Again, we have just shown that two propositions about logic are true:
(*) Every sound argument has no false premises.
(**) If an argument is valid and has no false premises, then it is sound.
These two ideas may seem obvious to you and not in need of proof – if you understand the definitions. If so, good. But the two little proofs we've given illustrate a standard method we'll use quite often.
To understand this method, first notice what propositions (*) and (**) have in common. (*) is about any sound argument and (**) is about any sound argument having no false premises. We might say that they have "general" or "arbitrary" subjects.
Furthermore, each proposition goes on to make a "claim" about its subject. So, for example, (*) has arbitrary subject of sound arguments and claims that each has no false premise.
So, our standard method of proof for such arbitrary subject statements is this:
- Start with the supposition that A is an arbitrary example of the group of things described by the subject.
- Chase through the relevant definitions pertaining to that subject and so to A.
- Show that the claim must hold of A.
Notice that this is just what we did on the last page. To repeat from last page, (*) is the claim that any sound argument cannot have a false premise, thus:
(*) is about ALL sound arguments. So, we first consider an
arbitrary sound argument; suppose A
is any such argument.
So, by definition of sound, A
must be valid and must have all true premises.
Finally, because A's premises are true,
it cannot have a false premise.
Q.E.D.
*
In fact we won't always use the symbol "A" as our arbitrary member of the general subject group. (You may pick any symbol you want.) And we won't always be able to finish in three lines — sometimes chasing through the definitions can take awhile! — but we can always start and end as this little argument did.