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Inference Rules
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inference called ''modus ponens'' takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion. Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursive are important; i.e. rule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophy Of Logic
Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application. This involves questions about how logic is to be defined and how different logical systems are connected to each other. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines. According to a common characterization, philosophical logic is the part of the philosophy of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. But other theorists draw the distinction between the philosophy of logic and philosophical logic differently or not at all. Metalogic is closely related to the philosophy of logic as the discipline investigating the properties of formal logical systems, like consi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schema (logic)
In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language. The logical form of an argument is called the argument form of the argument. History The importance of the concept of form to logic was already recognized in ancient times. Aristotle, in the '' Prior Analytics'', was probably the first to employ variable letters to represent valid inferences. Therefore, Jan Łukasiewicz claims that the introduction of variables was "one of Aristotle's greatest inventions." According ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...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 Lemberg in Austria-Hungary (now Lviv, Ukraine; pl, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-valued Logic
In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating ''true'', ''false'' and some indeterminate third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for ''true'' and ''false''. Emil Leon Post is credited with first introducing additional logical truth degrees in his 1921 theory of elementary propositions. The conceptual form and basic ideas of three-valued logic were initially published by Jan Łukasiewicz and Clarence Irving Lewis. These were then re-formulated by Grigore Constantin Moisil in an axiomatic algebraic form, and also extended to ''n''-valued logics in 1945. Pre-discovery Around 1910, Charles Sanders Peirce defined a many-valued logic system. He never published it. In fact, he did not even number the three pages of notes where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lewis Carroll
Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ''Through the Looking-Glass'' (1871). He was noted for his facility with word play, logic, and fantasy. His poems ''Jabberwocky'' (1871) and ''The Hunting of the Snark'' (1876) are classified in the genre of literary nonsense. Carroll came from a family of high-church Anglicanism, Anglicans, and developed a long relationship with Christ Church, Oxford, where he lived for most of his life as a scholar and teacher. Alice Liddell, the daughter of Christ Church's dean Henry Liddell, is widely identified as the original inspiration for ''Alice in Wonderland'', though Carroll always denied this. An avid puzzler, Carroll created the word ladder puzzle (which he then called "Doublets"), which he published in his weekly column for ''Vanity Fair ( ... [...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. They 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, and implication. 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 classical compositional semantics wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deductive Reasoning
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and "Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is ''sound'' if it is ''valid'' and all its premises are true. Some theorists define deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning. Psychology is interested in deductive reasoning as a psychological process, i.e. how people ''actually'' draw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deduction Theorem
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs—to prove an implication ''A'' → ''B'', assume ''A'' as an hypothesis and then proceed to derive ''B''—in systems that do not have an explicit inference rule for this. Deduction theorems exist for both propositional logic and first-order logic. The deduction theorem is an important tool in Hilbert-style deduction systems because it permits one to write more comprehensible and usually much shorter proofs than would be possible without it. In certain other formal proof systems the same conveniency is provided by an explicit inference rule; for example natural deduction calls it implication introduction. In more detail, the propositional logic deduction theorem states that if a formula B is deducible from a set of assumptions \Delta \cup \ then the implication A \to B is deducible from \Delta ; in symbols, \Delta \cup \ \vdash B implies \Delta \vdash A \to B . In the sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequent
In mathematical logic, a sequent is a very general kind of conditional assertion. : A_1,\,\dots,A_m \,\vdash\, B_1,\,\dots,B_n. A sequent may have any number ''m'' of condition formulas ''Ai'' (called " antecedents") and any number ''n'' of asserted formulas ''Bj'' (called "succedents" or "consequents"). A sequent is understood to mean that if all of the antecedent conditions are true, then at least one of the consequent formulas is true. This style of conditional assertion is almost always associated with the conceptual framework of sequent calculus. Introduction The form and semantics of sequents Sequents are best understood in the context of the following three kinds of logical judgments: Unconditional assertion. No antecedent formulas. * Example: ⊢ ''B'' * Meaning: ''B'' is true. Conditional assertion. Any number of antecedent formulas. Simple conditional assertion. Single consequent formula. * Example: ''A1'', ''A2'', ''A3'' ⊢ ''B'' * Meaning: IF ''A1'' AND ''A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert System
:''In mathematical physics, ''Hilbert system'' is an infrequently used term for a physical system described by a C*-algebra.'' In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob FregeMáté & Ruzsa 1997:129 and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well. Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference. Hilbert systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference. Systems of natural deduction take the opposite tack, including many deduction rules but very few or no axiom schemes. The most commonly studied Hilbert systems have either just one rule of inference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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