Back to: A special way of thinking about infinity in mathematics

No third truth value in constructive logic

The proof is short enough to present here if you are curious to see it. You may find it a bit mind boggling at first, but there are no hidden details; this is the complete proof, as written out in English rather than in the more concise but esoteric language of mathematical symbols.

First we need some background in the way constructive logic works.

In constructive logic, the only way to show a statement to be true is to prove it.

The only way to show that a statement is false is to suppose it is true, work out the consequences of that, and derive a contradiction.

It is possible to go on, and set out a complete set of logical inference rules. In fact we only need two here, but they are common sense, and they'd probably obscure the main point. I'll give them at the end.

Now we can prove the result.

Let A be some statement we'd like to assert to have an intermediate truth value. Maybe something like, that gold is yellow in colour. (Substitute your own favourite here).

A: Gold is yellow.

Let B be the statement that A is false. So, putting it as closely as possible in ordinary language, B reads

B: The statement that gold is yellow leads to a contradiction.

Let C be the statement that A has an intermediate truth value

C: The statement that gold is yellow can't be proved, and there is no way to derive a contradiction from it.

Will we be able to hold this position C, or will we be forced to back down? If we have to back down, then A has no third truth value. So we expect we will have to back down, but how is it going to happen? We need to assert C and see where it leads us.

So, let's throw caution to the winds, and assert:

C!

The first part of C asserts that A leads to a contradiction, because that is what a constructivist means if he/she says a statement can't be proved.

So that means that we are committed to asserting B.

The second part of C asserts that there is no way to derive a contradiction from A. So if we assert C, we are also commited to asserting that B is false, in other words, that B leads to a contradiction.

Putting those together, we find that, as a result of asserting C, we are committed to asserting B. We are also committed to saying that B leads to a contradiction.

So we are committed to asserting a contradiction.

The only way out is to back down, and stop trying to assert C.

There was nothing special about A. Substitute any favourite candidate for a statement with an intermediate truth value, and it will fall foul of the identical sequence of logical steps.

We did everything perfectly as regards logical inference, so the only thing that can give is to stop trying to assert C.

So constructive logic has no third truth value.

Q.E.D. - if you have managed to follow the argument. You may need to give it a fair bit of time to sink in.

There is nothing wrong with a third truth value per se

In fact, there is a whole mathematical field that studies intermediate truth values, called Fuzzy Logic.

You could even have a constructive study of fuzzy logic, so long as the fuzzy logic was what you were studying, and you keep the fuzziness there, and keep to clear crisp distinctions of truth and falsity in your description of how the logic works.

What we have proved

What we have seen here is that intermediate truth values can't be used in constructive logic, with its interpretation of true as only assertable if you can prove something, and false, only if you can derive a contradiction from it.

The logical rules used to prove this result

The first rule is:

If one asserts both S and T, then one is committed to asserting each one individually.

The second one is:

If one asserts that S is true, and asserts that S leads to T then one is committed to asserting T.

(the way we used it, S is the statement B, and T is the contradiction.).

I said they were common sense, but one should spell them out to do it properly, so there they are.

Back to: A special way of thinking about infinity in mathematics