The aim of the talk is to present some results about the characterization of the expressiveness of three-valued propositional languages that add operators to the weak Kleene language. This is part of a project which tries to characterize all the expansions of the weak Kleene language. In this presentation, we will concentrate on those languages which contain all the constant operators and are not included in the strong Kleene language. The characterizations use properties that can easily be checked using the truth tables. As an example of the theorems, let us consider the weak Kleene language and add an operator \(\circ\) such that \(\circ 0= \circ 1 = 1\) and \(\circ \frac{1}{2} = 0\) and a strong negation \(\sim\) such that \(\sim 0 = 1\), \(\sim 1 = \sim \frac{1}{2} = 0\), plus the constant operators. In a paracomplete interpretation, the operator \(\circ\) expresses determinedness, in a paraconsistent reading, it expresses consistency. Call that language \(K_{w}^{\circ\sim}\). How do we know whether a three-valued operator with truth table \(f(x_{1},\ldots,x_{n})\) can be defined in \(K_{w}^{\circ\sim}\)? The following proposition gives one characterization, but we need to introduce some notation first. \(c_{\frac{1}{2}}\) is the constant function with value \(\frac{1}{2}\). \(d(f)\) is the set of funcions we obtain from \(f\) when some of its variables are replaced by constants. \(T_{01}\) is the set of functions that preserve the set \(\left\{ 0,1\right\}\). Then \(f\in K_{w}^{\circ\sim}\) if, and only if, \(f\) satisfies the following conditions:

(1) for all \(g\in d(f)\), if \(g\neq c_{\frac{1}{2}}\), then \(g\in T_{01}\).

(2) if \(f\neq c_{\frac{1}{2}}\) and \(f(a_{1},\ldots ,a_{n})=\frac{1}{2}\) for some \(a_{1},\ldots ,a_{n}\in \{0,\frac{1}{2},1\}\), then there is \(a_{i}\) such that \(a_{i}=\frac{1}{2}\) and \(f(x_{1},\ldots ,x_{i-1},\frac{1}{2},x_{i+1},\ldots ,x_{n})=c_{\frac{1}{2}}\).

We provide natural deduction systems for the minimal theory of definite descriptions grounded in Lambert’s axiom within the context of neutral free logic. This framework accommodates non-denoting terms by introducing a third truth value for the case of formulae that include them, in addition to the traditional true and false values. We adhere to Pavlović and Gratzl’s approach [1] regarding quantifiers, atomic formulas, and simple terms. However, we diverge by utilising natural deduction systems instead of sequent calculi. Additionally, we enhance their foundational framework by integrating identity and definite descriptions into the system. We focus on the normalisation theorem for the natural deduction systems indicated above.

Bibliography

1. E. Pavlovi’{c} and N. Gratzl,Neutral Free Logic: Motivation, ProofTheory and Models,Journal of Philosophical Logic,vol. 52 (2023), no. 2, pp. 519–554.

Multivalued logics are commonly described in terms of valuations into an algebra of truth values. In [1], by lifting the consideration of valuations into subsets of the same algebra, Priest defines what he calls the Plurivalent version of a logic. This process essentially constitutes taking the power-algebra [2] of truth values from the original logic and memberhood of the originally designated elements as the new designation. In the simplest example of this construction Priest can identify his own logic of paradox (LP), and a logic known as analytic logic (AL) as plurivalent logics on classical truth values of the two valued Boolean algebra [1]. Furthermore we note that, by changing the designation notion, strong Kleene three valued logic \((\mathrm{K}_3)\) can also be identified by the same algebra through similar memberhood conditions.

Continuing work towards general methods for logics with team semantics [3] I will in this presentation consider the power-algebras of arbitrary Boolean algebras and establish a sound and complete labelled natural deduction system for entailments of memberhood statements. This gives rise to the definition and presentation of substructural versions of the aforementioned logics, with proof systems for which additional rules can be added to obtain the original logic.

Bibliography

1. Graham Priest,Plurivalent logics,The Australasian Journal of Logic,vol. 11 (2014), no. 1.
2. Chris Brink,Power structures,Algebra Universalis,vol. 30 (1993), pp. 177–216.
3. Fredrik Engstr{ö}m, Orvar Lorimer Olsson,The propositional logic of teams,arXiv preprint, (2023), arXiv.2303.14022