Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here  focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics,  can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each individual finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can  find for every formula φ a formula f(φ) such that for every formula ψ, φ → []ψ is provable in L if and only if f(φ) → ψ is provable in L?  ansr this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic.
['Wesley Halcrow Holliday']
I am writing a research paper on a idea Partiality and Adjointness in Modal Logic
[{"from": "human", "value": "I am writing a research paper on a idea Partiality and Adjointness in Modal Logic"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial \u201cpossibilities\u201d rather than total \u201cworlds.\u201d There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here  focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics,  can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each individual finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can  find for every formula \u03c6 a formula f(\u03c6) such that for every formula \u03c8, \u03c6 \u2192 []\u03c8 is provable in L if and only if f(\u03c6) \u2192 \u03c8 is provable in L?  ansr this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic."}]