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Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of an object language. For example, an interpretation function could take the predicate symbol T and assign it the extension \. All our interpretation does is assign the extension \ to the non-logical symbol T, and does not make a claim about whether T is to stand for tall and \mathrm for Abraham Lincoln. On the other hand, an interpretation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meaning (philosophy)
In philosophymore specifically, in its sub-fields semantics, semiotics, philosophy of language, metaphysics, and metasemanticsmeaning "is a relationship between two sorts of things: signs and the kinds of things they intend, express, or signify". The types of meanings vary according to the types of the thing that is being represented. There are: *the things, which might have meaning; *things that are also signs of other things, and therefore are always meaningful (i.e., natural signs of the physical world and ideas within the mind); *things that are necessarily meaningful, such as words and nonverbal symbols. The major contemporary positions of meaning come under the following partial definitions of meaning: *psychological theories, involving notions of thought, intention, or understanding; *logical theories, involving notions such as intension, cognitive content, or sense, along with extension, reference, or denotation; *message, content, information, or communication; * trut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well Formed Formula
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. The abbreviation wff is pronounced "woof", or sometimes "wiff", "weff", or "whiff". A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic. Introduction A key use of formulas is in propositional logic and predicate logic such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven. Although th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-logical Symbol
In logic, the formal languages used to create expressions consist of symbols, which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes also called logical and non-logical constants). The non-logical symbols of a language of first-order logic consist of predicates and individual constants. These include symbols that, in an interpretation, may stand for individual constants, variables, functions, or predicates. A language of first-order logic is a formal language over the alphabet consisting of its non-logical symbols and its logical symbols. The latter include logical connectives, quantifiers, and variables that stand for statements. A non-logical symbol only has meaning or semantic content when one is assigned to it by means of an interpretation. Consequently, a sentence containing a non-logical symbol lacks meaning except under an interpretation, so a sentence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Variable
In mathematical logic, a propositional variable (also called a sentence letter, sentential variable, or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order logics. Uses Formulas in logic are typically built up recursively from some propositional variables, some number of logical connectives, and some logical quantifiers. Propositional variables are the atomic formulas of propositional logic, and are often denoted using capital roman letters such as P, Q and R. ;Example In a given propositional logic, a formula can be defined as follows: * Every propositional variable is a formula. * Given a formula ''X'', the negation ''¬X'' is a formula. * Given two formulas ''X'' and ''Y'', and a binary connective ''b'' (such as the logical conjunction ∧), the expression ''(X b Y)'' is a formula. (Note the parenth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-to-one
In systems analysis, a one-to-many relationship is a type of cardinality (data modeling), cardinality that refers to the relationship between two wikt:entity, entities (see also entity–relationship model). For example, take a car and an owner of the car. The car can only be owned by one owner at a time or not owned at all, and an owner could own zero, one, or multiple cars. One owner could have many cars, one-to-many. In a relational database, a one-to-many relationship exists when one record is related to many records of another table. A one-to-many relationship is not a property of the data, but rather of the relationship itself. One-to-many often refer to a primary key to foreign key relationship between two tables, where the record in the first table can relate to multiple records in the second table. A foreign key is one side of the relationship that shows a row or multiple rows, with one of those rows being the primary key already listed on the first table. This is also cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Consequence
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more statements. A Validity (logic), valid logical argument is one in which the Consequent, 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 logical truth, necessary and Formalism (philosophy of mathematics), formal, by wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistency
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when there is no formula \varphi such that \varphi \in \langle A \rangle and \lnot \varphi \in \langle A \rangle. A ''trivial'' theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
From If To Is
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From may refer to: People *Isak From (born 1967), Swedish politician *Martin Severin From (1825–1895), Danish chess master * Sigfred From (1925–1998), Danish chess master Media * ''From'' (TV series), a sci-fi-horror series that debuted on Epix in 2022 * "From" (Fromis 9 song) (2024) * "From", a song by Big Thief from U.F.O.F. (2019) * "From", a song by Yuzu (2010) * "From", a song by Bon Iver from Sable, Fable (2025) Other * From, a preposition * From (SQL), computing language keyword * From: (email message header), field showing the sender of an email * FromSoftware, a Japanese video game company * Full range of motion, the travel in a range of motion Range of motion (or ROM) is the linear or angular distance that a moving object may normally travel while properly attached to another. In biomechanics and strength training, ROM refers to the angular distance and direction a joint can move be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graham Priest
Graham Priest (born 1948) is a philosopher and logician who is distinguished professor of philosophy at the CUNY Graduate Center, as well as a regular visitor at the University of Melbourne, where he was Boyce Gibson Professor of Philosophy and also at the University of St Andrews. Life Priest was educated at St John's College, Cambridge and the London School of Economics. His thesis advisor was John Lane Bell. He also holds a DLitt from the University of Melbourne. Priest was elected a corresponding fellow of the Australian Academy of the Humanities in 1995. In addition to his work in philosophy and logic, Priest practised karate-do. He is 3rd dan, International Karate-do Shobukai; 4th dan, shitō-ryū, and an Australian National kumite referee and kata judge. Presently, he practices tai chi. Philosophical work Priest is known for his defence of dialetheism, his in-depth analyses of the logical paradoxes (holding the thesis that there is a uniform treatment for many ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Logic
Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class shares characteristic properties: Gabbay, Dov, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press. # Law of excluded middle and double negation elimination # Law of noncontradiction, and the principle of explosion # Monotonicity of entailment and idempotency of entailment # Commutativity of conjunction # De Morgan duality: every logical operator is dual to another While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics. Shapiro, Stewart (2000). Classical Logic. In St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Satisfiable
In mathematical logic, a formula is ''satisfiable'' if it is true under some assignment of values to its variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, while the formula x+1=x is not satisfiable over the integers. The dual concept to satisfiability is validity; a formula is ''valid'' if every assignment of values to its variables makes the formula true. For example, x+3=3+x is valid over the integers, but x+3=y is not. Formally, satisfiability is studied with respect to a fixed logic defining the syntax of allowed symbols, such as first-order logic, second-order logic or propositional logic. Rather than being syntactic, however, satisfiability is a semantic property because it relates to the ''meaning'' of the symbols, for example, the meaning of + in a formula such as x+1=x. Formally, we define an interpretation (or model) to be an assignment of values to the variables and an assignment of meaning to all other non-logical symbols, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective \lor can be used to join the two atomic formulas P and Q, rendering the complex formula P \lor Q . Common connectives include negation, disjunction, conjunction, implication, and equivalence. In standard systems of classical logic, these connectives are interpreted as truth functions, though they receive a variety of alternative interpretations in nonclassical logics. Their classical interpretations are similar to the meanings of natural language expressions such as English "not", "or", "and", and "if", but not identical. Discrepancies between natural language connectives and those of classical logic have motivated nonclassical approaches to natural language meaning as well as approaches which pair a classi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
