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Propositional LogicPropositional 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
NegationNegation Heather is not
happy. ~H
The show is not over. ~S
P________~P__ T F F T
Negate falsehoods.Affirm truths.Suspend judgmenton propositions withunknown truth valuesGet 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 Q P&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)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.
DisjunctionDisjunction 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 PonensModus 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.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 TollensModus 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 AntecedentFallacy 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 thesprinklers
on?.
Mt. Ararat or bust!
Hypothetical SyllogismHypothetical 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 SyllogismDisjunctive 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 TestTruth 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 loveher with this truthtable.
TTFF
Setting up Truth TablesSetting 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.
A C T T T F F T F F
Truth Table Set UpTruth 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.
A B C T T T T T F
T F T TF F
F T T F T F F F T F F
F
Entering Truth values for Compound Propositions
Entering Truth values for Compound Propositions P Q PvQ P&Q P Q ~Q T T T T T F T F T F F T F T T F T F F F F F 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)TT for MP(If p, q; p;q)
Components Compound P Q P Q p2 c p1 T T T T
F F FT T FF T
No line has true premises and false conclusion, so the argument is valid.
Denying the AntecedentDenying 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.
DilemmaDilemma 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 thanone.
Truth TableTruth Table P Q R PvQ P R QR
c p1 p2 p3 T T 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!