The Language of Argument

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C o n d i t i o n a l s

derivation can be run in the other direction as well. But if each of these expres-

sions is derivable from the other, this suggests that they are equivalent. We use

this background argument as a justification for the following definition:

If p, then q = (by definition) not both p and not q.

We can put this into symbolic notation using “⊃” (called a horseshoe) to sym-

bolize the conditional connective:

p ⊃ q = (by definition) ~(p & ~q)

Given this definition, we can now construct the truth table for propositional

conditionals. It is simply the truth table for “~(p & ~q)”:

p q ~(p & ~q) p ⊃ q ~p ∨ q

T T T T T

T F F F F

F T T T T

F F T T T

Notice that “~(p & ~q)” is also truth-functionally equivalent to the expres-

sion “~p ∨ q.” We have cited it here because “~p ∨ q” has traditionally been

used to define “p ⊃ q.” For reasons that are now obscure, when a conditional

is defined in this truth-functional way, it is called a material conditional.

Let’s suppose, for the moment, that the notion of a material conditional

corresponds exactly to our idea of a propositional conditional. What would

follow from this? The answer is that we could treat conditionals in the same

way in which we have treated conjunction, disjunction, and negation. A

propositional conditional would be just one more kind of truth-functionally

compound proposition capable of definition by truth tables. Furthermore,

the validity of arguments that depend on this notion (together with con-

junction, disjunction, and negation) could be settled by appeal to truth table

techniques. Let us pause for a moment to examine this.

One of the most common patterns of reasoning is called modus ponens. It

looks like this:

If p, then q. p ⊃ q

p p

∴ q ∴ q

The truth table definition of a material conditional shows at once that this

pattern of argument is valid:

Premise Premise Conclusion

p q p ⊃ q q

T T T T OK

T F F F

F T T T

F F T F

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