The Language of Argument

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C H A P T E R 6 ■ P r o p o s i t i o n a l L o g i c

TESTING FOR VALIDITY

What is the point of all this? In everyday life, we rarely run into an expression

as complicated as the one in our example at the end of the previous section.

Our purpose here is to sharpen our sensitivity to how truth- functional

connectives work and then to express our insights in clear ways. This is

important because the validity of many arguments depends on the logical

features of these truth-functional connectives. We can now turn directly to

this subject.

Earlier we saw that every argument with the form “p & q; ∴ p” will be

valid. This is obvious in itself, but we saw that this claim could be justified

by an appeal to truth tables. A truth table analysis shows us that an argu-

ment with this form can never have an instance in which the premise is true

and the conclusion is false. We can now apply this same technique to argu-

ments that are more complex. In the beginning, we will examine arguments

that are still easy to follow without the use of technical help. In the end,

we will consider some arguments that most people cannot follow without

guidance.

Consider the following argument:

Valerie is either a doctor or a lawyer.

Valerie is neither a doctor nor a stockbroker.

∴ Valerie is a lawyer.

We can use the following abbreviations:

D = Valerie is a doctor.

L = Valerie is a lawyer.

S = Valerie is a stockbroker.

Given that “A,” “B,” and “C” are true propositions and “X,” “Y,” and “Z” are

false propositions, determine the truth values of the following compound

propositions:

1. ~X ∨ Y 9. ~(A ∨ (Z ∨ X))


2. ~(X ∨ Y) 10. ~(A ∨ ~(Z ∨ X))


3. ~(Z ∨ Z) 11. ~A ∨ ~(Z ∨ X)


4. ~(Z ∨~Z) 12. ~Z ∨ (Z & A)


5. ~ ~(A ∨ B) 13. ~(Z ∨ (Z & A))


6. (A ∨ Z) & B 14. ~((Z ∨ Z) & A)


7. (A ∨ X) & (B ∨ Z) 15. A ∨ ((~B & C) ∨~(~B ∨ ~(Z ∨ B)))


8. (A & Z) ∨ (B & Z) 16. A & ((~B & C) n ~(~B ∨ ~(Z ∨ B)))

Exercise XII

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