1 4 3C o n d i t i o n a l s“E & B.” Their truth-functional equivalence is too obvious to need proof.
Similar oddities arise for all discounting terms, such as “although,”
“whereas,” and “however.”
It might seem that if formal analysis cannot distinguish an “and” from
a “but,” then it can hardly be of any use at all. This is not true. A formal
analysis of an argument will tell us just one thing: whether the argument
is valid or not. If we expect the analysis to tell us more than this, we will
be sorely disappointed. It is important to remember two things: (1) We ex-
pect deductive arguments to be valid, and (2) usually we expect much more
than this from an argument. To elaborate on the second point, we usually
expect an argument to be sound as well as valid; we expect the premises to
be true. Beyond this, we expect the argument to be informative, intelligible,
convincing, and so forth. Validity, then, is an important aspect of an argu-
ment, and formal analysis helps us evaluate it. But validity is not the only
aspect of an argument that concerns us. In many contexts, it is not even our
chief concern.
We can now look at our analysis of conditionals, for here we find some
striking differences between the logician’s analysis and everyday use. The
following argument forms are both valid:- p 2. ~p
∴ q ⊃ p ∴ p ⊃ q
Check the validity of the argument forms above using truth tables.Exercise XXVThough valid, both argument forms seem odd—so odd that they have actu-
ally been called paradoxical. The first argument form seems to say this: If a
proposition is true, then it is implied by any proposition whatsoever. Here is
an example of an argument that satisfies this argument form and is therefore
valid:
Lincoln was president.
∴ If the moon is made of cheese, Lincoln was president.
This is a peculiar argument to call valid. First, we want to know what the
moon has to do with Lincoln’s having been president. Beyond this, how can
his having been president depend on a blatant falsehood? We can give these
questions even more force by noticing that even the following argument is
valid:
Lincoln was president.
∴ If Lincoln was not president, then Lincoln was president.97364_ch06_ptg01_111-150.indd 143 15/11/13 10:15 AM
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