Neighborhood Semantics for Modal Logic by Eric Pacuit

Author:Eric Pacuit
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

When (so there are no antecedents), the above rule reduces to the rule of Necessitation . When , the above rule is the monotonicity rule discussed in Sect. 2.3. Then, Lemma 2.6 can be rephrased as:

A modal logic is monotonic iff is a derived rule.

This suggests the following characterization of normal and regular modal logics.

Proposition 2.9

Suppose that is a modal logic that contains . Then, is regular iff is a derived rule for all .

is normal iff is a derived rule for all .

Exercise 2.24

Prove Proposition 2.9.

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