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Studia Logica
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Studia Logica
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Identity in modal logic theorem proving

Authors: Francis Jeffry Pelletier;

Identity in modal logic theorem proving

Abstract

In this paper the `relational' translation from quantified modal logics with identity into first-order predicate logic is repeated. In this well- known method, the Kripke semantics of the modal \(\square\)-operator: \(\square p\) is true in a world if \(p\) is true in all accessible worlds, is turned into a translation rule: \(\text{tr}(\square p,w)\to \forall v R(w,v)\Rightarrow\text{tr}(p,v)\). Predicate symbols get an extra argument which records the actual world in form of a `world term'. The method which is actually reported in the paper is somewhat more complicated because an extra predicate \(W\) is used to give the `world variables' the sort `world'. The intention behind this \(W\) predicate is to prevent the identification of world terms with domain terms. (This is in fact not necessary because the terms of the two different sorts always occur in different arguments of the predicates. In any reasonable calculus they never get mixed.) By some unmentioned reason the terms in the translated first-order modal logic version do not get extra world arguments. Therefore equations \(a= b\) can either be left as they are during the translation, and therefore they lose their modal context, or they are translated into \(I(a,b,w)\) where \(I\) is a world-dependent equality predicate. In this case the \(I\) predicate has to be axiomatized accordingly. It is not mentioned how the existential quantifier is treated in the theorem prover THINKER the author uses to prove the translated formulae, but it is very likely that both presented versions for dealing with equality cause serious problems if the existentially quantified variables occur in different modal contexts.

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Keywords

Mechanization of proofs and logical operations, equality in modal logic, relational translation, Modal logic (including the logic of norms)

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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
2
Average
Average
Average
bronze