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Conclusion

The Paradox of Knowability is a paradox if the logical constants occurring in its formalization are understood according to the realist explanation of their meaning; but in a discussion between realists and anti-realists one cannot assume that antirealists understand them in this way, for the paradox is intended to be an argument by which the former try to convince the latter to abandon their views on the meaning of the logical constants, and such an argument cannot be convincing if, in order to be formulated, it requires anti-realists to give up preventively their views. Vice versa, the paradox completely vanishes when the logical constants occurring in its formalization are understood according to the BHK explanation, since it is now necessary to distinguish two notions of truth: internal intuitionistic truth, which coincides with knowledge, and intuitive truth, essentially consisting in correspondence to external reality; in the former sense it is obvious that every truth is known, in the latter it is equally obvious—also for the anti-realist—that not every truth is known, and also that not every truth is knowable. From this point of view, the view of the paradox as an argument against anti-realism is the result of a wrong way of conceiving the rules of a rational discussion between classicist/realist and intuitionist/anti-realist.

In conclusion, the Paradox of Knowability leaves the debate between realists and anti-realists at the same point it was before its discovery. The crucial point of the debate is which notion between truth and evidence should be adopted as the key notion of the theory of meaning, or—if we accept the (in my opinion misleading) idea that meaning is to be explained in any case in terms of truth-conditions—which notion of truth, between bivalent and non-bivalent truth, the theory of meaning should be built on; in this case, the criterion for distinguishing realism from anti-realism cannot be the acceptance or refusal of the intuitive principle (K), but the acceptance or refusal of the principle of bivalence, according to Dummett's original suggestion.

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