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Absorption (logic)
Absorption is a Validity (logic), valid argument form and rule of inference of propositional logic. The rule states that if P implies Q, then P implies P and Q. The rule makes it possible to introduce Logical conjunction, conjunctions to formal proof, proofs. It is called the law of absorption because the term Q is "absorbed" by the term P in the consequent.Russell and Whitehead, ''Principia Mathematica'' The rule can be stated: :\frac where the rule is that wherever an instance of "P \to Q" appears on a line of a proof, "P \to (P \land Q)" can be placed on a subsequent line. Formal notation The ''absorption'' rule may be expressed as a sequent: : P \to Q \vdash P \to (P \land Q) where \vdash is a metalogical symbol meaning that P \to (P \land Q) is a logical consequence#Syntactic_consequence, syntactic consequence of (P \rightarrow Q) in some formal system, logical system; and expressed as a truth-functional tautology (logic), tautology or theorem of propositional calculus ... [...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] |
Absorption Law
In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations. Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if: :''a'' ¤ (''a'' ⁂ ''b'') = ''a'' ⁂ (''a'' ¤ ''b'') = ''a''. A set equipped with two commutative and associative binary operations \scriptstyle \lor ("join") and \scriptstyle \land ("meet") that are connected by the absorption law is called a lattice; in this case, both operations are necessarily idempotent. Examples of lattices include Heyting algebras and Boolean algebras,See Boolean algebra (structure)#Axiomatics for a proof of the absorption laws from the distributivity, identity, and boundary laws. in particular sets of sets with ''union'' and ''intersection'' operators, and ordered sets with ''min'' and ''max'' operations. In classical logic, and in particular Boolean algebra, the operations OR and AND, which are also denoted by \scriptstyle \lor and \scriptstyle \land, sat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution (logic)
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, one has 2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3). One says that multiplication ''distributes'' over addition. This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted \,\land\,) and the logical or (denoted \,\lor\,) distributes over the other. Definition Given a set S and two binary operators \,*\, and \,+\, on S, *the operation \,*\, is over (or with respect to) \,+\, if, given any elements x, y, \text z of S, x * (y + z) = (x * y) + (x * z); *the operation \,*\, is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunction Introduction
Conjunction introduction (often abbreviated simply as conjunction and also called and introduction or adjunction) is a valid rule of inference of propositional logic. The rule makes it possible to introduce a conjunction into a logical proof. It is the inference that if the proposition P is true, and the proposition Q is true, then the logical conjunction of the two propositions P and Q is true. For example, if it is true that "it is raining", and it is true that "the cat is inside", then it is true that "it is raining and the cat is inside". The rule can be stated: :\frac where the rule is that wherever an instance of "P" and "Q" appear on lines of a proof, a "P \land Q" can be placed on a subsequent line. Formal notation The ''conjunction introduction'' rule may be written in sequent notation: : P, Q \vdash P \land Q where P and Q are propositions expressed in some formal system, and \vdash is a metalogical symbol meaning that P \land Q is a syntactic consequence if P an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Excluded Middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law (or principle) of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'': "no third ossibilityis given". It is a tautology. The principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, but provable in the future ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Implication (rule Of Inference)
In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that ''P implies Q'' is logically equivalent to ''not-P or Q'' and that either form can replace the other in logical proofs. In other words, if P is true, then Q must also be true, while if Q is true, then P cannot be true either; additionally, when P is not true, Q may be either true or false. P \to Q \Leftrightarrow \neg P \lor Q Where "\Leftrightarrow" is a metalogical symbol representing "can be replaced in a proof with," and P and Q are any given logical statements. To illustrate this, consider the following statements: * P: Sam ate an orange for lunch * Q: Sam ate a fruit for lunch Then, to say, "Sam ate an orange for lunch" "Sam ate a fruit for lunch" (P \to Q). Logically, if Sam did not eat a fruit for lunch, then Sam also cannot have eaten an orange for lunch (b ... [...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] |
Propositional Calculus
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 Quantifier (logic), 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" (logical conjunction, conjunction), "or" (lo ... [...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] |
Tautology (logic)
In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be Contingency (philosophy), logically contingent. Such a formula can be made either true or false based on the values assigned to its propositi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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