Dual Intuitionistic Logic and a Variety of Negations: The Logic of Scientific Research

Abstract

We consider a logic which is semantically dual (in some precise sense of the term) to intuitionistic. This logic can be labeled as "falsification logic": it embodies the Popperian methodology of scientific discovery. Whereas intuitionistic logic deals with constructive truth and non-constructive falsity, and Nelson's logic takes both truth and falsity as constructive notions, in the falsification logic truth is essentially non-constructive as opposed to falsity that is conceived constructively. We also briefly clarify the relationships of our falsification logic to some other logical systems.

Logics of Formal Inconsistency

“…2 Another recent proposal by Y. Shramko also defends the paraconsistent character of falsificationism (cf. [Shramko, 2005]). …”

“…Similarly, paracomplete logics could have a contribution to make for the study of verificationism in science. The logical approach to such questions has recently been vindicated by studies such as [Shramko, 2005].…”

Structuring Co-constructive Logic for Proofs and Refutations

“…An immediate difficulty for constructing proof-theoretic semantics is due to an inability of intuitionistic logic to adequately deal with refutation and falsity as pointe out in [40,39]. The primacy of proof (in contrast to refutation) is problematic for the development of a semantics of proofs, where we do not want to rely upon a counter-model as stand-in for refutation.…”

“…10 This builds upon analysis of the relationship between intuitionistic and co-intuitionistic logic as discussed in [4,5,7,12,16,35,37,39,52]; and systems of proof and refutation discussed in [41,42,43,44,45,53,54]. Unlike the construction suggested here, the typical relationship between intuitionistic and co-intuitionistic logic in the literature is not without significant 6 Note that this feature is not forced on intuitionistic sequents, though restrictions on rules or dependency relations are required to ensure that only intuitionistic derivations are valid in sequents with multiple conclusions, see [9] for discussion.…”

On Displaying Negative Modalities

“…As far as I can tell, term kite of negations was coined by Dunn in [12]. Dunn's kite of negations was also generalized by what Yaroslav Shramko calls a uniform kite of negations [31]. Now, bi-intuitionistic temporalized modal logic might sound like it is trying a bit too much to get away from classical logic.…”