Y Safle

When Mr Spock says something is logical in Star Trek, what he often seems to mean is that something is common sense in som ultilitarian way of thinking. Thus “The needs of the many outweigh the needs of the few” is a clear utilitarian statement.

What he does not do is couch everything he says in terms of formal logic. And there is one very good reason why that may be a good thing! There is a principle in logic known as “ex falso sequitur quodlibet” - from a contradiction, everything follows. What this principle tells us is that if there is a single inconsistency of the form “P is both true and false” (a negation inconsistency), then every other proposition can be validly implied. If such a contradiction is allowed, then essentially everything is true.

Here is the proof:

1. P and P’ (that is, P is both true and false)

2. P (therefore P is true)

3. P or Q (Either P is true or Q is true)

4. P’ (P is false from 1. above)

5. Q (from 3. If P is false, Q is true).

But that is nonsense, you say - because perhaps Q has nothing to do with P. Hold that thought, because the other problem is what propositions are actually both true and false?

Well let us suppose that I am standing in the doorway of a room. One foot is in the room, and one foot is out. Am I in the room or not? You could argue that it is both true and false that I am in the room. Let us apply the proposition to the proof above:

1. I am in the room AND I am not in the room

2. I am in the room

3. It is therefore true that either I am in the room or black is white.

4. But I am not in the room

5. Therefore, from 3 and 4 I can see that black is white.

(And, as Douglas Adams pointed out, I shall now go and get myself killed on a zebra crossing).

But the point here is not that we can really prove anything, but that we don’t instinctively think logically. We allow that some things can be kind of true and kind of false, and we don’t actuall accept step 3 in the above argument. If a proposition has nothing to do with another, then why should its negation inconsistency imply the truth (and simultanously the falsehood) of the other proposition?

Human brains are wired up to be “paraconsistent”. Paraconsistency is a type of logic that ignores or removes the ex falso proof, often by insisting on the relevance of conjoined propositions. Whether we strictly need paraconsistent logic is a whole larger debate, but we need to bear in mind that our own thinking is more relevance based than strictly logical. And that is, perhaps, a good thing.