HOME  TheInfoList.com 
Internal Consistency Of The Bible In classical deductive logic , a CONSISTENT theory is one that does not contain a contradiction . The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model , i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic , although in contemporary mathematical logic the term SATISFIABLE is used instead. The syntactic definition states a theory T {displaystyle T} is consistent if and only if there is no formula {displaystyle phi } such that both {displaystyle phi } and its negation {displaystyle lnot phi } are elements of the set T {displaystyle T} [...More...]  "Internal Consistency Of The Bible" on: Wikipedia Yahoo 

List Of Logic Symbols In logic , a set of symbols is commonly used to express logical representation. The following table lists many common symbols together with their name, pronunciation, and the related field of mathematics. Additionally, the third column contains an informal definition, the fourth column gives a short example, the fifth and sixth give the unicode location and name for use in HTML HTML documents. The last column provides the LaTeX LaTeX symbol [...More...]  "List Of Logic Symbols" on: Wikipedia Yahoo 

Mathematical Logic MATHEMATICAL LOGIC is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics , the foundations of mathematics , and theoretical computer science . The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory , model theory , recursion theory , and proof theory . These areas share basic results on logic, particularly firstorder logic , and definability . In computer science (particularly in the ACM Classification ) mathematical logic encompasses additional topics not detailed in this article; see Logic Logic in computer science for those. Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics [...More...]  "Mathematical Logic" on: Wikipedia Yahoo 

List Of Mathematical Symbols This is a list of symbols found within all branches of mathematics to express a formula or to represent a constant . When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. Further, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of wellformed formulas . In short, convention dictates the meaning [...More...]  "List Of Mathematical Symbols" on: Wikipedia Yahoo 

Formulas In science , a FORMULA is a concise way of expressing information symbolically, as in a mathematical or chemical formula . The informal use of the term FORMULA in science refers to the general construct of a relationship between given quantities . The plural of formula can be spelled either as formulas or formulae (from the original Latin). In mathematics , a formula is an entity constructed using the symbols and formation rules of a given logical language . For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion ; but, having done this once in terms of some parameter (the radius for example), mathematicians have produced a formula to describe the volume: This particular formula is: V = 4 3 r 3 {displaystyle V={frac {4}{3}}pi r^{3}} . Having obtained this result, the volume of any sphere can be computed as long as its radius is known [...More...]  "Formulas" on: Wikipedia Yahoo 

Negation In logic , NEGATION, also called LOGICAL COMPLEMENT, is an operation that takes a proposition p to another proposition "not p", written ¬p, which is interpreted intuitively as being true when p is false, and false when p is true. Negation is thus a unary (singleargument) logical connective . It may be applied as an operation on propositions , truth values , or semantic values more generally. In classical logic , negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic , according to the Brouwer–Heyting–Kolmogorov interpretation , the negation of a proposition p is the proposition whose proofs are the refutations of p [...More...]  "Negation" on: Wikipedia Yahoo 

Independence (mathematical Logic) In mathematical logic , INDEPENDENCE refers to the unprovability of a sentence from other sentences. A sentence σ is INDEPENDENT of a given firstorder theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false. Sometimes, σ is said (synonymously) to be UNDECIDABLE from T; this is not the same meaning of "decidability " as in a decision problem . A theory T is INDEPENDENT if each axiom in T is not provable from the remaining axioms in T. A theory for which there is an independent set of axioms is INDEPENDENTLY AXIOMATIZABLE. CONTENTS * 1 Usage note * 2 Independence results in set theory * 3 Applications to physical theory * 4 See also * 5 Notes * 6 References USAGE NOTESome authors say that σ is independent of T when T simply cannot prove σ, and do not necessarily assert by this that T cannot refute σ [...More...]  "Independence (mathematical Logic)" on: Wikipedia Yahoo 

Axiom An AXIOM or POSTULATE is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.' The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy , an axiom is a statement that is so evident or wellestablished, that it is accepted without controversy or question. As used in modern logic , an axiom is simply a premise or starting point for reasoning [...More...]  "Axiom" on: Wikipedia Yahoo 

Presburger Arithmetic PRESBURGER ARITHMETIC is the firstorder theory of the natural numbers with addition , named in honor of Mojżesz Presburger , who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely. The axioms include a schema of induction. Presburger arithmetic is much weaker than Peano arithmetic , which includes both addition and multiplication operations. Unlike Peano arithmetic, Presburger arithmetic is a decidable theory . This means it is possible to algorithmically determine, for any sentence in the language of Presburger arithmetic, whether that sentence is provable from the axioms of Presburger arithmetic [...More...]  "Presburger Arithmetic" on: Wikipedia Yahoo 

Recursively Enumerable In computability theory , traditionally called recursion theory, a set S of natural numbers is called RECURSIVELY ENUMERABLE, COMPUTABLY ENUMERABLE, SEMIDECIDABLE, PROVABLE or TURINGRECOGNIZABLE if: * There is an algorithm such that the set of input numbers for which the algorithm halts is exactly S.Or, equivalently, * There is an algorithm that enumerates the members of S. That means that its output is simply a list of the members of S: s1, s2, s3, ... . If necessary, this algorithm may run forever.The first condition suggests why the term semidecidable is sometimes used; the second suggests why computably enumerable is used. The abbreviations R.E. and C.E. are often used, even in print, instead of the full phrase. In computational complexity theory , the complexity class containing all recursively enumerable sets is RE . In recursion theory , the lattice of r.e. sets under inclusion is denoted E {displaystyle {mathcal {E}}} [...More...]  "Recursively Enumerable" on: Wikipedia Yahoo 

Primitive Recursive Arithmetic PRIMITIVE RECURSIVE ARITHMETIC, or PRA, is a quantifier free formalization of the natural numbers . It was first proposed by Skolem as a formalization of his finitist conception of the foundations of arithmetic , and it is widely agreed that all reasoning of PRA is finitist. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ε0 , which is the prooftheoretic ordinal of Peano arithmetic . PRA's proof theoretic ordinal is ωω, where ω is the smallest transfinite ordinal. PRA is sometimes called SKOLEM ARITHMETIC. The language of PRA can express arithmetic propositions involving natural numbers and any primitive recursive function , including the operations of addition , multiplication , and exponentiation . PRA cannot explicitly quantify over the domain of natural numbers [...More...]  "Primitive Recursive Arithmetic" on: Wikipedia Yahoo 

Zermelo–fraenkel Set Theory In mathematics, ZERMELO–FRAENKEL SET THEORY, named after mathematicians Ernst Zermelo Ernst Zermelo and Abraham Fraenkel , is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets free of paradoxes such as Russell\'s paradox . Zermelo–Fraenkel set theory Zermelo–Fraenkel set theory with the historically controversial axiom of choice included is commonly abbreviated ZFC, where C stands for choice. Many authors use ZF to refer to the axioms of Zermelo–Fraenkel set theory Zermelo–Fraenkel set theory with the axiom of choice excluded. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics . ZFC is intended to formalize a single primitive notion, that of a hereditary wellfounded set , so that all entities in the universe of discourse are such sets [...More...]  "Zermelo–fraenkel Set Theory" on: Wikipedia Yahoo 

Firstorder Logic FIRSTORDER LOGIC—also known as FIRSTORDER PREDICATE CALCULUS and PREDICATE LOGIC—is a collection of formal systems used in mathematics , philosophy , linguistics , and computer science . Firstorder logic Firstorder logic uses quantified variables over nonlogical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists X such that X is Socrates and X is a man" and there exists is a quantifier while X is a variable. This distinguishes it from propositional logic , which does not use quantifiers or relations. A theory about a topic is usually a firstorder logic together with a specified domain of discourse over which the quantified variables range, finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold for those things [...More...]  "Firstorder Logic" on: Wikipedia Yahoo 

Witness (mathematics) In mathematical logic , a WITNESS is a specific value t to be substituted for variable x of an existential statement of the form ∃x φ(x) such that φ(t) is true. CONTENTS * 1 Examples * 2 Henkin witnesses * 3 Relation to game semantics * 4 See also * 5 References EXAMPLESFor example, a theory T of arithmetic is said to be inconsistent if there exists a proof in T of the formula "0=1". The formula I(T), which says that T is inconsistent, is thus an existential formula. A witness for the inconsistency of T is a particular proof of "0 = 1" in T. Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which S is an nplace relation on natural numbers, R is an nplace recursive relation , and ↔ indicates logical equivalence (if and only if): " S(x1, ..., xn) ↔ ∃y R(x1, . [...More...]  "Witness (mathematics)" on: Wikipedia Yahoo 

Ωconsistency In mathematical logic , an ωCONSISTENT (or OMEGACONSISTENT, also called NUMERICALLY SEGREGATIVE) THEORY is a theory (collection of sentences ) that is not only (syntactically) consistent (that is, does not prove a contradiction ), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. The name is due to Kurt Gödel Kurt Gödel , who introduced the concept in the course of proving the incompleteness theorem [...More...]  "Ωconsistency" on: Wikipedia Yahoo 

Paraconsistent Logic A PARACONSISTENT LOGIC is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent (or "inconsistencytolerant") systems of logic. Inconsistencytolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle ); however, the term paraconsistent ("beside the consistent") was not coined until 1976, by the Peruvian philosopher Francisco Miró Quesada [...More...]  "Paraconsistent Logic" on: Wikipedia Yahoo 