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Converse Implication
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposition ''All S are P'', the converse is ''All P are S''. Either way, the truth of the converse is generally independent from that of the original statement.Robert Audi, ed. (1999), ''The Cambridge Dictionary of Philosophy'', 2nd ed., Cambridge University Press: "converse". Implicational converse Let ''S'' be a statement of the form ''P implies Q'' (''P'' → ''Q''). Then the converse of ''S'' is the statement ''Q implies P'' (''Q'' → ''P''). In general, the truth of ''S'' says nothing about the truth of its converse, unless the antecedent ''P'' and the consequent ''Q'' are logically equivalent. For example, consider the true statement "If I am a human, then I am mortal." The converse of that statement is "If I am mortal, then I am ... [...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] |
Gunther Schmidt
Gunther Schmidt (born 1939, Rüdersdorf) is a German mathematician who works also in informatics. Life Schmidt began studying Mathematics in 1957 at Göttingen University. His academic teachers were in particular Kurt Reidemeister, Wilhelm Klingenberg and Karl Stein. In 1960 he transferred to Ludwig-Maximilians-Universität München where he studied functions of several complex variables with Karl Stein. Schmidt wrote a thesis on analytic continuation of such functions. In 1962 Schmidt began work at TU München with students of Robert Sauer, in the beginning in labs and tutorials, later in mentoring and administration. Schmidt's interests turned toward programming when he collaborated with Hans Langmaack on rewriting and the braid group in 1969. Friedrich L. Bauer and Klaus Samelson were establishing software engineering at the university and Schmidt joined their group in 1974. In 1977 he submitted his Habilitation "Programs as partial graphs". He became a professor in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse (logic)
In logic, an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence. More specifically, given a conditional sentence of the form P \rightarrow Q , the inverse refers to the sentence \neg P \rightarrow \neg Q . Since an inverse is the contrapositive of the converse, inverse and converse are logically equivalent to each other. For example, substituting propositions in natural language for logical variables, the inverse of the following conditional proposition :"If it's raining, then Sam will meet Jack at the movies." would be :"If it's not raining, then Sam will not meet Jack at the movies." The inverse of the inverse, that is, the inverse of \neg P \rightarrow \neg Q , is \neg \neg P \rightarrow \neg \neg Q , and since the double negation of any statement is equivalent to the original statement in classical logic, the inverse of the inverse is logically equivalent to the original conditional P \rightarrow Q . Thus it is p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle (300s BCE). Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular evidence to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce, contradistinguishing abduction from induction. Various fields study how inference is done in practice. Human inference (i.e. how humans draw conclusions) is traditionally studied within the fields of logic, argumentation studies, and cognitive psychology; artificial intelligence researchers develop automated inference systems to emulate human inference. Statistical inference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse (semantics)
In linguistics, converses or relational antonyms are pairs of words that refer to a relationship from opposite points of view, such as ''parent/child'' or ''borrow/lend''. The relationship between such words is called a converse relation. Converses can be understood as a pair of words where one word implies a relationship between two objects, while the other implies the existence of the same relationship when the objects are reversed. Converses are sometimes referred to as complementary antonyms because an "either/or" relationship is present between them. One exists only because the other exists. List of converse words * ''Own'' and ''belong'' are relational opposites i.e. "A owns B" is the same as "B belongs to A." * ''Win'' and ''lose'' i.e. if someone wins, someone must lose. * '' Fraction'' and ''whole'' i.e. if there is a fraction, there must be a whole. * ''Above'' and ''below'' * ''Employer'' and ''employee'' * ''Parent'' and ''child'' * ''Teacher'' and ''student'' * ''Buy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contraposition
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as Proof by contrapositive, proof by contraposition. The contrapositive of a statement has its Antecedent (logic), antecedent and consequent Inverse (logic), inverted and Conversion (logic), flipped. Material conditional, Conditional statement P \rightarrow Q. In Logical connective, formulas: the contrapositive of P \rightarrow Q is \neg Q \rightarrow \neg P . If ''P'', Then ''Q''. — If not ''Q'', Then not ''P''. ''"''If ''it is raining,'' then ''I wear my coat" —'' "If ''I don't wear my coat,'' then ''it isn't raining."'' The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true. The contrapositive ( \neg Q \rightarrow \neg P ) can be compared with three other statements: ;Inverse (logic), Inversion (the inverse), \neg P \rightarrow \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Proposition
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments using categorical statements (i.e., syllogisms) forms an important branch of deductive reasoning that began with the Ancient Greeks. The Ancient Greeks such as Aristotle identified four primary distinct types of categorical proposition and gave them standard forms (now often called ''A'', ''E'', ''I'', and ''O''). If, abstractly, the subject category is named ''S'' and the predicate category is named ''P'', the four standard forms are: *All ''S'' are ''P''. (''A'' form, \forall _\rightarrow P_xequiv \forall neg S_\lor P_x/math>) *No ''S'' are ''P''. (''E'' form, \forall _\rightarrow \neg P_xequiv \forall neg S_\lor \neg P_x/math>) *Some ''S'' are ''P''. (''I'' form, \exists _\land P_x/math>) *Some ''S'' are not ''P''. (''O'' form, \ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of philosophy within the Lyceum and the wider Aristotelian tradition. His writings cover many subjects including physics, biology, zoology, metaphysics, logic, ethics, aesthetics, poetry, theatre, music, rhetoric, psychology, linguistics, economics, politics, meteorology, geology, and government. Aristotle provided a complex synthesis of the various philosophies existing prior to him. It was above all from his teachings that the West inherited its intellectual lexicon, as well as problems and methods of inquiry. As a result, his philosophy has exerted a unique influence on almost every form of knowledge in the West and it continues to be a subject of contemporary philosophical discussion. Little is known about his life. Aristotle was born in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists''"'' is a quantifier, while ''x'' is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of ax ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (defined by Aristotle in his 350 BCE book '' Prior Analytics''), a syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across. For example, knowing that all men are mortal (major premise) and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form: All men are mortal. Socrates is a man. Therefore, Socrates is mortal.In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. From the Middle Ages onwards, ''categorical syllogism'' and ''syllogism'' were usually used interchangeably. This a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution Of Terms
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments using categorical statements (i.e., syllogisms) forms an important branch of deductive reasoning that began with the Ancient Greeks. The Ancient Greeks such as Aristotle identified four primary distinct types of categorical proposition and gave them standard forms (now often called ''A'', ''E'', ''I'', and ''O''). If, abstractly, the subject category is named ''S'' and the predicate category is named ''P'', the four standard forms are: *All ''S'' are ''P''. (''A'' form, \forall _\rightarrow P_xequiv \forall neg S_\lor P_x/math>) *No ''S'' are ''P''. (''E'' form, \forall _\rightarrow \neg P_xequiv \forall neg S_\lor \neg P_x/math>) *Some ''S'' are ''P''. (''I'' form, \exists _\land P_x/math>) *Some ''S'' are not ''P''. (''O'' form, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James H
James is a common English language surname and given name: *James (name), the typically masculine first name James * James (surname), various people with the last name James James or James City may also refer to: People * King James (other), various kings named James * Saint James (other) * James (musician) * James, brother of Jesus Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Arts, entertainment, and media * ''James'' (2005 film), a Bollywood film * ''James'' (2008 film), an Irish short film * ''James'' (2022 film), an Indian Kannada-language film * James the Red Engine, a character in ''Thomas the Tank En ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
