The Language of Argument

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B a s i c P r o p o s i t i o n a l C o n n e c t i v e s

Using these abbreviations, the argument and its counterpart argument

form look like this:

D ∨ L p ∨ q

~(D ∨ S) ~(p ∨ r)

∴ L ∴ q

The expression on the right gives the argument form of the argument presented

on the left. To test the argument for validity, we ask whether the argument form

is valid. The procedure is cumbersome, but perfectly mechanical:

Premise Premise Conclusion

p q r (p ∨ q) (p ∨ r) ~(p ∨ r) q

T T T T T F T

T T F T T F T

T F T T T F F

T F F T T F F

F T T T T F T

F T F T F T T OK

F F T F T F F

F F F F F T F

Notice that there is only one combination of truth values for which both

premises are true, and in that case the conclusion is true as well. So the

original argument is valid because it is an instance of a valid argument

form—that is, an argument form with no substitution instances for which

true premises are combined with a false conclusion.

This last truth table may need some explaining. First, why do we get eight

rows in this truth table where before we got only four? The answer to this

is that we need to test the argument form for every possible combination of

truth values for the component propositions. With two variables, there are

four possible combinations: (TT), (TF), (FT), and (FF). With three variables,

there are eight possible combinations: (TTT), (TTF), (TFT), (TFF), (FTT),

(FTF), (FFT), and (FFF). The general rule is this: If an argument form has

n variables, the truth table used in its analysis must have 2n rows. For four

variables there will be sixteen rows; for five variables, thirty-two rows; for

six variables, sixty-four rows; and so on. You can be sure that you capture

all possible combinations of truth values by using the following pattern in

constructing the columns of your truth table under each individual variable:

First column Second column Third column...

First half Ts, First quarter Ts, First eighth Ts,

second half Fs. second quarter Fs, second eighth Fs,

and so on. and so on.

A glance at the earlier examples in this chapter will show that we have been

using this pattern, and it is the standard way of listing the possibilities.

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