Formal Test for Validity

EVALUATIONS

Evaluations An evaluation is an assignment of truth-values to sentence letters. For example: A = T B = T C = F D = T E = F...

Evaluating WFFs To evaluate a WFF is to determine whether it is true or false according to an evaluation. Let’s consider ((Q & ~P) → R) Here’s our evaluation: Q = T, P = T, R = F.

Evaluation: Stage 1 PQR Write down sentence letters.

Evaluation: Stage 1 PQR TTF Insert truth-values from evaluation.

Evaluation: Stage 2 PQR((Q&~P)→R) TTF Copy down the formula to evaluate.

Evaluation: Stage 3 PQR((Q&~P)→R) TTFT Copy the truth-values of each variable.

Evaluation: Stage 3 PQR((Q&~P)→R) TTFTT Copy the truth-values of each variable.

Evaluation: Stage 3 PQR((Q&~P)→R) TTFTTF Copy the truth-values of each variable.

Evaluation: Stage 4 PQR((Q&~P)→R) TTFTTF Find a connective to evaluate.

Evaluation: Stage 4 PQR((Q&~P)→R) TTFTTF Need these truth values.

Evaluation: Stage 4 PQR((Q&~P)→R) TTFTTF Need these truth values.

Evaluation: Stage 4 PQR((Q&~P)→R) TTFTTF Need this truth value.

Evaluation: Stage 4 PQR((Q&~P)→R) TTFTFTF Need this truth value.

Evaluation: Stage 4 PQR((Q&~P)→R) TTFTFTF Need these truth values.

Evaluation: Stage 4 PQR((Q&~P)→R) TTFTFFTF Need these truth values.

Evaluation: Stage 4 PQR((Q&~P)→R) TTFTFFTF Need these truth values.

Evaluation: Stage 4 PQR((Q&~P)→R) TTFTFFTTF Need these truth values.

In-Class Exercises Evaluation: P = F, Q = F, R = T ~(~P & ~Q) ~(P → ~Q) ((P & ~Q) & R)

FULL TRUTH-TABLES

Possibilities for One Sentence Letter φ…φ……φ… T F

Possibilities for Two Sentence Letters φψ…φ…ψ……φ…ψ… TT TF FT FF

Possibilities for Three Sentence Letters φψχ…φ…ψ…χ……φ…ψ…χ… TTT TTF TFT TFF FTT FTF FFT FFF

~(~P & ~Q) PQ~(~P&~Q) TT TF FT FF

Copy Whole Column PQ~(~P&~Q) TTT TFT FTF FFF

Copy Whole Column PQ~(~P&~Q) TTTT TFTF FTFT FFFF

Evaluate Each Row PQ~(~P&~Q) TTFTT TFFTF FTTFT FFTFF

Evaluate Each Row PQ~(~P&~Q) TTFTFT TFFTTF FTTFFT FFTFTF

~(~P & ~Q) PQ~(~P&~Q) TTFTFFT TFFTFTF FTTFFFT FFTFTTF

~(~P & ~Q) PQ~(~P&~Q) TTTFTFFT TFTFTFTF FTTTFFFT FFFTFTTF

(~(~P & ~Q) ↔ (P v Q)) So “~(~P & ~Q)” has the same truth-table as “(P v Q).” Why is that? Suppose I say: “you didn’t do your homework and you didn’t come to class on time.” When is this statement false? When either you did your homework or you came to class on time.

In-Class Exercise Write a full truth-table for: ~(P → ~Q)

PQ~(P→~Q) TT TF FT FF

~(P → ~Q) PQ~(P→~Q) TTTT TFTF FTFT FFFF

~(P → ~Q) PQ~(P→~Q) TTTFT TFTTF FTFFT FFFTF

~(P → ~Q) PQ~(P→~Q) TTTFFT TFTTTF FTFTFT FFFTTF

~(P → ~Q) PQ~(P→~Q) TTTTFFT TFFTTTF FTFFTFT FFFFTTF

(~(P → ~Q) ↔ (P & Q)) So “~(P → ~Q)” has the same truth-table as “(P & Q).” Why is that? Suppose I say: “If you eat this spicy food, you will cry.” You might respond by saying “No, that’s not true: I will eat the spicy food and I will not cry.”

In-Class Exercise Write a full truth-table for: (P & (~Q & R))

PQR(P&(~Q&R)) TTT TTF TFT TFF FTT FTF FFT FFF

(P & (~Q & R)) PQR(P&(~Q&R)) TTTTTT TTFTTF TFTTFT TFFTFF FTTFTT FTFFTF FFTFFT FFFFFF

(P & (~Q & R)) PQR(P&(~Q&R)) TTTTFTT TTFTFTF TFTTTFT TFFTTFF FTTFFTT FTFFFTF FFTFTFT FFFFTFF

(P & (~Q & R)) PQR(P&(~Q&R)) TTTTFTFT TTFTFTFF TFTTTFTT TFFTTFFF FTTFFTFT FTFFFTFF FFTFTFTT FFFFTFFF

(P & (~Q & R)) PQR(P&(~Q&R)) TTTTFFTFT TTFTFFTFF TFTTTTFTT TFFTFTFFF FTTFFFTFT FTFFFFTFF FFTFFTFTT FFFFFTFFF

(P & (~Q & R)) PQR(P&(~Q&R)) TTTTFFTFT TTFTFFTFF TFTTTTFTT TFFTFTFFF FTTFFFTFT FTFFFFTFF FFTFFTFTT FFFFFTFFF

TRUTH-TABLES AND VALIDITY

The Truth-Table Test for Validity We know that an argument is deductively valid when we know that if it is true, then its conclusion must be true. We can use truth-tables to show that certain arguments are valid.

The Test Suppose we want to show that the following argument is valid: (P → Q) ~Q Therefore, ~P We begin by writing down all the possible truth- values for the sentence letters in the argument.

Write Down All the Possibilities PQ TT TF FT FF

Write Truth-Table for Premises PQ(P → Q)~Q TTTF TFFT FTTF FFTT

Write Truth-Table for Conclusion PQ(P → Q)~Q~P TTTFF TFFTF FTTFT FFTTT

Look at Lines Where Premises are True PQ(P → Q)~Q~P TTT*FF TF TF FTT T FFTTT