6 research outputs found
Carnap's problem for intuitionistic propositional logic
We show that intuitionistic propositional logic is \emph{Carnap categorical}:
the only interpretation of the connectives consistent with the intuitionistic
consequence relation is the standard interpretation. This holds relative to the
most well-known semantics with respect to which intuitionistic logic is sound
and complete; among them Kripke semantics, Beth semantics, Dragalin semantics,
and topological semantics. It also holds for algebraic semantics, although
categoricity in that case is different in kind from categoricity relative to
possible worlds style semantics.Comment: Keywords: intuitionistic logic, Carnap's problem, nuclear semantics,
algebraic semantics, logical constants, consequence relations, categoricity.
Versions: 3rd version has minor additions, and correction of an error in 2nd
version (not in 1st version
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Locales, Nuclei, and Dragalin Frames
It is a classic result in lattice theory that a poset is a complete lattice iff it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) iff it can be realized as fixpoints of a nucleus on the locale of upsets of a poset. He also showed how to generate a nucleus on upsets by adding a structure of “paths” to a poset, forming what we call a Dragalin frame. This allowed Dragalin to introduce a semantics for intuitionistic logic that generalizes Beth and Kripke semantics. He proved that every spatial locale (locale of open sets of a topological space) can be realized as fixpoints of the nucleus generated by a Dragalin frame. In this paper, we strengthen Dragalin’s result and prove that every locale—not only spatial locales—can be realized as fixpoints of the nucleus generated by a Dragalin frame. In fact, we prove the stronger result that for every nucleus on the upsets of a poset, there is a Dragalin frame based on that poset that generates the given nucleus. We then compare Dragalin’s approach to generating nuclei with the relational approach of Fairtlough and Mendler [11], based on what we call FM-frames. Surprisingly, every Dragalin frame can be turned into an equivalent FM-frame, albeit on a different poset. Thus, every locale can be realized as fixpoints of the nucleus generated by an FM-frame. Finally, we consider the relational approach of Goldblatt [13] and characterize the locales that can be realized using Goldblatt frames