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Friedman Translation
In mathematical logic, the Friedman translation is a certain transformation of intuitionistic formulas. Among other things it can be used to show that the Π02-theorems of various first-order theories of classical mathematics are also theorems of intuitionistic mathematics. It is named after its discoverer, Harvey Friedman. Definition Let ''A'' and ''B'' be intuitionistic formulas, where no free variable of ''B'' is quantified in ''A''. The translation ''AB'' is defined by replacing each atomic subformula ''C'' of ''A'' by . For purposes of the translation, ⊥ is considered to be an atomic formula as well, hence it is replaced with (which is equivalent to ''B''). Note that ¬''A'' is defined as an abbreviation for hence Application The Friedman translation can be used to show the closure of many intuitionistic theories under the Markov rule, and to obtain partial conservativity results. The key condition is that the \Delta^0_0 sentences of the logic be decidable, allowing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intuitionistic Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-formed Formula
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic. Introduction A key use of formulas is in propositional logic and predicate logic such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven. Although the term "formula" may be used for written marks (for instance, on a piece of paper or ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Theories
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their properties. Preliminaries For every natural mathematical structure there is a signature σ listing the constants, functions, and relations of the theory together with their arities, so that the object is naturally a σ-structure. Given a signature σ there is a unique first-order language ''L''σ that can be used to capture the first-order expressible facts about the σ-structure. There are two common ways to specify theories: #List or describe a set of sentences in the language ''L''σ, called the axioms of the theory. #Give a set of σ-structures, and define a theory to be the set of sentences in ''L''σ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite ... [...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 Stanford Encyclop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harvey Friedman
__NOTOC__ Harvey Friedman (born 23 September 1948)Handbook of Philosophical Logic, , p. 38 is an American mathematical logician at Ohio State University in Columbus, Ohio. He has worked on reverse mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary. In recent years this has advanced to a study of Boolean relation theory, which attempts to justify large cardinal axioms by demonstrating their necessity for deriving certain propositions considered "concrete". Friedman earned his Ph.D. from the Massachusetts Institute of Technology in 1967, with a dissertation on ''Subsystems of Analysis''. His advisor was Gerald Sacks. Friedman received the Alan T. Waterman Award in 1984. He also assumed the title of Vising Scientist at IBM. He delivered the Tarski Lectures in 2007. In 1967, Friedman was listed in the ''Guinness Book of World Records'' for being the world's youngest professor when he taught at Stanford University at age ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Variable
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol. In computer programming, the term free variable refers to variables used in a function that are neither local variables nor parameters of that function. The term non-local variable is often a synonym in this context. A bound variable, in contrast, is a variable that has been ''bound'' to a specific value or range of values in the domain of discourse or universe. This may be achieved through the use of logical quantifiers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bound Variable
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a Mathematical notation, notation (symbol) that specifies places in an expression (mathematics), expression where Substitution (logic), substitution may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol. In computer programming, the term free variable refers to variable (programming), variables used in a function (computer science), function that are neither local variables nor parameter (computer programming), parameters of that function. The term non-local variable is often a synonym in this context. A bound variable, in contrast, is a variable that ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Rule
Markov's principle, named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but not in intuitionistic constructive mathematics. However, many particular instances of it are nevertheless provable in a constructive context as well. History The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often with a realizability perspective on the notion of mathematical function. In computability theory In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then for some input it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable. In intuitionistic logic In predicate logic, a predicate ''P'' over some d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conservative Extension
In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the original. More formally stated, a theory T_2 is a ( proof theoretic) conservative extension of a theory T_1 if every theorem of T_1 is a theorem of T_2, and any theorem of T_2 in the language of T_1 is already a theorem of T_1. More generally, if \Gamma is a set of formulas in the common language of T_1 and T_2, then T_2 is \Gamma-conservative over T_1 if every formula from \Gamma provable in T_2 is also provable in T_1. Note that a conservative extension of a consistent theory is consistent. If it were not, then by the principle of explosion, every formula in the language of T_2 would be a theorem of T_2, so every formula in the language of T_1 would be a theorem of T_1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Arithmetic
In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism.Troelstra 1973:18 It is named after Arend Heyting, who first proposed it. Axiomatization As with first-order Peano arithmetic , the intended model of this theory are the natural numbers and the theories characterize addition and multiplication. Heyting arithmetic adopts the axioms of Peano arithmetic, including the signature with zero "0" and the successor "S", but uses intuitionistic logic for inference. In particular, the principle of the excluded middle does not hold in general. Metalogic and theorems As with other theories over intuitionistic logic, various instances of can be proven. For instance, proves equality "=" is decidable for all numbers, :\vdash \forall n. \forall m. \big((n = m)\lor\neg(n = m)\big) In fact, since equality is the only predicate symbol in Heyting arithmetic, it then follows that, for any quantifier-free formula \phi, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ( la, Arithmetice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
