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Investigations in arrow logic. (English) Zbl 0874.03022

Marx, Maarten (ed.) et al., Arrow logic and multi-modal logic. Stanford, CA: CSLI Publications. Studies in Logic, Language and Information. 35-61 (1996).
Summary: We give an extensive overview of arrow logics of pair-frames with respect to the following four properties:
\(\bullet\) finite axiomatizability with an orthodox derivation system,
\(\bullet\) decidability,
\(\bullet\) Craig interpolation, and
\(\bullet\) Beth definability.
Since the universe of a pair-frame is a binary relation, we can consider classes of such frames satisfying conditions like reflexivity, symmetry and transitivity. Our main result is that an arrow logic of pair-frames has any of the above properties if and only if it contains frames with non-transitive universes. Transitivity of the universe is closely connected to associativity of composition, because composition is associative on a pair-frame if its universe is a transitive relation. So the conclusion of this paper can also be stated as follows: we can give arrow logic the intuitive pair-frame semantics and have all the four above-mentioned positive properties if we are willing to give up associativity of composition.
Arrow logic comes in two versions: the “abstract arrows” version and the concrete “ordered pairs” version. Our results point in the direction that the abstract arrows version seems to have no advantages (like decidability etc.) over the ordered pairs version.
For the entire collection see [Zbl 0859.00018].

MSC:

03B45 Modal logic (including the logic of norms)
03B70 Logic in computer science
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