|
Classical Modal Logic
In modal logic, a classical modal logic L is any modal logic containing (as axiom or theorem) the duality of the modal operators :\Diamond A \leftrightarrow \lnot\Box\lnot A that is also closed under the rule :\frac. Alternatively, one can give a dual definition of L by which L is classical if and only if it contains (as axiom or theorem) :\Box A \leftrightarrow \lnot\Diamond\lnot A and is closed under the rule :\frac. 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 In logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautology (logic), tautologies; * All instances of the Kripke_semantics, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B) and it is closed ... is regular and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Modal Logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality, causation. For instance, in epistemic modal logic, the well-formed_formula, formula \Box P can be used to represent the statement that P is known. In deontic modal logic, that same formula can represent that P is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula \Box P \rightarrow P as a Tautology_(logic), tautology, representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false. Modal logics are formal systems that include unary operation, unary operators such as \Diamond and \Box, representing possibility and necessi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Duality (mathematics)
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dual of is . In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called ''primal''). Such involutions sometimes have fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the ''standard duality in projective geometry''. In mathematical contexts, ''duality'' has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Deductive Closure
In mathematical logic, a set of Well-formed formula, logical formulae is deductively closed if it contains every formula that can be deductive reasoning, logically deduced from ; formally, if always implies . If is a set of formulae, the deductive closure of is its smallest superset that is deductively closed. The deductive closure of a Theory (mathematical logic), theory is often denoted or . Some authors do not define a theory as deductively closed (thus, a theory is defined as any set of Sentence (logic), sentences), but such theories can always be 'extended' to a deductively closed set. A theory may be referred to as a ''deductively closed theory'' to emphasize it is defined as a deductively closed set. Deductive closure is a special case of the more general mathematical concept of Closure (mathematics)#Closure_operator, closure — in particular, the deductive closure of is exactly the closure of with respect to the operation of logical consequence (). Examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Normal Modal Logic
In logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautology (logic), tautologies; * All instances of the Kripke_semantics, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B) and it is closed under: * Detachment rule (''modus ponens''): A\to B, A \in L implies B \in L; * Necessitation rule: A \in L implies \Box A \in L. 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 (modal logic), S5, are normal (and hence are extensions of K). However a number of deontic logic, deontic and epistemic logics, for example, are non-normal, often because they give up the Kripke schema. Every normal modal logic is regular modal logic, regular and hence classical modal logic, classical. Common normal modal logics The following table lists several common normal modal systems. The notation refers to the table at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Semantics (mathematical Logic)
In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositional intuitionistic logic. MV-algebras are the algebraic semantics of Łukasiewicz logic. See also * Algebraic semantics (computer science) * Lindenbaum–Tarski algebra Further reading * (2nd published by ASL in 2009open accessat Project Euclid * * * Good introduction for readers with prior exposure to non-classical logics but without much background in order theory and/or universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Neighborhood Semantics
Neighborhood semantics, also known as Scott–Montague semantics, is a formal semantics for modal logics. It is a generalization, developed independently by Dana Scott and Richard Montague, of the more widely known relational semantics for modal logic. Whereas a relational frame \langle W,R\rangle consists of a set ''W'' of worlds (or states) and an accessibility relation ''R'' intended to indicate which worlds are alternatives to (or, accessible from) others, a neighborhood frame \langle W,N\rangle still has a set ''W'' of worlds, but has instead of an accessibility relation a ''neighborhood function'' : N : W \to 2^ that assigns to each element of ''W'' a set of subsets of ''W''. Intuitively, each family of subsets assigned to a world are the propositions necessary at that world, where 'proposition' is defined as a subset of ''W'' (i.e. the set of worlds at which the proposition is true). Specifically, if ''M'' is a model on the frame, then : M,w\models\square \varphi \L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Regular Modal Logic
In modal logic, a regular modal logic is a modal logic containing (as axiom or theorem) the duality of the modal operators: \Diamond A \leftrightarrow \lnot\Box\lnot A and closed under the rule \frac. Every normal modal logic In logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautology (logic), tautologies; * All instances of the Kripke_semantics, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B) and it is closed ... is regular, and every regular modal logic is classical. References *Chellas, Brian. ''Modal Logic: An Introduction''. Cambridge University Press, 1980. Logic Modal logic {{logic-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
