Reply #156: The conventional logic collapses a number of distinct ideas

[struggle4progress]

in order to simplify statements. In this sense, it represents an ideal state of affairs, where everything really is either true or false: it reflects, somehow, not what you actually know, but what you might hope to know under the best conditions. If one takes an argument, allegedly about concrete operations, which uses conventional logic, then with some regularity you will find that you are being asked to believe something that has no operational meaning: it is merely some idealization of the state of affairs; in mathematics, for example, you will find yourself asked to believe things that you cannot verify by computation.

By insisting that the logical connectives reflect concrete operations, this unsatisfactory state of affairs is eliminated: an argument, alleged about concrete operations, leads to a conclusion that can be understood in terms of concrete operations. The reason most people dislike it is simply that the cost of this realism is fairly substantial: many complicated arguments, cast in the standard logic, become much more complicated statements (typically involving plenty of negations and double negations) in concrete operational terms.

The explanation of "P => Q" as "~P or Q" is very ancient: typically this notion of implication is called "material implication" and it goes back to the ancient Greeks. Whether material implication is a satisfactory explanation of implication has been a subject of continuing controversy at times: IIRC, there was such a debate in the late Victorian era, for example. In fact,
"(P => Q) <=> (~P or Q)" seems equivalent to the law of the excluded middle for P (if you believe it for all Q):

if you believe material implication, then since you certainly believe "P => P" you will believe "~P or P"; if you believe "~P or Q" and assume P, then you certainly believe Q, so believe "(~P or Q) => (P => Q)"; and if you believe "~P or P" and believe "P => Q" you certainly believe "~P or Q"

That is, if one wants a concrete operational logic, "(~P or Q) => (P => Q)" is perfectly acceptable but the Chrysippian equivalence "(~P or Q) <=> (P => Q)" is a much stronger statement

I think anyone who wants to claim to have purged his/her own thinking of mystical metaphysical speculation ought to use this logic, to be consistent with the claim, but in fact it's a gigantic nuisance, so almost nobody will. On the other hand, to write effective computer code, one must in some sense think this way. Whether one really ought to purge all mystical metaphysics from one's thinking is, of course, an entirely different question.