# On the expressiveness of expansions of the weak Kleene language

## Jose Martinez-Fernandez

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 \(f(x_{1},\ldots ,x_{i-1},\frac{1}{2},x_{i+1},\ldots ,x_{n})=c_{\frac{1}{2}}\).