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文件名称：The Proof Theory and Semantics of Intuitionistic Modal Logic
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更新时间：2013-06-30 13:39:41
informal classical mathematics
Possible world semantics underlies many of the applications of modal logic in
computer science and philosophy. The standard theory arises from interpreting the
semantic de nitions in the ordinary meta-theory of informal classical mathematics.
If, however, the same semantic de nitions are interpreted in an intuitionistic metatheory
then the induced modal logics no longer satisfy certain intuitionistically
invalid principles. This thesis investigates the intuitionistic modal logics that arise
in this way.
Natural deduction systems for various intuitionistic modal logics are presented.
From one point of view, these systems are self-justifying in that a possible world
interpretation of the modalities can be read o directly from the inference rules. A
technical justi cation is given by the faithfulness of translations into intuitionistic
rst-order logic. It is also established that, in many cases, the natural deduction
systems induce well-known intuitionisticmodal logics, previously given by Hilbertstyle
axiomatizations.
The main bene t of the natural deduction systems over axiomatizations is their
susceptibility to proof-theoretic techniques. Strong normalization (and con
uence)
results are proved for all of the systems. Normalization is then used to establish
the completeness of cut-free sequent calculi for all of the systems, and decidability
for some of the systems.
Lastly, techniques developed throughout the thesis are used to establish that
those intuitionistic modal logics proved decidable also satisfy the nite model
property. For the logics considered, decidability and the nite model property
presented open problems