A Fundamental Non-Classical Logic

Abstract

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation. Finally, we discuss ways of extending these representations to lattices with a conditional or implication operation.

Interpolation and Quantifiers in Ortholattices

“…They also show that a predicate logic extension to orthologic admits interpolants, although the proof of both theorems are non-constructive and contain no discussion of algorithms or space and time complexity. Recently, orthologic was used in practice in a proof assistant [15], for modelling of epistemic logic [20,21] and for normalizing formulas in software verification [14].…”

Carnap’s Problem for Intuitionistic Propositional Logic

“…There does not have to be a smallest element. 20 Set-based semantics S for this language are defined as before, and standard interpretations are again unique.…”

“…However, as in the case of ∨ , there seems to be no way to prove categoricity for ∨ in these logics, at least not along the lines we have tried so far. (20) Open problem Are the logics ∧∨ and (more interestingly) ¬∧∨ categorical for set-based semantics?…”

“…One can weaken the logics Φ defined in Section 4 by constraining the inference rules in various ways. A case in point is [20], which studies a weaker logic in L ¬∧∨ , called F , with restrictions on the ∨E and ¬I rules. Among other things, distributivity no longer holds.…”