Wednesday, October 20, 2004

The function of logic

What is logic? Until last night I believed that the sole purpose of logic was to guarantee consistency. Suppose we have a set of propositions (statements that are either true or false):

{Pi, i = 1..N}

We then assign each proposition a truth value:

{Pi = Ti, i = 1..N}

where Ti is true or false. Logic is the tool we use to determine whether or not the assigned truth values lead to a contradiction.

For example, if P1 = ~ P2 then logic tells us that T1 = ~ T2. If T1 = T2 then there is a contradiction.

If we forego logic, we must accept that for any proposition, P, we regard as true, its negative, ~P, might also be true. Clearly, in this case, knowledge would be impossible.

However, last night I wondered what would happen if we weakened logic so that the only logical operation was the NOT (~) operator. This allows us to meaningfully accept propositions as true without the fear that the negated proposition would also be true. But does it make knowledge possible?

My conclusion is that it allows facts, but does not permit comprehension or perception of structure. This is because structure is a relationship between facts. I may declare that "A > B", and its negation is "A <= B", but this is not the same thing as "A < B".

Another example: visually, a line is a sequence of points. In some sense we are saying that in a 2-D coordinate system (point P1 has no Y component) AND (point P2 has no Y component) AND (point P3 has no Y component) AND... If logical AND does not apply, we cannot even perceive structure.

This is not surprising since, as Gottlob Frege and Bertrand Russell proved, all mathematics can be derived from logic. Set theory (classification) and sorting are all consequences of logic, and without these we can have no perception.

If one gives up on logic, one sacrifices structure and perception.


Robin Zebrowski said...

Running late so I can't post much, but you might want to look into Josiah Royce's discussion of logic (it *might* be CS Peirce who said this, but offhand I *think* it was Royce.) Basically, he demonstrated that if you start from negation, and nothing else, you can actually derive a lot. Negation is a self-reinstating principle, meaning that to deny negation actually negates... Wait, I think it really was Peirce. (My brain is full. I'm sorry - this just strikes me as something you might want to look into because it's pretty cool.)

Doctor Logic said...

Thanks, Robin.

I'll check out those references.