sick. He is sick. Therefore, he drank tequila. T  S (Really, it was a bad S burrito.)  T P  Q Q An invalid form.  P

Modus Tollens If the engine is running, there is gas in the tank. There is no gas in the tank. Therefore, the engine is not running. E  G ~G ~E P  Q ~Q___ ~P A valid deductive argument form.

Fallacy of Denying the Antecedent If it rains, the street gets wet. It didn’t rain. So the street isn’t wet. R  S ~R___ ~S An invalid form. Who left the sprinklers on?. Mt. Ararat or bust!

Hypothetical Syllogism If the first native is a politician, then he lies. If he lies, then he denies being a politician. Therefore, if the first native is a politician, then he denies being a politician. P  Q Q  R  P  R A valid deductive form.

Disjunctive Syllogism Either Obamacare will kill kids, or Bachman spoke falsely. Obamacare will not kill kids. Therefore, Bachman spoke falsely. O v B ~O B P v Q ~P__  Q A valid deductive argument form.

Truth Table Validity Test A truth table represents all possible truth value combinations of the premises and conclusion of an argument. Valid forms never have a false conclusion with true premises. I’ll prove I love her with this truth table. TTFF

Setting up Truth Tables If the argument contains two simple propositions (as in, If A then C, A,  C) then four rows are needed (place two Ts and two Fs under A, and T, F, T, F under C. AC TT TF FT FF

Truth Table Set Up If the argument contains three simple propositions (as in: If A then B; If B then C;  If A then C) then eight rows are needed. Place four Ts then four Fs under A; two Ts, two Fs, two Ts, two Fs under B; and T,F,T,F,T,F,T,F under C. ABC TTT TTF TFT TFF FTT FTF F FT FFF

Entering Truth values for Compound Propositions PQ PvQP&Q P  Q~Q TT T T T F T F T F F T F T T F T F F F FF T T [Refer to this chart when entering truth values for disjunctions, conditionals, and negations. The truth values of the compounds are a function of the truth values of the simple components (P and Q).]

TT for MP(If p, q; p;  q) Components Compound PQP  Q p2 c p1 TT T TF F FT T FF T No line has true premises and false conclusion, so the argument is valid.

Denying the Antecedent If God exists, morality is objective. God does not exist. So, morality is merely subjective. G  O ~G___ ~O P Q P  Q ~P ~Q prem 1 p2 c T T T F F T F F F T F T T T F F F T T T Line three has true prem.s & false conclusion; so invalid.

Dilemma Either you get married or you break it off. If you get married, you will be miserable. If you break it off you will be miserable. (Unexpressed Conclusion?) Damn, two can live cheaper than one.

Truth Table PQRPvQ P  R Q  R c p1 p2 p3 TT T T T T T T F T F F T F T T T T x T F F T F T F T T T T T x F T F T T F F F T F T T F F F F T T Valid!