So – Math geekery.
I was playing with booleans here. By asking a very simple set of questions I derived the XNOR function.
T = True; F = False;
Is T T? T
Is T F? F
Is F T? F
Is F F? T
Which returns the results of the XNOR (Exclusive Neither Or) also known as “Logical Biconditional”.
Further… In this case, since applying T to a value returns the value itself, T is considered the Identity Operator. Applying F to a value returns the inverted/negated value, which means you could consider F to be the Negation Operator.
One can then look at this as T and F being functions that return values based on their input. As such you can feed them functionally as entities and proceed from there into an set of algebraic axioms of some sort (which I’m sure has already been done) but I think it’s fun to try to figure this out.
If one sees this as operational mathematics, one realizes that the identity and negation operators are merely positive and negative one. In this sense, then, F != 0… If it did, F(F) = F, but we see that it returns 1 (in a multiplicative identity approach) so therefore, it’s -1, which then negates to a +1: -1(-1) = 1;
But if you attempt to do it in an additive approach, you’d have to do something differently, and I haven’t worked that out yet, since this is just lunch time meanderings.
When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a theorem and the other its reciprocal. Thus whenever a theorem and its reciprocal are true we have a biconditional. A simple theorem gives rise to an implication whose antecedent is the hypothesis and whose consequent is the thesis of the theorem.
And as always I’m probably wrong on something here, because I’m just fucking around with things, not actually doing any formal study, which means I’m relying on my intuition and vague knowledge, which, as we all know, is a dangerous thing. Leading to such things as falsities in thoughts. 😉