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If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...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 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] |
Jan Łukasiewicz
Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. His work centred on philosophical logic, mathematical logic and history of logic. He thought innovatively about traditional propositional logic, the principle of non-contradiction and the law of excluded middle, offering one of the earliest systems of many-valued logic. Contemporary research on Aristotelian logic also builds on innovative works by Łukasiewicz, which applied methods from modern logic to the formalization of Aristotle's syllogistic. The Łukasiewicz approach was reinvigorated in the early 1970s in a series of papers by John Corcoran and Timothy Smiley that inform modern translations of '' Prior Analytics'' by Robin Smith in 1989 and Gisela Striker in 2009. Łukasiewicz is regarded as one of the most important historians of logic. Life He was born in Lwów in Austria-Hungary (now Lviv, Ukr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Diagram
An Euler diagram (, ) is a diagrammatic means of representing Set (mathematics), sets and their relationships. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are similar to another set diagramming technique, Venn diagrams. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram shows only relevant relationships. The first use of "Eulerian circles" is commonly attributed to Swiss mathematician Leonhard Euler (1707–1783). In the United States, both Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement of the 1960s. Since then, they have also been adopted by other curriculum fields such as reading as well as organizations and businesses. Euler diagrams consist of simple closed shapes in a two-dimensional plane that each depict a set or category. How or whether these shapes overlap demonstrates the relationships between the sets. Ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes).Lyons, John. "Semantics, vol. I." Cambridge: Cambridge (1977). p.158 and on. Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what the mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. Basic terminology In modern usage, a definition is something, typically expressed in words, that at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definitions
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes).Lyons, John. "Semantics, vol. I." Cambridge: Cambridge (1977). p.158 and on. Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what the mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. Basic terminology In modern usage, a definition is something, typically expressed in words, tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consonant Gemination
In phonetics and phonology, gemination (; from Latin 'doubling', itself from '' gemini'' 'twins'), or consonant lengthening, is an articulation of a consonant for a longer period of time than that of a singleton consonant. It is distinct from stress. Gemination is represented in many writing systems by a doubled letter and is often perceived as a doubling of the consonant.William Ham, ''Phonetic and Phonological Aspects of Geminate Timing'', p. 1–18 Some phonological theories use 'doubling' as a synonym for gemination, while others describe two distinct phenomena. Consonant length is a distinctive feature in certain languages, such as Japanese. Other languages, such as Greek, do not have word-internal phonemic consonant geminates. Consonant gemination and vowel length are independent in languages like Arabic, Japanese, Hungarian, Malayalam, and Finnish; however, in languages like Italian, Norwegian, and Swedish, vowel length and consonant length are interdependent. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euphony
Phonaesthetics (also spelled phonesthetics in North America) is the study of the beauty and pleasantness associated with the sounds of certain words or parts of words. The term was first used in this sense, perhaps by during the mid-20th century and derives . Speech sounds have many aesthetic qualities, some of which are subjectively regarded as ''euphonious'' (pleasing) or ''cacophonous'' (displeasing). Phonaesthetics remains a budding and often subjective field of study, with no scientifically or otherwise formally established definition; today, it mostly exists as a marginal branch of psychology, phonetics, or poetics. More broadly, the British linguist David Crystal has regarded phonaesthetics as the study of "phonaesthesia" (i.e., sound symbolism and phonesthemes): that not just words but even certain sound combinations carry meaning. For example, he shows that English speakers tend to associate unpleasantness with the sound ''sl-'' in such words as ''sleazy'', ''slime'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Halmos
Paul Richard Halmos (; 3 March 1916 – 2 October 2006) was a Kingdom of Hungary, Hungarian-born United States, American mathematician and probabilist who made fundamental advances in the areas of mathematical logic, probability theory, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor. He has been described as one of The Martians (scientists), The Martians. Early life and education Born in the Kingdom of Hungary into a History of the Jews in Hungary, Jewish family, Halmos immigrated to the United States at age 13. He obtained his B.A. from the University of Illinois at Urbana-Champaign, University of Illinois, majoring in mathematics while also fulfilling the requirements for a degree in philosophy. He obtained the degree after only three years, and was 19 years old when he graduated. He then began a Ph.D. in philosophy, still at the Champaign–Urbana campus. However, after failin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John L
"John L" is a song by English rock band Black Midi, released in 2021 as the lead single from their second studio album, ''Cavalcade (Black Midi album), Cavalcade''. The song describes the story of a powerful leader, the titular John L, who is eventually betrayed and killed by his followers. It was released on March 23, with the B-side Despair and a music video directed by Nina McNeely. A 12-inch release for the single was made available for pre-order on the same day and released on April 9. The song is one of few on ''Cavalcade'' to have writing credits for guitarist Matt Kwasniewski-Kelvin, written before his departure from the band but recorded after. Composition and recording "John L" is an Avant-garde music, avant-garde progressive rock song described by ''Guitar World'' as "[featuring] dissonant piano chimes, weaving hypnotic vocals, a cacophony of string sounds, and an edge-of-the-seat dynamic range, spanning from complete silence to raucous, high-energy midsections." ''Mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truth-function
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: the input and output of a truth function are all truth values; a truth function will always output exactly one truth value, and inputting the same truth value(s) will always output the same truth value. The typical example is in propositional logic, wherein a compound statement is constructed using individual statements connected by logical connectives; if the truth value of the compound statement is entirely determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and any logical connectives used are said to be truth functional. Classical propositional logic is a truth-functional logic, in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunction
In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula S \lor W , assuming that S abbreviates "it is sunny" and W abbreviates "it is warm". In classical logic, disjunction is given a truth functional semantics according to which a formula \phi \lor \psi is true unless both \phi and \psi are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an ''inclusive'' interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Theory
Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
