T1.3 p4

T1.3: 4 of 8

Mostly we learn logic so we can better reason about the world. But sometimes we reason about logic too.

Let's begin with a very easy example. We can show — just by going through the definition of 'sound' — that

(*) Every sound argument has no false premises.

How do we prove this? It may seem too obvious for words (once you understand the definition). But bear with this thinking...

(*) is about ALL sound arguments. So, first we consider any sound argument; suppose A is any such argument.

By definition of sound, A must be be valid and have all true premises.

Finally, because A's premises are true, it cannot have a false premise.

Q.E.D.^*

That's it: we just suppose that we have a sound argument, call it "A", and chase through the definition of "sound".

Let's look at a demonstration to underscore the step-by-step proof process.

We will show that

(**) If an argument is valid and has no false premises, then it is sound.

sign to pause the demonstration.

We need to think about any argument A that

(1. and 2. are given in the "If" clause of the statement (**) we would like to prove.)

So, suppose that A is any valid argument with no false premises.

Then because A has no false premises, it follows that all its premises are true.

Finally, then, A is valid and has only true premises. So, A is "sound" by our definition of that term.

Q.E.D. (because we've gotten to the "then" clause.)