Ideas from 'A Completeness Theorem in Modal Logic' by Saul A. Kripke [1959], by Theme Structure
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4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
10163

Propositional modal logic has been proved to be complete




Full Idea:
At the age of 19 Saul Kripke published a completeness proof of propositional modal logic.




From:
report of Saul A. Kripke (A Completeness Theorem in Modal Logic [1959]) by Feferman / Feferman  Alfred Tarski: life and logic Int V

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic
10760

With possible worlds, S4 and S5 are sound and complete, but S1S3 are not even sound




Full Idea:
Kripke gave a possible worlds semantics to a whole range of modal logics, and S4 and S5 turned out to be both sound and complete with this semantics. Hence more systems could be designed. S1S3 failed in soundness, leading to 'impossible worlds'.




From:
report of Saul A. Kripke (A Completeness Theorem in Modal Logic [1959]) by Marcus Rossberg  Firstorder Logic, 2ndorder, Completeness §4

4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
16189

The variable domain approach to quantified modal logic invalidates the Barcan Formula




Full Idea:
Kripke's variable domain approach to quantified modal logic famously invalidates the Barcan Formula.




From:
report of Saul A. Kripke (A Completeness Theorem in Modal Logic [1959]) by Ori Simchen  The Barcan Formula and Metaphysics §3




A reaction:
[p.9 and p.16] In a single combined domain all the possibilia must be present, but with variable domains objects in remote domains may not exist in your local domain. BF is committed to those possible objects.

15132

The Barcan formulas fail in models with varying domains




Full Idea:
Kripke showed that the Barcan formula ∀x□A⊃□∀xA and its converse fail in models which require varying domains.




From:
report of Saul A. Kripke (A Completeness Theorem in Modal Logic [1959]) by Timothy Williamson  Truthmakers and Converse Barcan Formula §1




A reaction:
I think this is why I reject the Barcan formulas for metaphysics  because the domain of metaphysics should be seen as varying, since some objects are possible in some contexts and not in others. Hmm…
