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Strict Implication
In logic, a strict conditional (symbol: \Box, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic. For any two propositions ''p'' and ''q'', the formula ''p'' → ''q'' says that ''p'' materially implies ''q'' while \Box (p \rightarrow q) says that ''p'' strictly implies ''q''. Strict conditionals are the result of Clarence Irving Lewis's attempt to find a conditional for logic that can adequately express indicative conditionals in natural language. They have also been used in studying Molinist theology. Avoiding paradoxes The strict conditionals may avoid paradoxes of material implication. The following statement, for example, is not correctly formalized by material implication: : If Bill Gates has graduated in Medicine, then Elvis never died. This condition should clearly be false: the degree of Bi ... [...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] |
Counterfactual Conditional
Counterfactual conditionals (also ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood. Counterfactuals are one of the most studied phenomena in philosophical logic, formal semantics, and philosophy of language. They were first discussed as a problem for the material conditional analysis of conditionals, which treats them all as trivially true. Starting in the 1960s, philosophers and linguists developed the now-classic possible world approach, in which a counterfactual's truth hinges on its consequent holding at certain possible worlds w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Import-Export (logic)
Import and export or import/export may refer to: * Import and export of goods ** International trade ** Import/export regulations, trade regulations of such goods ** Import/export tariffs, taxes on the trade in such goods * Import and export of data in computing, the moving of data between applications ** Import and export of formats, data conversion from one file type to another * ''Import/Export'', a 2007 Austrian film * An ''import statement'' used as a form or analog of an include directive Many programming languages and other computer files have a directive, often called include (sometimes copy or import), that causes the contents of the specified file to be inserted into the original file. These included files are called copybooks ... in some programming languages * Import–export (logic), a form of deductive argument in classical logic {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Semantics
Dynamic semantics is a framework in logic and natural language semantics that treats the meaning of a sentence as its potential to update a context. In static semantics, knowing the meaning of a sentence amounts to knowing when it is true; in dynamic semantics, knowing the meaning of a sentence means knowing "the change it brings about in the information state of anyone who accepts the news conveyed by it." In dynamic semantics, sentences are mapped to functions called ''context change potentials'', which take an input context and return an output context. Dynamic semantics was originally developed by Irene Heim and Hans Kamp in 1981 to model anaphora, but has since been applied widely to phenomena including presupposition, plurals, questions, discourse relations, and modality. Dynamics of anaphora The first systems of dynamic semantics were the closely related ''File Change Semantics'' and ''discourse representation theory'', developed simultaneously and independently by Iren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counterfactual Conditional
Counterfactual conditionals (also ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood. Counterfactuals are one of the most studied phenomena in philosophical logic, formal semantics, and philosophy of language. They were first discussed as a problem for the material conditional analysis of conditionals, which treats them all as trivially true. Starting in the 1960s, philosophers and linguists developed the now-classic possible world approach, in which a counterfactual's truth hinges on its consequent holding at certain possible worlds w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corresponding Conditional
In logic, the corresponding conditional of an argument (or derivation) is a material conditional whose antecedent is the conjunction of the argument's (or derivation's) premises and whose consequent is the argument's conclusion. An argument is valid if and only if its corresponding conditional is a logical truth. It follows that an argument is valid if and only if the negation of its corresponding conditional is a contradiction. Therefore, the construction of a corresponding conditional provides a useful technique for determining the validity of an argument. Example Consider the argument A: Either it is hot or it is cold It is not hot Therefore it is cold This argument is of the form: Either P or Q Not P Therefore Q or (using standard symbols of propositional calculus): P Q P ____________ Q The corresponding conditional C is: IF ((P or Q) and not P) THEN Q or (using standard symbols): ((P Q) P) Q and the argument A is valid just in case the corresponding condit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion (computer Science)
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science. Most computer programming languages support recursion by allowing a function to call itself from within its own code. Some functional programming languages (for instance, Clojure) do not define any looping constructs but rely solely on recursion to repeatedly call code. It is proved in computability theory that these recursive-only languages are Turing complete; this means that they are as powerful (they can be used to solve the same problems) as imperative languages based on control structures such as and . Repeatedly calling a function from within itself may cause the call stack to have a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arrow (computer Science)
In computer science, arrows or bolts are a type class used in programming to describe computations in a pure and declarative fashion. First proposed by computer scientist John Hughes as a generalization of monads, arrows provide a referentially transparent way of expressing relationships between ''logical'' steps in a computation. Unlike monads, arrows don't limit steps to having one and only one input. As a result, they have found use in functional reactive programming, point-free programming, and parsers among other applications. Motivation and history While arrows were in use before being recognized as a distinct class, it wasn't until 2000 that John Hughes first published research focusing on them. Until then, monads had proven sufficient for most problems requiring the combination of program logic in pure code. However, some useful libraries, such as the Fudgets library for graphical user interfaces and certain efficient parsers, defied rewriting in a monadic form. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Arithmetic
In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism.Troelstra 1973:18 It is named after Arend Heyting, who first proposed it. Axiomatization As with first-order Peano arithmetic , the intended model of this theory are the natural numbers and the theories characterize addition and multiplication. Heyting arithmetic adopts the axioms of Peano arithmetic, including the signature with zero "0" and the successor "S", but uses intuitionistic logic for inference. In particular, the principle of the excluded middle does not hold in general. Metalogic and theorems As with other theories over intuitionistic logic, various instances of can be proven. For instance, proves equality "=" is decidable for all numbers, :\vdash \forall n. \forall m. \big((n = m)\lor\neg(n = m)\big) In fact, since equality is the only predicate symbol in Heyting arithmetic, it then follows that, for any quantifier-free formula \phi, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpretability
In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other. Informal definition Assume ''T'' and ''S'' are formal theories. Slightly simplified, ''T'' is said to be ''interpretable'' in ''S'' if and only if the language of ''T'' can be translated into the language of ''S'' in such a way that ''S'' proves the translation of every theorem of ''T''. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logical structure of formulas. This concept, together with weak interpretability, was introduced by Alfred Tarski in 1953. Three other related concepts are cointerpretability, logical tolerance, and cotolerance, introduced by Giorgi Japaridze in 1992–93. See also * Interpretation (logic) * Interpretation (model theory) * Interpretability logic References * Japaridze, G., and De Jongh, D. (1998) "The logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relevance Logic
Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but not universally, called ''relevant logic'' by British and, especially, Australian logicians, and ''relevance logic'' by American logicians. Relevance logic aims to capture aspects of implication that are ignored by the " material implication" operator in classical truth-functional logic, namely the notion of relevance between antecedent and conditional of a true implication. This idea is not new: C. I. Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic grants paradoxes of material implication such as the principle that a falsehood implies any proposition. Hence "if I'm a donkey, then two and two is four" is true when translated as a material implication, yet it seems intuitiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
