### Logic And Conditionals

Posted:

**Tue Jun 10, 2008 4:35 pm UTC**I did a lot of thinking about logic and set theory yesterday, in which I stumbled upon the realization of the parallels between the two. My question this time! Suppose we are trying to prove the statement "if p, then q". Most books I've read claim that that statement is true for every case except "p and not q", which I have accepted for my work. However, I still wonder. What would happen if we allowed statements to have a value that is neither true nor false, but "undetermined"? If I have a single sample of p and q, I cannot say that "if p then q" is really true, but I can't quite say it's false. My only knowledge from that single sample lets me claim that "for some p, q". So rather than making the claim "if p then q" is true, I call it "undetermined".

The only way to actually prove an if-then statement true would be to: 1) check every single case of p and see whether or not there is a q, which for most situations is impossible (i.e. "if an integer is even, then it is the sum of two primes) 2) take a set of axioms that make assumed statements about all elements of a given set, and use implications to make new statements about that set or it's subsets

The only way to prove one false: 1) find a single contradictory case 2) use the set of axioms to prove either "for some p, not q"

All other cases leave the statement "undetermined". While I haven't the faintest idea of how (I've looked at the actual paper; I don't think I understood the first sentence) of how Godel proved what he did about algebraic systems or really what it says, *exactly*, but doesn't he basically prove that for any algebraic system, these "undetermined" statements must exist?

I suppose my next step would be to construct some truth tables for other statements, but including the idea of the "undetermined" status of a variable. For instance, if p is undetermined and q is true, then "p and q" would be undetermined. Anyway, would doing such an investigation be worth my time? These fora seem to have a handful of people who have a good deal more experience with math in general, so if someone else has already done this investigation and nothing particularly interesting popped out (as I figured will happen) I would like to know.

The only way to actually prove an if-then statement true would be to: 1) check every single case of p and see whether or not there is a q, which for most situations is impossible (i.e. "if an integer is even, then it is the sum of two primes) 2) take a set of axioms that make assumed statements about all elements of a given set, and use implications to make new statements about that set or it's subsets

The only way to prove one false: 1) find a single contradictory case 2) use the set of axioms to prove either "for some p, not q"

All other cases leave the statement "undetermined". While I haven't the faintest idea of how (I've looked at the actual paper; I don't think I understood the first sentence) of how Godel proved what he did about algebraic systems or really what it says, *exactly*, but doesn't he basically prove that for any algebraic system, these "undetermined" statements must exist?

I suppose my next step would be to construct some truth tables for other statements, but including the idea of the "undetermined" status of a variable. For instance, if p is undetermined and q is true, then "p and q" would be undetermined. Anyway, would doing such an investigation be worth my time? These fora seem to have a handful of people who have a good deal more experience with math in general, so if someone else has already done this investigation and nothing particularly interesting popped out (as I figured will happen) I would like to know.