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Principle Of Distributivity
The principle of distributivity states that the algebraic distributive law is valid, where both logical conjunction and logical disjunction are distributive over each other so that for any propositions ''A'', ''B'' and ''C'' the equivalences :A \land (B \lor C) \iff (A \land B) \lor (A \land C) and :A \lor (B \land C) \iff (A \lor B) \land (A \lor C) hold. The principle of distributivity is valid in classical logic, but both valid and invalid in quantum logic. The article " Is Logic Empirical?" discusses the case that quantum logic is the correct, empirical logic, on the grounds that the principle of distributivity is inconsistent In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences o ... with a reasonable interpretation of quantum phenomena. References Abstract algebra Principles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Law
In mathematics, the distributive property of binary operations is a generalization of 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). Therefore, one would say 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Validity (logic)
In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be truth, true and the conclusion nevertheless to be False (logic), false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called well-formed formula, well-formed formulas (also called ''wffs'' or simply ''formulas''). The validity of an argument can be tested, proved or disproved, and depends on its logical form. Arguments In logic, an argument is a set of related statements expressing the ''premises'' (which may consists of non-empirical evidence, empirical evidence or may contain some axiomatic truths) and a ''necessary conclusion based on the relationship of the premises.'' An argument is ''valid'' if and only if it would be contradicto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Conjunction
In logic, mathematics and linguistics, ''and'' (\wedge) is the Truth function, truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or \times or \cdot in which \wedge is the most modern and widely used. The ''and'' of a set of operands is true if and only if ''all'' of its operands are true, i.e., A \land B is true if and only if A is true and B is true. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English language, English "Conjunction (grammar), and"; * In programming languages, the Short-circuit evaluation, short-circuit and Control flow, control structure; * In set theory, Intersection (set theory), intersection. * In Lattice (order), lattice theory, logical conjunction (Infimum and supremum, greatest lower bound). Notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical 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] |
Proposition
A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky is blue" expresses the proposition that the sky is blue. Unlike sentences, propositions are not linguistic expressions, so the English sentence "Snow is white" and the German "Schnee ist weiß" denote the same proposition. Propositions also serve as the objects of belief and other propositional attitudes, such as when someone believes that the sky is blue. Formally, propositions are often modeled as functions which map a possible world to a truth value. For instance, the proposition that the sky is blue can be modeled as a function which would return the truth value T if given the actual world as input, but would return F if given some alternate world where the sky is green. However, a number of alternative formalizations have be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Equivalence
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related. Logical equivalences In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these. General logical equivalences Logical equivalences involving conditional statements :#p \rightarrow q \equiv \neg p \vee q :#p \rightarrow q \equiv \neg q \rightarrow \neg p :#p \vee q \equiv \neg p \rightarrow q :#p \wedge q \equiv \neg (p \rightarrow \neg q) :#\neg (p \rightarrow q) \equiv p \wedge \neg q :#(p \righta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Logic
Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class shares characteristic properties: Gabbay, Dov, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press. # Law of excluded middle and double negation elimination # Law of noncontradiction, and the principle of explosion # Monotonicity of entailment and idempotency of entailment # Commutativity of conjunction # De Morgan duality: every logical operator is dual to another While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics. Shapiro, Stewart (2000). Classical Logic. In St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Logic
In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The formal system takes as its starting point an observation of Garrett Birkhoff and John von Neumann, that the structure of experimental tests in classical mechanics forms a Boolean algebra, but the structure of experimental tests in quantum mechanics forms a much more complicated structure. A number of other logics have also been proposed to analyze quantum-mechanical phenomena, unfortunately also under the name of "quantum logic(s)". They are not the subject of this article. For discussion of the similarities and differences between quantum logic and some of these competitors, see '. Quantum logic has been proposed as the correct logic for propositional inference generally, most notably by the philosopher Hilary Putnam, at least at one point in his career. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Is Logic Empirical?
"Is Logic Empirical?" is the title of two articles (one by Hilary Putnam and another by Michael Dummett) that discuss the idea that the algebraic properties of logic may, or should, be empirically determined; in particular, they deal with the question of whether empirical facts about quantum phenomena may provide grounds for revising classical logic as a consistent logical rendering of reality. The replacement derives from the work of Garrett Birkhoff and John von Neumann on quantum logic. In their work, they showed that the outcomes of quantum measurements can be represented as binary propositions and that these quantum mechanical propositions can be combined in a similar way as propositions in classical logic. However, the algebraic properties of this structure are somewhat different from those of classical propositional logic in that the principle of distributivity fails. The idea that the principles of logic might be susceptible to revision on empirical grounds has many ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistency
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when there is no formula \varphi such that \varphi \in \langle A \rangle and \lnot \varphi \in \langle A \rangle. A ''trivial'' theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpretation Of Quantum Mechanics
An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments. However, there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, local or non-local, which elements of quantum mechanics can be considered real, and what the nature of measurement is, among other matters. While some variation of the Copenhagen interpretation is commonly presented in textbooks, many other interpretations have been developed. Despite nearly a century of debate and experiment, no consensus has been reached among physicists and philosophers of physics concerning which interpretation best "represents" reality. History The definition of quantum t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
