Empirical Negation

Abstract

An extension of intuitionism to empirical discourse, a project most seriously taken up by Dummett and Tennant, requires an empirical negation whose strength lies somewhere between classical negation ('It is unwarranted that. . . ') and intuitionistic negation ('It is refutable that. . . '). I put forward one plausible candidate that compares favorably to some others that have been propounded in the literature. A tableau calculus is presented and shown to be strongly complete.

Empirical Negation, Co-Negation and the Contraposition Rule II: Proof-Theoretical Investigations

“…Changing perspective, from an intuitionistic viewpoint there is a certain advantage in considering Beth semantics. There is a relatively simple proof of intuitionistic completeness (proving completeness with only intuitionistically accepted principles) for intuitionistic logic [3,11]. The intuitionistic completeness proof for Kripke semantics [12] gives a more refined result, but is comparatively more involved.…”

“…In the first paper, we semantically investigated how some logics with nonstandard negation (IPC ∼ [1,2], TCC ω [4], daC [9] and CC ω [10]) are related to each other. In particular, we noted how the difference between IPC ∼ and TCC ω can be understood as the difference between Kripke and Beth semantics.…”

Empirical Negation, Co-negation and Contraposition Rule I: Semantical Investigations

“…We shall observe that this difference in rules can be eliminated, by replacing [RP] with [RC] and an additional axiom. This will give a completeness proof of daC with respect to the semantics of CC ω , and thus the semantics of Došen [5]. It will also provide a more unified viewpoint of the logics related to CC ω as defined by extra axioms with no change in rules.…”

Structuring Co-constructive Logic for Proofs and Refutations

“…8 In logics without this restriction, such as classical logic, this fails by contraction, so there may exist a disjunction α ∨ β without any means of deciding which formula is proved, which is problematic if we pursue a semantics of proof. …”

“…7 Furthermore, as Shramko et al [40] point out, if we define falsity in terms of negation, then we are led to a reliance, not only on a syntactic feature (negation), but also on truth, and, as Dummett [10] is aware, this, leads to bivalence under commonly-held assumptions regarding the nature of proofs [see author reference omitted for discussion]. 8 To get ahead of ourselves, this property is mirrored in co-intuitionistic logic by the "conjunction property", where α ∧ β iff α or β. 9 Lafont [14, Appendix B.1] argues that 'classical logic is inconsistent, not from a logical viewpoint (false is not provable), but from an algorithmic one', since proofs can not be considered algorithmically, and so 'classical logic has no denotational semantics, except the trivial one which identifies all the proofs of the same type'.…”