Abstract
The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of first-degree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the mathbf{Dunn}calculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. In terms of the mathbf{Dunn} calculus, we first introduce two different interpolation methods, each of which uniformly shows that the Dunn logics have the interpolation property. One of the methods is closely related to Maehara’s method but the other method, which we call the pruned tableau method, is novel to this paper. We provide various reasons to prefer the pruned tableau method to the Maehara-style method. We then turn our attention to extensions of Dunn logics with so-called appropriate implication connectives. Although these logics have been considered at various places in the literature, a study of the interpolation properties of these logics is lacking. We use the pruned tableau method to uniformly show that these extended Dunn logics have the interpolation property and explain that the same result cannot be obtained via the Maehara-style method. Finally, we show how the pruned tableau method constructs interpolants for functionally complete extensions of the Dunn logics.
Highlights
We extend our Dunn calculus with tableau rules for ⊃ and invoke the pruned tableau method to prove, in a uniform way, that the extended Dunn logics over L⊃ all have the interpolation property
We used the Dunn calculus to define two distinct uniform interpolation methods for Dunn logics and their extensions: the Maehara-style method, which is inspired by Maehara’s interpolation method for CL and the pruned tableau method, which is novel to this paper
The pruned tableau method is simpler than the Maehara-style method as it does not construct interpolants via an inductive argument and as the constructed interpolants typically have lower sentential complexity than those of the Maehara-style method
Summary
The following conditions were laid down by Dunn [11] to equip the logic of first-degree entailment FDE [8,9,11] with an intuitive semantics based on the values T (true and not false), B (both truth and false), N (neither true nor false) and F (false and not true). Pietz and Rivieccio [17] presented Exactly True Logic (ETL), an interesting variation upon FDE that is obtained by preserving exact truth, i.e. the value T, over all 4-valued valuations of L Both [17] and Wintein and Muskens [25] study ETL to quite some extent, no investigation of its interpolation property is to be found in the literature.
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