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Tautology (rule Of Inference)
In propositional logic, tautology is either of two commonly used rules of replacement. The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs. They are: The principle of idempotency of disjunction: : P \lor P \Leftrightarrow P and the principle of idempotency of conjunction: : P \land P \Leftrightarrow P Where "\Leftrightarrow" is a metalogical symbol representing "can be replaced in a logical proof with." Formal notation Theorems are those logical formulas \phi where \vdash \phi is the conclusion of a valid proof,Logic in Computer Science, ''p. 13'' while the equivalent semantic consequence \models \phi indicates a tautology. The ''tautology'' rule may be expressed as a sequent: : P \lor P \vdash P and : P \land P \vdash P where \vdash is a metalogical symbol meaning that P is a syntactic consequence of P \lor P, in the one case, P \land P in the other, in some logical system; or as a rule of inference: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" ( ... [...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] |
Proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the non-linguistic bearer of truth or falsity which makes any sentence that expresses it either true or false. While the term "proposition" may sometimes be used in everyday language to refer to a linguistic statement which can be either true or false, the technical philosophical term, which differs from the mathematical usage, refers exclusively to the non-linguistic meaning behind the statement. The term is often used very broadly and can also refer to various related concepts, both in the history of philosophy and in contemporary analytic philosophy. It can generally be used to refer to some or all of the following: The primary bearers of truth values (such as "true" and "false"); the objects of belief and other propositional attitudes (i.e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–1927, it appeared in a second edition with an important ''Introduction to the Second Edition'', an ''Appendix A'' that replaced ✸9 and all-new ''Appendix B'' and ''Appendix C''. ''PM'' is not to be confused with Russell's 1903 ''The Principles of Mathematics''. ''PM'' was originally conceived as a sequel volume to Russell's 1903 ''Principles'', but as ''PM'' states, this became an unworkable suggestion for practical and philosophical reasons: "The present work was originally intended by us to be comprised in a second volume of ''Principles of Mathematics''... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found application to a wide variety of disciplines, including ecology, theology, education, physics, biology, economics, and psychology, among other areas. In his early career Whitehead wrote primarily on mathematics, logic, and physics. His most notable work in these fields is the three-volume ''Principia Mathematica'' (1910–1913), which he wrote with former student Bertrand Russell. ''Principia Mathematica'' is considered one of the twentieth century's most important works in mathematical logic, and placed 23rd in a list of the top 100 English-language nonfiction books of the twentieth century by Modern Library. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, artificial intelligence, cognitive science, computer science and various areas of analytic philosophy, especially philosophy of mathematics, philosophy of language, epistemology, and metaphysics.Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003. He was one of the early 20th century's most prominent logicians, and a founder of analytic philosophy, along with his predecessor Gottlob Frege, his friend and colleague G. E. Moore and his student and protégé Ludwig Wittgenstein. Russell with Moore led the British "revolt against idealism". Together with his former teacher A. N. Whitehead, Russell wrote ''Principia Mathematica'', a milestone in the development of classical logic, and a major attempt to reduce the whole ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Inference
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. rules suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal System
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined abstraction, system of abstract thought. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Background Each formal system is described by primitive Symbol (formal), symbols (which collectively form an Alphabet (computer science), alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?Beall, JC and Restall, Greg, Logical Consequence' The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical conse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semantic Consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?Beall, JC and Restall, Greg, Logical Consequence' The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical consequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Replacement
In logic, a rule of replacementMoore and Parker is a transformation rule that may be applied to only a particular segment of an logical expression, expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for Well-formed formula, logical expressions in the system. Whereas a rule of inference is always applied to a whole logical expression, a rule of replacement may be applied to only a particular segment. Within the context of a logical proof, logically equivalent expressions may replace each other. Rules of replacement are used in propositional logic to manipulate propositions. Common rules of replacement include de Morgan's laws, Commutative property, commutation, Associative property, association, Distribution (logic), distribution, double negation, Transposition (logic), transposition, Material implication (rule of inference), material implication, logical equivalence, Exportation (logic), exportation, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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