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Modal Algebra
In algebra and logic, a modal algebra is a structure \langle A,\land,\lor,-,0,1,\Box\rangle such that *\langle A,\land,\lor,-,0,1\rangle is a Boolean algebra, *\Box is a unary operation on ''A'' satisfying \Box1=1 and \Box(x\land y)=\Box x\land\Box y for all ''x'', ''y'' in ''A''. Modal algebras provide models of propositional modal logics in the same way as Boolean algebras are models of classical logic. In particular, the variety of all modal algebras is the equivalent algebraic semantics of the modal logic ''K'' in the sense of abstract algebraic logic, and the lattice of its subvarieties is dually isomorphic to the lattice of normal modal logics. Stone's representation theorem can be generalized to the Jónsson–Tarski duality, which ensures that each modal algebra can be represented as the algebra of admissible sets in a modal general frame. A Magari algebra (or diagonalizable algebra) is a modal algebra satisfying \Box (-\Box x \lor x) = \Box x. Magari algebras corresp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Modal Logic
In logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautologies; * All instances of the 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, are normal (and hence are extensions of K). However a number of deontic and epistemic logic Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applica ...s, for example, are non-normal, often because they give up the Kripke schema. Every normal modal logic is regular and hen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of ''implication'' such that (''c'' ∧ ''a'') ≤ ''b'' is equivalent to ''c'' ≤ (''a'' → ''b''). From a logical standpoint, ''A'' → ''B'' is by this definition the weakest proposition for which modus ponens, the inference rule ''A'' → ''B'', ''A'' ⊢ ''B'', is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by to formalize intuitionistic logic. As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when ''a'' → ''b'' is defined as ¬''a'' ∨ ''b'', as is every complete distributive lattice satisfying a one-sided infinite distributive law when ''a'' → ''b'' is taken to be the supremum of the set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior Algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras. Definition An interior algebra is an algebraic structure with the signature :⟨''S'', ·, +, ′, 0, 1, I⟩ where :⟨''S'', ·, +, ′, 0, 1⟩ is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities: # ''x''I ≤ ''x'' # ''x''II = ''x''I # (''xy'')I = ''x''I''y''I # 1I = 1 ''x''I is called the interior of ''x''. The dual of the interior operator is the closure operator C defined by ''x''C = ((''x''′)I)′. ''x''C is called the closure of ''x''. By the principle of duality, the closure operator satisfies the identities: # ''x''C ≥ ''x'' # ''x''CC = ''x''C # (''x'' + ''y'')C = ''x''C + ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provability Logic
Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic. Examples There are a number of provability logics, some of which are covered in the literature mentioned in . The basic system is generally referred to as GL (for Gödel– Löb) or L or K4W (W stands for well-foundedness). It can be obtained by adding the modal version of Löb's theorem to the logic K (or K4). Namely, the axioms of GL are all tautologies of classical propositional logic plus all formulas of one of the following forms: * Distribution axiom: * Löb's axiom: And the rules of inference are: * ''Modus ponens'': From ''p'' → ''q'' and ''p'' conclude ''q''; * Necessitation: From \vdash ''p'' conclude \vdash . History The GL model was pioneered by Robert M. Solovay in 1976. Since then, until his death in 1996, the prime inspirer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Frame
In logic, general frames (or simply frames) are Kripke frames with an additional structure, which are used to model modal and intermediate logics. The general frame semantics combines the main virtues of Kripke semantics and algebraic semantics: it shares the transparent geometrical insight of the former, and robust completeness of the latter. Definition A modal general frame is a triple \mathbf F=\langle F,R,V\rangle, where \langle F,R\rangle is a Kripke frame (i.e., R is a binary relation on the set F), and V is a set of subsets of F that is closed under the following: *the Boolean operations of (binary) intersection, union, and complement, *the operation \Box, defined by \Box A=\. They are thus a special case of fields of sets with additional structure. The purpose of V is to restrict the allowed valuations in the frame: a model \langle F,R,\Vdash\rangle based on the Kripke frame \langle F,R\rangle is admissible in the general frame \mathbf, if :\\in V for every propositional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theorem
In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure. Examples Algebra * Cayley's theorem states that every group is isomorphic to a permutation group. * Representation theory studies properties of abstract groups via their representations as linear transformations of vector spaces. *Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. *: A variant, Stone's representation theorem for distributive lattices, states that every distributive lattice is isomorphic to a sublattice of the power set lattice of some set. *: Another variant, Stone's duality, states that there exists a duality (in the sense of an arrow-reversing equivalence) between the categories of Boolean algebras and that of Stone spaces. * The Poincaré–Birkhoff–Witt theorem states that every Lie algebra embeds into the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jónsson–Tarski Duality
In logic, general frames (or simply frames) are Kripke frames with an additional structure, which are used to model modal and intermediate logics. The general frame semantics combines the main virtues of Kripke semantics and algebraic semantics: it shares the transparent geometrical insight of the former, and robust completeness of the latter. Definition A modal general frame is a triple \mathbf F=\langle F,R,V\rangle, where \langle F,R\rangle is a Kripke frame (i.e., R is a binary relation on the set F), and V is a set of subsets of F that is closed under the following: *the Boolean operations of (binary) intersection, union, and complement, *the operation \Box, defined by \Box A=\. They are thus a special case of fields of sets with additional structure. The purpose of V is to restrict the allowed valuations in the frame: a model \langle F,R,\Vdash\rangle based on the Kripke frame \langle F,R\rangle is admissible in the general frame \mathbf, if :\\in V for every propositional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone's Representation Theorem
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone. Stone was led to it by his study of the spectral theory of operators on a Hilbert space. Stone spaces Each Boolean algebra ''B'' has an associated topological space, denoted here ''S''(''B''), called its Stone space. The points in ''S''(''B'') are the ultrafilters on ''B'', or equivalently the homomorphisms from ''B'' to the two-element Boolean algebra. The topology on ''S''(''B'') is generated by a (closed) basis consisting of all sets of the form \, where ''b'' is an element of ''B''. This is the topology of pointwise convergence of nets of homomorphisms into the two-element Boolean algebra. For every Boolean algebra ''B'', ''S''(''B'') is a compact totally disco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These ''lattice-like'' structures all admi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
