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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.

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Evaluations1
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
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
Evaluation: Stage 1

Write down sentence letters.


Evaluation stage 11
Evaluation: Stage 1

Insert truth-values from evaluation.


Evaluation stage 2
Evaluation: Stage 2

Copy down the formula to evaluate.


Evaluation stage 3
Evaluation: Stage 3

Copy the truth-values of each variable.


Evaluation stage 31
Evaluation: Stage 3

Copy the truth-values of each variable.


Evaluation stage 32
Evaluation: Stage 3

Copy the truth-values of each variable.


Evaluation stage 4
Evaluation: Stage 4

Find a connective to evaluate.


Evaluation stage 41
Evaluation: Stage 4

Need these truth values.


Evaluation stage 42
Evaluation: Stage 4

Need these truth values.


Evaluation stage 43
Evaluation: Stage 4

Need this truth value.


Evaluation stage 44
Evaluation: Stage 4

Need this truth value.


Evaluation stage 45
Evaluation: Stage 4

Need these truth values.


Evaluation stage 46
Evaluation: Stage 4

Need these truth values.


Evaluation stage 47
Evaluation: Stage 4

Need these truth values.


Evaluation stage 48
Evaluation: Stage 4

Need these truth values.


In class exercises
In-Class Exercises

Evaluation: P = F, Q = F, R = T

  • ~(~P & ~Q)

  • ~(P → ~Q)

  • ((P & ~Q) & R)













P q p v q
(~(~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
In-Class Exercise

Write a full truth-table for:

~(P → ~Q)







P q p q
(~(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 exercise1
In-Class Exercise

Write a full truth-table for:

(P & (~Q & R))









The truth table test for 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
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
Write Down All the Possibilities





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