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Autoepistemic Logic
The autoepistemic logic is a formal logic for the representation and reasoning of knowledge about knowledge. While propositional logic can only express facts, autoepistemic logic can express knowledge and lack of knowledge about facts. The stable model semantics, which is used to give a semantics to logic programming with negation as failure, can be seen as a simplified form of autoepistemic logic. Syntax The syntax of autoepistemic logic extends that of propositional logic by a modal operator \BoxTo clarify, the modal operator \Box is a medium white square; this is not a browser rendering issue indicating knowledge: if F is a formula, \Box F indicates that F is known. As a result, \Box \neg F indicates that \neg F is known and \neg \Box F indicates that F is not known. This syntax is used for allowing reasoning based on knowledge of facts. For example, \neg \Box F \rightarrow \neg F means that F is assumed false if it is not known to be true. This is a form of negation as failur ... [...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] |
Possible World Semantics
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] |
Lecture Notes In Computer Science
''Lecture Notes in Computer Science'' is a series of computer science books published by Springer Science+Business Media since 1973. Overview The series contains proceedings, post-proceedings, monographs, and Festschrifts. In addition, tutorials, state-of-the-art surveys, and "hot topics" are increasingly being included. The series is indexed by DBLP. See also *''Monographiae Biologicae'', another monograph series published by Springer Science+Business Media *''Lecture Notes in Physics'' *''Lecture Notes in Mathematics'' *''Electronic Workshops in Computing ''Electronic Workshops in Computing'' (eWiC) is a publication series by the British Computer Society. The series provides free online access for conferences and workshops in the area of computing. For example, the EVA London Conference proceeding ...'', published by the British Computer Society References External links * Publications established in 1973 Computer science books Series of non-fiction books Springer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modal Logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other systems by adding unary operators \Diamond and \Box, representing possibility and necessity respectively. For instance the modal formula \Diamond P can be read as "possibly P" while \Box P can be read as "necessarily P". Modal logics can be used to represent different phenomena depending on what kind of necessity and possibility is under consideration. When \Box is used to represent epistemic necessity, \Box P states that P is epistemically necessary, or in other words that it is known. When \Box is used to represent deontic necessity, \Box P states that P is a moral or legal obligation. In the standard relational semantics for modal logic, formulas are assigned truth values relative to a ''possible world''. A formula's truth value at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-monotonic Logic
A non-monotonic logic is a formal logic whose conclusion relation is not monotonic. In other words, non-monotonic logics are devised to capture and represent defeasible inferences (cf. defeasible reasoning), i.e., a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusion(s) based on further evidence. Most studied formal logics have a monotonic entailment relation, meaning that adding a formula to a theory never produces a pruning of its set of conclusions. Intuitively, monotonicity indicates that learning a new piece of knowledge cannot reduce the set of what is known. A monotonic logic cannot handle various reasoning tasks such as reasoning by default (conclusions may be derived only because of lack of evidence of the contrary), abductive reasoning (conclusions are only deduced as most likely explanations), some important approaches to reasoning about knowledge (the ignorance of a conclusion must be retracted when the concl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Logic Network
A probabilistic logic network (PLN) is a conceptual, mathematical, and computational approach to uncertain inference; inspired by logic programming, but using probabilities in place of crisp (true/false) truth values, and fractional uncertainty in place of autoepistemic logic, crisp known/unknown values. In order to carry out effective reasoning in real-world circumstances, artificial intelligence software must robustly handle uncertainty. However, previous approaches to uncertain inference do not have the breadth of scope required to provide an integrated treatment of the disparate forms of cognitively critical uncertainty as they manifest themselves within the various forms of pragmatic inference. Going beyond prior probabilistic approaches to uncertain inference, PLN is able to encompass within uncertain logic such ideas as induction, abductive reasoning, abduction, analogy, fuzziness and speculation, and reasoning about time and causality. PLN was developed by Ben Goertzel, Mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncertain Inference
Uncertain inference was first described by C. J. van Rijsbergen as a way to formally define a query and document relationship in Information retrieval. This formalization is a logical implication with an attached measure of uncertainty. Definitions Rijsbergen proposes that the measure of uncertainty of a document ''d'' to a query ''q'' be the probability of its logical implication, i.e.: :P(d \to q) A user's query can be interpreted as a set of assertions about the desired document. It is the system's task to infer, given a particular document, if the query assertions are true. If they are, the document is retrieved. In many cases the contents of documents are not sufficient to assert the queries. A knowledge base of facts and rules is needed, but some of them may be uncertain because there may be a probability associated to using them for inference. Therefore, we can also refer to this as ''plausible inference''. The plausibility of an inference d \to q is a function of the pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
S5 (modal Logic)
In logic and philosophy, S5 is one of five systems of modal logic proposed by Clarence Irving Lewis and Cooper Harold Langford in their 1932 book ''Symbolic Logic''. It is a normal modal logic, and one of the oldest systems of modal logic of any kind. It is formed with propositional calculus formulas and tautologies, and inference apparatus with substitution and modus ponens, but extending the syntax with the modal operator ''necessarily'' \Box and its dual ''possibly'' \Diamond. The axioms of S5 The following makes use of the modal operators \Box ("necessarily") and \Diamond ("possibly"). S5 is characterized by the axioms: *K: \Box(A\to B)\to(\Box A\to\Box B); *T: \Box A \to A, and either: * 5: \Diamond A\to \Box\Diamond A; * or both of the following: :* 4: \Box A\to\Box\Box A, and :* B: A\to\Box\Diamond A. The (5) axiom restricts the accessibility relation R of the Kripke frame to be Euclidean, i.e. (wRv \land wRu) \implies vRu . Kripke semantics In terms of Kripke seman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?Beall, JC and Restall, Greg, Logical Consequence' The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical conse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Formula
In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula. A propositional formula is constructed from simple propositions, such as "five is greater than three" or propositional variables such as ''p'' and ''q'', using connectives or logical operators such as NOT, AND, OR, or IMPLIES; for example: : (''p'' AND NOT ''q'') IMPLIES (''p'' OR ''q''). In mathematics, a propositional formula is often more briefly referred to as a "proposition", but, more precisely, a propositional formula is not a proposition but a formal expression that ''denotes'' a proposition, a formal object under discussion, just like an expression such as "" is not a value, but denotes a value. In some contexts, maintaining the distincti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modal Operator
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. In the case of alethic modal logic, a modal operator can be said to be truth-functional in another sense, namely, that of being sensitive only to the distribution of truth-values across possible worlds, actual or not. Finally, a modal operator is "intuitively" characterized by expressing a modal attitude (such as necessity, possibility, belief, or knowledge) about the proposition to which the operator is applied. See also Garson, James, "Modal Logic", The Stanford Encyclopedia of Philosophy (Summer 2021 Edition), Edward N. Zalta (ed.), URL = Syntax for modal operators The syntax rules for modal operators \Box and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
