Abstract
Let ⊥, →, and □ be primitive, and let us have a countable supply of propositional letters. By a (modal) logic we understand a proper subset of the set of all formulas containing every tautology and being closed under modus ponens and substitution. A logic is regular if it contains every instance of □A ∧ □B ↔ □(A ∧ B) and is closed under the ruleA regular logic is normal if it contains □⊤. The smallest regular logic we denote by C (the same as Lemmon's C2), the smallest normal one by K. If L and L' are logics and L ⊆ L′, then L is a sublogic of L', and L' is an extension of L; properly so if L ≠ L'. A logic is quasi-regular (respectively, quasi-normal) if it is an extension of C (respectively, K).A logic is Post complete if it has no proper extension. The Post number, denoted by p(L), is the number of Post complete extensions of L. Thanks to Lindenbaum, we know thatThere is an obvious upper bound, too:Furthermore,.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.