## Semantics of Modality

Definitions of Definition

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*Intension* is the relation of a set of attributes stipulated as intrinsically true of a sign; judged by the degree of mereological equivalence of the relation.

*Extension *is the function of a set of objects denoted as true by implication of the intension of a set; judged by the correspondence of object to its function.

Modality of Extension

*Epistemic necessity* is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is probabilistically judged as always true of some denoted term in a particular doxastic system Đ.

*Epistemic possibility* is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is probabilistically judged as sometimes true of some denoted term in a particular doxastic system Đ.

*Epistemic impossibility* is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is probabilistically judged as never true of some denoted term in a particular doxastic system Đ.

*Metaphysical necessity* is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is true in all possible worlds.

*Metaphysical possibility* is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is actualizable; and therefore true in some possible worlds.

*Metaphysical impossibility* is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is non-actualizable or logically contradictory; and therefore true in no possible worlds.

Modality of Intension

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*Semantic necessity* is the modal quantification of any intensional relation, whereby some class of definiens ϕ^{n}(x_{n}) is stipulated as always true of some definiendum ^{┌}ϕ^{┐} in a particular intensional system Ł.

*Semantic possibility* is the modal quantification of any intensional relation, whereby some class of definiens ϕ^{n}(x_{n}) is stipulated as sometimes true of some definiendum ^{┌}ϕ^{┐ }if and only if there is no contradiction with the class of definiens ϕ^{n}(x_{n}) that are stipulated as always true of the definiendum ^{┌}ϕ^{┐ }in a particular intensional system Ł.

*Semantic impossibility* is the modal quantification of any intensional relation, whereby some class of definiens ϕ^{n}(x_{n}) is stipulated as never true of some definiendum ^{┌}ϕ^{┐ }in a particular intensional system Ł.