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The weakest classical system is sometimes referred to as E and is non-normal. Both algebraic and neighborhood semantics characterize familiar classical modal systems that are weaker than the weakest normal modal logic K. Every regular modal logic is classical, and every normal modal logic is regular and hence classical.
Modal logic is a kind of logic used to represent statements about necessity and possibility.It plays a major role in philosophy and related fields as a tool for understanding concepts such as knowledge, obligation, and causation.
In logic and philosophy, S5 is one of five systems of modal logic proposed by Clarence Irving Lewis and Cooper Harold Langford in their 1932 book Symbolic Logic. It is a normal modal logic, and one of the oldest systems of modal logic of any kind. It is formed with propositional calculus formulas and tautologies, and inference apparatus with ...
A non-normal modal logic is a variant of modal logic that deviates from the basic principles of normal modal logics. Normal modal logics adhere to the distributivity axiom ( ( p → q ) → ( p → q ) {\displaystyle \Box (p\to q)\to (\Box p\to \Box q)} ) and the necessitation principle which states that "a tautology must be necessarily true ...
The smallest logic satisfying the above conditions is called K. Most modal logics commonly used nowadays (in terms of having philosophical motivations), e.g. C. I. Lewis's S4 and S5, are normal (and hence are extensions of K). However a number of deontic and epistemic logics, for example, are non-normal, often because they give up the Kripke ...
Informally, the number of transitions in the 'longest chain' of transitions in the first-order formula is the modal depth of the formula. The modal depth of the formula used in the example above is two. The first-order formula indicates that the transitions from to and from to are needed to verify the validity of the formula. This is also the ...
In quantified modal logic, the Buridan formula and the converse Buridan formula (more accurately, schemata rather than formulas) (i) syntactically state principles of interchange between quantifiers and modalities; (ii) semantically state a relation between domains of possible worlds.
Hoare logic, algorithmic logic, weakest preconditions, and dynamic logic are all well suited to discourse and reasoning about sequential behavior. Extending these logics to concurrent behavior however has proved problematic. There are various approaches but all of them lack the elegance of the sequential case.