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Kripke Semantics
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: \ ... [...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 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] |
Possible World
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their metaphysical status has been a subject of controversy in philosophy, with modal realists such as David Lewis arguing that they are literally existing alternate realities, and others such as Robert Stalnaker arguing that they are not. Logic Possible worlds are one of the foundational concepts in modal and intensional logics. Formulas in these logics are used to represent statements about what ''might'' be true, what ''should'' be true, what one ''believes'' to be true and so forth. To give these statements a formal interpretation, logicians use structures containing possible worlds. For instance, in the relational semantics for classical propositional modal logic, the formula \Diamond P (read as "possibly P") is actually true if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Other properties related to completeness The property converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property. Forms of completeness Expressive completeness A formal language is ''expressively complete'' if it can express the subject matter for which it is intended. Functional completeness A set of logical connectives associated with a formal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Relation
A symmetric relation is a type of binary relation. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: : \forall a, b \in X(a R b \Leftrightarrow b R a) , where the notation ''aRb'' means that . An example is the relation "is equal to", because if is true then is also true. If ''R''T represents the converse of ''R'', then ''R'' is symmetric if and only if . Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation. Examples In mathematics * "is equal to" ( equality) (whereas "is less than" is not symmetric) * "is comparable to", for elements of a partially ordered set * "... and ... are odd": :::::: Outside mathematics * "is married to" (in most legal systems) * "is a fully biological sibling of" * "is a homophone of" * "is a co-worker of" * "is a teammate of" Relationship to asymmetric and antisymmetric relations By definition, a nonempty relation cannot be both symmetric and asymmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serial Relation
In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation. Bertrand Russell used serial relations in ''The Principles of Mathematics'' (1903) as he explored the foundations of order theory and its applications. The term ''serial relation'' was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness, irreflexivity, and transitivity. A serial relation ''R'' is an endorelation on a set ''U''. As stated by Russell, \forall x \exists y \ xRy , where the universal and existential quantifiers refer to ''U''. In contemporary language of relations, this property defines a total relation. But a total relation may be heterogeneous. Serial relations are of historic interest. For a relation ''R'', let denote the "successor neigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Relation
In mathematics, a binary relation on a set (mathematics), set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Every partial order and every equivalence relation is transitive. For example, less than and equality (mathematics), equality among real numbers are both transitive: If and then ; and if and then . Definition A homogeneous relation on the set is a ''transitive relation'' if, :for all , if and , then . Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where is the infix notation for . Examples As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy is also an ancestor of Carrie. On the other hand, "is the birth mother of" is not a transitive relation, because if Alice is the birth mother of Brenda, and Brenda is the birth mother of Claire, then it does ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Relation
In mathematics, a partial order or total order < on a is said to be dense if, for all and in for which , there is a in such that . That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element between them", since all elements of a total order are comparable. Example The s as a linearly ordered set are a densely ordered set in this sense, ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symbolic Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deontic Logic
Deontic logic is the field of philosophical logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts. It can be used to formalize imperative logic, or directive modality in natural languages. Typically, a deontic logic uses ''OA'' to mean ''it is obligatory that A'' (or ''it ought to be (the case) that A''), and ''PA'' to mean ''it is permitted (or permissible) that A'', which is defined as PA\equiv \neg O\neg A. In natural language, the statement "You may go to the zoo OR the park" should be understood as Pz\land Pp instead of Pz\lor Pp, as both options are permitted by the statement. When there are multiple agents involved in the domain of discourse, the deontic modal operator can be specified to each agent to express their individual obligations and permissions. For example, by using a subscript O_i for agent a_i, O_iA means that "It i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 applications in many fields, including philosophy, theoretical computer science, artificial intelligence, economics, and linguistics. While philosophers since Aristotle have discussed modal logic, and Medieval philosophers such as Avicenna, Ockham, and Duns Scotus developed many of their observations, it was C. I. Lewis who created the first symbolic and systematic approach to the topic, in 1912. It continued to mature as a field, reaching its modern form in 1963 with the work of Saul Kripke. Historical development Many papers were written in the 1950s that spoke of a logic of knowledge in passing, but the Finnish philosopher G. H. von Wright's 1951 paper titled ''An Essay in Modal Logic'' is seen as a founding document. It was not unti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Relation
In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations. Etymology The word ''reflexive'' is originally derived from the Medieval Latin ''reflexivus'' ('recoiling' reflex.html" ;"title="f. ''reflex">f. ''reflex'' or 'directed upon itself') (c. 1250 AD) from the classical Latin ''reflexus-'' ('turn away', 'reflection') + ''-īvus'' (suffix). The word entered Early Modern English in the 1580s. The sense of the word meaning 'directed upon itself', as now used in mathematics, surviving mostly by its use in philosophy and grammar (cf. ''Reflexive verb'' and ''Reflexive pronoun''). The first e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
