Caleb M.H. Camrud

A very natural and philosophically important subclassical logic is FDE (for first-degree entailment). This account of logical consequence can be seen as going beyond the standard two-valued account (of “just true” and “just false”) to a four-valued account (adding the additional values of “both true and false” and “neither true nor false”). A natural question arises: What account of logical consequence arises from considering further (positive) combinations of such values? A partial answer was given by Priest in 2014; Shramko and Wansing had also given a partial result some years earlier, although in a different (more algebraic) context. In this note we generalize Priest’s (and indirectly Shramko and Wansing’s) result to show that even if one considers ordinal-many (positive) combinations of the previous values, for any ordinal, the resulting consequence relation (the resulting logic) remains FDE.

In classical modal semantics, a binary accessibility relation connects worlds. In this paper, we present a uniform and systematic treatment of modal semantics with a continuous accessibility relation alongside the continuous accessibility modal logics that they model. We develop several such logics for a variety of philosophical applications. Our main conclusions are as follows. Modal logics with a continuous accessibility relation are sound and complete in their natural classes of models. The class of Kripke frames where a continuous accessibility relation has a magnitude characterizing its degree of accessibility is not modally definable, and this has unappreciated significance to completeness proofs for such logics, revealing a methodological advantage of using classical multimodal semantics over fuzzy modal semantics. There is a pseudometric space modal logic that is complete in the class of pseudometric spaces, a natural semantic setting for quantitative modal reasoning about similarity. There is a metric space modal logic that is complete in the class of metric spaces, a natural semantic setting for quantitative modal reasoning about neighborhoods and counterfactual stability. There is a real line continuous temporal logic that is canonical for real lines, a natural semantic setting for quantitative modal reasoning about time.

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form $$\phi ^\mathcal {M}\le r$$, and the open diagram, which encapsulates strict inequalities of the form $$\phi ^\mathcal {M}< r$$. We show that the closed and open $$\Sigma _N$$ diagrams are $$\Pi ^0_{N+1}$$ and $$\Sigma ^0_N$$ respectively, and that the closed and open $$\Pi _N$$ diagrams are $$\Pi ^0_N$$ and $$\Sigma ^0_{N + 1}$$ respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal.