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Strict Conditional
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 graduated in medicine, then Elvis never died. This condition should clearly be false: the degree of Bill ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Consequent
A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called the consequent. In some contexts, the consequent is called the ''apodosis''.See Conditional sentence. Examples: * If P, then Q. Q is the consequent of this hypothetical proposition. * If X is a mammal, then X is an animal. Here, "X is an animal" is the consequent. * If computers can think, then they are alive. "They are alive" is the consequent. The consequent in a hypothetical proposition is not necessarily a consequence of the antecedent. * If monkeys are purple, then fish speak Klingon. "Fish speak Klingon" is the consequent here, but intuitively is not a consequence of (nor does it have anything to do with) the claim made in the antecedent that "monkeys are purple". See also * Antecedent (logic) * Conjecture * Necessity and s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Counterfactual Conditional
Counterfactual conditionals (also ''contrafactual'', ''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 certai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Indagationes Mathematicae
''Indagationes Mathematicae'' (Latin for "Mathematical Investigations") is a Dutch mathematics journal. The journal originates from the ''Proceedings of the Royal Netherlands Academy of Arts and Sciences'' (or ''Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen''), founded in 1895. From 1939, mathematics articles in this journal were published separately, under the alternative title ''Indagationes Mathematicae''. In 1951 the proceedings officially split into three journals, keeping the same name but distinguished from each other by being in separate series. They were Series A (Mathematical Sciences), Series B (Physical Sciences), and Series C (Biological and Medical Sciences). At that time, Series A became published by the North-Holland Publishing Company; the volumes from this time are now listed by the publisher as ''Indagationes Mathematicae (Proceedings)''. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 recursion, recursive problems by using function (computer science), 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 itse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Heyting Arithmetic
In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. It is named after Arend Heyting, who first proposed it. Axiomatization Heyting arithmetic can be characterized just like the first-order theory of Peano arithmetic , except that it uses the intuitionistic predicate calculus for inference. In particular, this means that the double-negation elimination principle, as well as the principle of the excluded middle , do not hold. Note that to say does not hold exactly means that the excluded middle statement is not automatically provable for all propositions—indeed many such statements are still provable in and the negation of any such disjunction is inconsistent. is strictly stronger than in the sense that all -theorems are also -theorems. Heyting arithmetic comprises the axioms of Peano arithmetic and the intended model is the collection of natural numbers . The signature includes zero "0" and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 * Conservative extension * Interpretation (logic) * Interpretation (model theory) * Interpretability logic References * Japaridze, G., a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Constructive Logic
Constructive logic is a family of logics where proofs must be constructive (i.e., proving something means one must build or exhibit it, not just argue it “must exist” abstractly). No “non-constructive” proofs are allowed (like the classic proof by contradiction without a witness). The main constructive logics are the following: 1. Intuitionistic logic Founder: L. E. J. Brouwer (1908, philosophy) formalized by A. Heyting (1930) and A. N. Kolmogorov (1932) Key Idea: Truth = having a proof. One cannot assert “P or not P” unless one can prove P or prove \neg \neg P. Features: * No law of excluded middle (P \lor \neg P is not generally valid). * No double negation elimination (\neg \neg\ P \to P is not valid generally). * Implication is constructive: a proof of P \to Q is a method turning any proof of P into a proof of Q. Used in: type theory, constructive mathematics. 2. Modal logics for constructive reasoning Founder(s): * K F. Gödel (1933) showed that intui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Conversational Implicature
In pragmatics, a subdiscipline of linguistics, an implicature is something the speaker suggests or implies with an utterance, even though it is not literally expressed. Implicatures can aid in communicating more efficiently than by explicitly saying everything we want to communicate. The philosopher H. P. Grice coined the term in 1975. Grice distinguished ''conversational'' implicatures, which arise because speakers are expected to respect general rules of conversation, and ''conventional'' ones, which are tied to certain words such as "but" or "therefore". Take for example the following exchange: : A (to passerby): I am out of gas. : B: There is a gas station 'round the corner. Here, B does not say, but ''conversationally implicates'', that the gas station is open, because otherwise his utterance would not be relevant in the context. Conversational implicatures are classically seen as contrasting with entailments: they are not necessary or logical consequences of what is said, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |