the material conditional function $a \to b$ returns a truth value of 1 whenever the antecedent is false. meanwhile the logical consequence function $a \implies b$ returns a truth value of 0 whenever the antecedent is false. neither of these are desirable.
consider the statement "if 2 > 5, then the eiffel tower is in bolivia". in classical logic by the material conditional this statement is vacuously true; anything that follows from from a falsehood, semantic or syntactic, is. this is based on the principle of explosion($\forall p((a \wedge (\neg a)) \to p)$). but this is clearly undesirable. we know the Eiffel tower is not in bolivia and setting another non-true thing to be true that is unrelated to the eiffel tower ought not make it true.
consider the statement "if 2 > 5 and 5 > 3, then 2 > 3". in classical logic by logical consequence this statement is false; any statement with a false antecedent is false. here we have a conjunction between the terms "2 > 5" and "5 > 3", and a conjunction returns a truth value of 1 just in case both terms are true. only the second term in this conjunction is true so the conjunction as a whole is false, meaning the statement as a whole is false under the logical consequence. but this too is undesirable. of course 2 is not greater than 5 but if it were it would also have to be greater than 3 because 5 is greater than 3.
now both of these functions have their merits. if we were to apply the first function to the second statement and vice versa we would get the kind of answers we want. but we have the problem of determining first without intuition which function to apply and second of having multiple conditionals running around in the same language. the second problem is more of an aesthetic concern really but the more powerful a language is, ie the more assumptions it makes and abstracts over, much like in programming, the less pure and useful it is for fine-grained operations. you'd never write a operating system in the wolfram language for instance. no, we only want multiple semi-redundant symbols if they are just shorthand for another more primitive symbol.
(of course both functions already mentioned are shorthand for more primitive symbols. detailed analysis to come.)
instead we need a new function that applies in every case and gives a truth value depending only on the relation of the terms rather than the truth value of the terms. what this function is i do not know. but it seems imperative if we are to have a rigorous foundation of human knowledge. perhaps i will spend my life working on it.
- (no subject)