Logic II: introduction to non-classical logics syllabus (original) (raw)

A fundamental non-classical logic

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; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations.

A dialectic contra-classical logic

Logica Universalis, 2023

The paper presents a contra-classical dialectic logic, inspired and motivated by Hegels dialectics. Its axiom schemes are (H1) ∶ ⊢ϕ→¬ϕ (H2) ∶ ¬⊢ϕ→ϕ (H3) ∶ ⊢(ϕ→ψ)→(ϕ→¬ψ) (H4) ∶ ⊢(ϕ→¬ψ)→(ϕ→ψ) (0.1) Thus, in a sense, this dialectic logic is a kind of "mirror image" of connexive logic. The informal interpretation of '→' emerging from the above four axiom schemes is not of a conditional (or implication); rather, it is the relation of determination in the presence of truth-value gaps: ϕ→ψ is read as ϕ determines ψ, namely, necessarily, if ϕ is true, then ψ is either true or false, not gappy. As far as I know, such a connective has not been considered before in the literature.

Controversies about the Introduction of Non-Classical Logics

BRAIN. Broad Research in Artificial Intelligence and Neuroscience , 2014

Logic is a set of well-formed formulae, along with an inference relation. But the Classical Logic is bivalent; for this reason, very limited to solve problems with uncertainty on the data. It is well-known that Artificial Intelligence requires Logic. Because its Classical version shows too many insufficiencies, it is very necessary to introduce more sophisticated tools, as may be Non-Classical Logics; amongst them, Fuzzy Logic, Modal Logic, Non-Monotonic Logic, Para-consistent Logic, and so on. All them in the same line: against the dogmatism and the dualistic vision of the world: absolutely true vs. absolutely false, black vs. white, good or bad by nature, Yes vs. No, 0 vs. 1, Full vs. Empty, etc. We attempt to analyze here some of these very interesting Classical and modern Non-Classical Logics.