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Modal Companion
In logic, a modal companion of a superintuitionistic (intermediate) logic ''L'' is a normal modal logic that interprets ''L'' by a certain canonical translation, described below. Modal companions share various properties of the original intermediate logic, which enables to study intermediate logics using tools developed for modal logic. Gödel–McKinsey–Tarski translation Let ''A'' be a propositional intuitionistic formula. A modal formula ''T''(''A'') is defined by induction on the complexity of ''A'': :T(p)=\Box p, for any propositional variable p, :T(\bot)=\bot, :T(A\land B)=T(A)\land T(B), :T(A\lor B)=T(A)\lor T(B), :T(A\to B)=\Box(T(A)\to T(B)). As negation is in intuitionistic logic defined by A\to\bot, we also have :T(\neg A)=\Box\neg T(A). ''T'' is called the Gödel translation or Gödel–McKinsey– Tarski translation. The translation is sometimes presented in slightly different ways: for example, one may insert \Box before every subformula. All such variants are prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Willem Blok
Willem Johannes "Wim" Blok (1947–2003) was a Dutch logician who made major contributions to algebraic logic, universal algebra, and modal logic. His important achievements over the course of his career include "a brilliant demonstration of the fact that various techniques and results that originated in universal algebra can be used to prove significant and deep theorems in modal logic." Blok began his career in 1973 as an algebraist investigating the varieties of interior algebras at the University of Illinois at Chicago. Following the 1976 completion of his Ph.D. on that topic, he continued on to study more general varieties of modal algebras. As an algebraist, Blok "was recognised by the modal logic community as one of the most influential modal logicians" by the end of the 1970s. He published many papers in the ''Reports on Mathematical Logic'', served as a member on their editorial board, and was one of their guest editors. Along with Don Pigozzi, Wim Blok co-authored the mono ... [...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 usua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\to Y, its inverse f^\colon Y\to X admits an explicit description: it sends each element y\in Y to the unique element x\in X such that . As an example, consider the real-valued function of a real variable given by . One can think of as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of is the function f^\colon \R\to\R defined by f^(y) = \frac . Definitions Let be a function whose domain is the set , and whose codomain is the set . Then is ''invertible'' if there exists a function from to such that g(f(x))=x for all x\in X and f(g(y))=y for all y\in Y. If is invertible, then there is exactly one function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kripke Frame
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise'). Semantics of modal logic The language of propositional modal logic consists of a countably infinite set of propositional variables, a set of truth-functional connectives (in this article \to and \neg), and the modal operator \Box ("necessarily"). The modal operator \Diamond ("possibly") is (classically) the dual of \Box and may be defined in terms of necessity like so: \Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^\complement = U \setminus A. Or formally: A^\complement = \. The absolute complement of is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection (set Theory)
In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is written using the symbol "\cap" between the terms; that is, in infix notation. For example: \\cap\=\ \\cap\=\varnothing \Z\cap\N=\N \\cap\N=\ The intersection of more than two sets (generalized intersection) can be written as: \bigcap_^n A_i which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols. Definition The intersection of two sets A and B, denoted by A \cap B, is the set of all objects that are members of both the sets A and B. In symbols: A \cap B = \. That is, x is an element of the intersection A \cap B if and only if x is both an element of A and an element of B. For example: * The intersection of the sets and is . * The number 9 is in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \,\sim\, on S, the of an element a in S, denoted by is the set \ of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S by \,\sim\,, and is denoted by S / \sim. When the set S has some structure (such as a group operation or a topology) and the equivalence relation \,\sim\, is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''inco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Set
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \,\sim\, on S, the of an element a in S, denoted by is the set \ of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S by \,\sim\,, and is denoted by S / \sim. When the set S has some structure (such as a group operation or a topology) and the equivalence relation \,\sim\, is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation \,\sim\, on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b, and c in X: * a \sim a ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or skeletal) preorder is a partial order, and a symmetric preorder is an equivalence relation. The name comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric. Because a preorder is a binary relation, the symbol \,\leq\, can be used as the notational device for the relation. However, because they are not necessarily antisymmetric, some of the ordinary intuition associated to the symbol \,\leq\, may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worthwh ... [...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 proposition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
