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

P Q R

Write down sentence letters.

Evaluation: Stage 1

P Q RT T F

Insert truth-values from evaluation.

Evaluation: Stage 2

P Q R ((Q & ~ P) → R)T T F

Copy down the formula to evaluate.

Evaluation: Stage 3

P Q R ((Q & ~ P) → R)T T F T

Copy the truth-values of each variable.

Evaluation: Stage 3

P Q R ((Q & ~ P) → R)T T F T T

Copy the truth-values of each variable.

Evaluation: Stage 3

P Q R ((Q & ~ P) → R)T T F T T F

Copy the truth-values of each variable.

Evaluation: Stage 4

P Q R ((Q & ~ P) → R)T T F T T F

Find a connective to evaluate.

Evaluation: Stage 4

P Q R ((Q & ~ P) → R)T T F T T F

Need these truth values.

Evaluation: Stage 4

P Q R ((Q & ~ P) → R)T T F T T F

Need these truth values.

Evaluation: Stage 4

P Q R ((Q & ~ P) → R)T T F T T F

Need this truth value.

Evaluation: Stage 4

P Q R ((Q & ~ P) → R)T T F T F T F

Need this truth value.

Evaluation: Stage 4

P Q R ((Q & ~ P) → R)T T F T F T F

Need these truth values.

Evaluation: Stage 4

P Q R ((Q & ~ P) → R)T T F T F F T F

Need these truth values.

Evaluation: Stage 4

P Q R ((Q & ~ P) → R)T T F T F F T F

Need these truth values.

Evaluation: Stage 4

P Q R ((Q & ~ P) → R)T T F T F F T T F

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…TF

Possibilities for Two Sentence Letters

φ ψ …φ…ψ…T TT FF TF F

Possibilities for Three Sentence Letters

φ ψ χ …φ…ψ…χ…T T TT T FT F TT F FF T TF T FF F TF F F

~(~P & ~Q)

P Q ~ (~ P & ~ Q)T TT FF TF F

Copy Whole Column

P Q ~ (~ P & ~ Q)T T TT F TF T FF F F

Copy Whole Column

P Q ~ (~ P & ~ Q)T T T TT F T FF T F TF F F F

Evaluate Each Row

P Q ~ (~ P & ~ Q)T T F T TT F F T FF T T F TF F T F F

Evaluate Each Row

P Q ~ (~ P & ~ Q)T T F T F TT F F T T FF T T F F TF F T F T F

~(~P & ~Q)

P Q ~ (~ P & ~ Q)T T F T F F TT F F T F T FF T T F F F TF F T F T T F

~(~P & ~Q)

P Q ~ (~ P & ~ Q)T T T F T F F TT F T F T F T FF T T T F F F TF F F T F T T F

(~(~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)

~(P → ~Q)

P Q ~ (P → ~ Q)T TT FF TF F

~(P → ~Q)

P Q ~ (P → ~ Q)T T T TT F T FF T F TF F F F

~(P → ~Q)

P Q ~ (P → ~ Q)T T T F TT F T T FF T F F TF F F T F

~(P → ~Q)

P Q ~ (P → ~ Q)T T T F F TT F T T T FF T F T F TF F F T T F

~(P → ~Q)

P Q ~ (P → ~ Q)T T T T F F TT F F T T T FF T F F T F TF F F F T T F

(~(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))

(P & (~Q & R))P Q R (P & (~ Q & R))T T TT T FT F TT F FF T TF T FF F TF F F

(P & (~Q & R))P Q R (P & (~ Q & R))T T T T T TT T F T T FT F T T F TT F F T F FF T T F T TF T F F T FF F T F F TF F F F F F

(P & (~Q & R))P Q R (P & (~ Q & R))T T T T F T TT T F T F T FT F T T T F TT F F T T F FF T T F F T TF T F F F T FF F T F T F TF F F F T F F

(P & (~Q & R))P Q R (P & (~ Q & R))T T T T F T F TT T F T F T F FT F T T T F T TT F F T T F F FF T T F F T F TF T F F F T F FF F T F T F T TF F F F T F F F

(P & (~Q & R))P Q R (P & (~ Q & R))T T T T F F T F TT T F T F F T F FT F T T T T F T TT F F T F T F F FF T T F F F T F TF T F F F F T F FF F T F F T F T TF F F F F T F F F

(P & (~Q & R))P Q R (P & (~ Q & R))T T T T F F T F TT T F T F F T F FT F T T T T F T TT F F T F T F F FF T T F F F T F TF T F F F F T F FF F T F F T F T TF F F F F T F F F

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) ~QTherefore, ~P

We begin by writing down all the possible truth-values for the sentence letters in the argument.

Write Down All the Possibilities

P QT TT FF TF F

Write Truth-Table for Premises

P Q (P → Q) ~QT T T FT F F TF T T FF F T T

Write Truth-Table for Conclusion

P Q (P → Q) ~Q ~PT T T F FT F F T FF T T F TF F T T T

Look at Lines Where Premises are True

P Q (P → Q) ~Q ~PT T T *F FT F *F T FF T T *F TF F T T T