When someone plays a game, they are minded to try, at least, to score points in the game (even if there is no final score but one can simply try for a higher score each time one plays), and often enough to win. But then would one be motivated to play a game where it could be indeterminate if one had scored/won, or even whether one had made a certain kind of move at all?

Or, specifically, imagine a "proof game," and ask: do I have a proof, or do I not have a proof, of X (where X is whatever one is looking to prove), as of some move? If a logician rejects the LEM in favor of provability, yet what about exclusive disjunctions over claims to have a proof in the first place? But if there are "indeterminately existent" proofs, then there would be indeterminately existent objects of proof, and then we would still seem to have a sort of "fuzzy access" to recognition-transcendent information. It does not seem like much has been gained, and it seems also as if much has been lost, for having to think this (would an intuitionist, on thematic grounds, be more comfortable with a mathematical ontology of indeterminate proofs that indeterminately support the existence of indeterminate transcendent objects, or with the no-nonsense LEM, which at least can be tailored to help in avoiding the claim to mathematical omniscience?).

More acutely: if we can will that we disjoin one assertion A exclusively from all assertions that satisfy not(A), have we not willed the LEM? Or, that is, is the LEM not "true" just because (but only when) the form of our will obeys it? Between, "Do X," and, "Not(Do X)," there is no LEM-free space; we cannot resolve moral dilemmas by discharging both obligations (supposedly). So again, game-theoretically, it is as if the LEM is "true" or else the game of logic is not a game of scoring points/winning. (But then again, a logical pluralist can be styled as someone who is not academically competitive about the system of logic they tend to use in their own work. And the analogy with arbitrarily high scoring might not be applicable: a pluralist working with a single logic can still "score points" by showing that, in the given system, a given proof is or is not valid; but for the pluralist, it is often a cakewalk to alter the structural premises of the system so as to make a proof valid; so then the question of validity, for the pluralist, is not as entrancing, neither then is scoring proof-points; it might be interesting but is not so accounted as revelatory of ultimate reality.)

  • From a founder of GTS:"GTS leads inevitably to the development of a logic in which moves connected with nested quantifiers can be informationally independent of each other. This logic is known as independence-friendly (IF) logic... In IF first-order logic, the law of excluded middle does not hold... By adding a sentence-initial contradictory negation, we obtain a stronger extended IF first-order logic." As usual, "consonance" depends on whether one wishes to minimize or to maximize, not how
    – Conifold
    Feb 3 at 10:55


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