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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.
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 ...
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.
When a Sahlqvist formula is used as an axiom in a normal modal logic, the logic is guaranteed to be complete with respect to the basic elementary class of frames the axiom defines. This result comes from the Sahlqvist completeness theorem [Modal Logic, Blackburn et al., Theorem 4.42]. But there is also a converse theorem, namely a theorem that ...
The Barcan formula is: . In English, the schema reads: If every x is necessarily F, then it is necessary that every x is F.It is equivalent to . The Barcan formula has generated some controversy because—in terms of possible world semantics—it implies that all objects which exist in any possible world (accessible to the actual world) exist in the actual world, i.e. that domains cannot grow ...
The modal depth of a formula indicates 'how far' one needs to look in a Kripke model when checking the validity of the formula. For each modal operator, one needs to transition from a world in the model to a world that is accessible through the accessibility relation. The modal depth indicates the longest 'chain' of transitions from a world to ...
A modal connective (or modal operator) is a logical connective for modal logic.It is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-truth-functional in the following sense: The truth-value of composite formulae sometimes depend on factors other than the actual truth-value of their components.
Classical logic is a 19th and 20th-century innovation. The name does not refer to classical antiquity, which used the term logic of Aristotle. Classical logic was the reconciliation of Aristotle's logic, which dominated most of the last 2000 years, with the propositional Stoic logic. The two were sometimes seen as irreconcilable.