Converse Ackermann Property and Minimal Negation

A logic S has the Converse Ackermann Property (CAP) if non-necessitive propositions are not derivable in S from necessitive ones. We show in this paper how to introduce minimal negation in positive logics with the CAP. Relational ternary semantics are provided for all the logics considered in this paper

Extensions of the basic constructive logic for weak consistency BKc1 defined with a falsity constant

The logic BKc1 is the basic constructive logic for weak consistency (i.e., absence of the negation of a theorem) in the ternary relational semantics without a set of designated points. In this paper, a number of extensions of B Kc1 defined with a propositional falsity constant are defined. It is also proved that weak consistency is not equivalent to negation-consistency or absolute consistency (i.e., non-triviality) in any logic included in positive contractionless intermediate logic LC plus the constructive negation of BKc1 and the (constructive) contraposition axioms

Admissibility of Ackermann’s rule δ in relevant logics

It is proved that Ackermann’s rule δ is admissible in a wide spectrum of relevant logics satisfying certain syntactical properties

A simple Henkin-style completeness proof for Gödel 3-valued logic G3

A simple Henkin-style completeness proof for Gödel 3-valued propositional logic G3 is provided. The idea is to endow G3 with an under-determined semantics (u-semantics) of the type defined by Dunn. The key concept in u-semantics is that of “under-determined interpretation” (u-interpretation). It is shown that consistent prime theories built upon G3 can be understood as (canonical) u-interpretations. In order to prove this fact we follow Brady by defining G3 as an extension of Anderson and Belnap’s positive fragment of First Degree Entailment Logic

Minimal Negation in the Ternary Relational Semantics

Minimal Negation is defined within the basic positive relevance logic in the relational ternary semantics: B+. Thus, by defining a number of subminimal negations in the B+ context, principles of weak negation are shown to be isolable. Complete ternary semantics are offered for minimal negation in B+. Certain forms of reductio are conjectured to be undefinable (in ternary frames) without extending the positive logic. Complete semantics for such kinds of reductio in a properly extended positive logic are offered

Reduced Routley-Meyer semantics for the logics characterized by natural implicative expansions of Kleene's strong 3-valued matrix

15 p.The aim of this paper is to provide a reduced Routley-Meyer semantics for the logics characterized by all natural implicative expansions of Kleene’s strong 3-valued matrix (with two designated values, as well as with only one) susceptible to be interpreted in Routley-Meyer semantics.S

A Variety of DeMorgan Negations in Relevant Logics

The present paper is inspired by Sylvan and Plumwood’s logicBM defined in “Non-normal relevant logics” and by their treatmentof negation with the ∗-operator in “The semantics of first-degree en-tailment”. Given a positive logic L including Routley and Meyer’sbasic positive logic and included in either the positive fragment of Eor in that of RW, we investigate the essential De Morgan negation ex-pansions of L and determine all the deductive relations they maintainto each other. A Routley-Meyer semantics is provided for each logicdefined in the paper

A class of simpler logical matrices for the variable-sharing property

In our paper “A general characterization of the variable-sharing property by means of logical matrices”, a general class of so-called “Relevant logical matrices”, RMLs, is defined. The aim of this paper is to define a class of simpler Relevant logical matrices RMLs′serving the same purpose that RMLs, to wit: any logic verified by an RML′has the variable-sharing property and related properties predicable of the logic of entailment E and of the logic of relevance R

Converse Ackermann property and constructive negation defined with a negation connective

The Converse Ackermann Property is the unprovability of formulas of the form (A -> B) -> C when C does contain neither -> nor ¬. Intuitively, the CAP amounts to rule out the derivability of pure non-necessitive propositions from non-necessitive ones. A constructive negation of the sort historically defined by, e.g., Johansson is added to positive logics with the CAP in the spectrum delimited by Ticket Entailment and Dummett’s logic LC

Relevance logics, paradoxes of consistency and the K rule II. A non-constructive negation

The logic B+ is Routley and Meyer’s basic positive logic. We define the logics BK+ and BK'+ by adding to B+ the K rule and to BK+ the characteristic S4 axiom, respectively. These logics are endowed with a relatively strong non-constructive negation. We prove that all the logics defined lack the K axiom and the standard paradoxes of consistency