Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 3.7.2 Getting at Truth – Sound and Valid Arguments

So now we know that we’ve got a set of inference rules that, given some set of premises P , let us prove:

• all, and

• only

the statements that follow from P .

We’ll say that an argument (proof) is valid provided that every one of its steps can be justified by a sound inference rule. Sometimes, when presented with such an argument, we’ll say, “Its reasoning is valid.”

But does this mean that the conclusion of a valid argument is necessarily true ? Unfortunately, no.

Give names to the following statements:

L: Lucy is a unicorn.

H: Lucy has a large horn on her head.

Suppose that we have the following premises:

[1] L Lucy is a unicorn

[2] L  H If Lucy is a unicorn then she has a large horn on her head.

Then, using Modus Ponens, we have:

[3] H Lucy has a large horn on her head.

But Lucy has no horn. The problem is that she’s a cat, not a unicorn. Our reasoning is valid. But we’ve proven something that isn’t true.

We’ll say that an argument (proof) is sound provided that it is valid and that its premises are true (in whatever world we are reasoning about). The Lucy argument that we just gave is valid but not sound.

To get at truth, we must construct sound arguments. They must start from premises that are true and their reasoning must be valid.

By the way, in case you’re a bit confused about terminology here, you’re not alone. There is an unfortunate (but so conventional that we cannot ignore it) use of the word “sound” to mean one thing when applied to inference rules and another thing when applied to entire arguments:

• An inference rule is sound just in case it preserves truth. In other words, it can derive only conclusions that follow from the premises.

• An argument (proof) is sound just in case truth is both introduced by the premises and preserved by the argument.

Sorry about that. But don’t worry. The key thing is that good arguments possess both properties. We are focusing on developing sound inference rules. When people go to apply those rules to help them reason about real problems, it’s up to them to choose premises that make sense in their problem domains.