|
Tolerance (in Logic)
In mathematical logic, a tolerant sequence is a sequence :T_1,...,T_n of formal theories such that there are consistent extensions :S_1,...,S_n of these theories with each S_{i+1} interpretable in S_i. Tolerance naturally generalizes from sequences of theories to trees of theories. Weak interpretability can be shown to be a special, binary case of tolerance. This concept, together with its dual concept of cotolerance, was introduced by Japaridze in 1992, who also proved that, for Peano arithmetic and any stronger theories with effective axiomatizations, tolerance is equivalent to \Pi_1-consistency. See also *Interpretability *Cointerpretability * Interpretability logic References G. Japaridze ''The logic of linear tolerance''. Studia Logica ''Studia Logica'' (full name: Studia Logica, An International Journal for Symbolic Logic), is a scienific journal publishing papers employing formal tools from Mathematics and Logic. The scope of papers published in Studia Lo ... [...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] |
Theory (mathematical Logic)
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element \phi\in T of a deductively closed theory T is then called a theorem of the theory. In many deductive systems there is usually a subset \Sigma \subseteq T that is called "the set of axioms" of the theory T, in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms. General theories (as expressed in formal language) When defining theories for foundational purposes, additional care must be taken, as normal set-theoretic language may not be appropriate. The construction of a theory begins by specifying a definite non-empty ''conceptual class'' \mathcal, the element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpretability
In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other. Informal definition Assume ''T'' and ''S'' are formal theories. Slightly simplified, ''T'' is said to be ''interpretable'' in ''S'' if and only if the language of ''T'' can be translated into the language of ''S'' in such a way that ''S'' proves the translation of every theorem of ''T''. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logical structure of formulas. This concept, together with weak interpretability, was introduced by Alfred Tarski in 1953. Three other related concepts are cointerpretability, logical tolerance, and cotolerance, introduced by Giorgi Japaridze in 1992–93. See also * Interpretation (logic) * Interpretation (model theory) * Interpretability logic References * Japaridze, G., and De Jongh, D. (1998) "The logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Interpretability
In mathematical logic, weak interpretability is a notion of translation of logical theories, introduced together with interpretability by Alfred Tarski in 1953. Let ''T'' and ''S'' be formal theories. Slightly simplified, ''T'' is said to be weakly interpretable in ''S'' if, and only if, the language of ''T'' can be translated into the language of ''S'' in such a way that the translation of every theorem of ''T'' is consistent with ''S''. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logical structure of formulas. A generalization of weak interpretability, tolerance, was introduced by Giorgi Japaridze Giorgi Japaridze (also spelled Giorgie Dzhaparidze) is a Georgian-American researcher in logic and theoretical computer science. He currently holds the title of Full Professor at the Computing Sciences Department of Villanova University. Japaridze i ... in 1992. See also * Interpretability logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giorgi Japaridze
Giorgi Japaridze (also spelled Giorgie Dzhaparidze) is a Georgian-American researcher in logic and theoretical computer science. He currently holds the title of Full Professor at the Computing Sciences Department of Villanova University. Japaridze is best known for his invention of computability logic, cirquent calculus, and Japaridze's polymodal logic. Research During 1985–1988 Japaridze elaborated the system GLP, known as Japaridze's polymodal logic. This is a system of modal logic with the "necessity" operators …, understood as a natural series of incrementally weak provability predicates for Peano arithmetic. In "The polymodal logic of provability" Japaridze proved the arithmetical completeness of this system, as well as its inherent incompleteness with respect to Kripke frames. GLP has been extensively studied by various authors during the subsequent three decades, especially after Lev Beklemishev, in 2004, pointed out its usefulness in understanding the proof theory of ... [...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] |
Interpretability
In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other. Informal definition Assume ''T'' and ''S'' are formal theories. Slightly simplified, ''T'' is said to be ''interpretable'' in ''S'' if and only if the language of ''T'' can be translated into the language of ''S'' in such a way that ''S'' proves the translation of every theorem of ''T''. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logical structure of formulas. This concept, together with weak interpretability, was introduced by Alfred Tarski in 1953. Three other related concepts are cointerpretability, logical tolerance, and cotolerance, introduced by Giorgi Japaridze in 1992–93. See also * Interpretation (logic) * Interpretation (model theory) * Interpretability logic References * Japaridze, G., and De Jongh, D. (1998) "The logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cointerpretability
In mathematical logic, cointerpretability is a binary relation on formal theories: a formal theory ''T'' is cointerpretable in another such theory ''S'', when the language of ''S'' can be translated into the language of ''T'' in such a way that ''S'' proves every formula whose translation is a theorem of ''T''. The "translation" here is required to preserve the logical structure of formulas. This concept, in a sense dual to interpretability In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other. Informal definition Assume ''T'' and ''S'' are formal theories. Slightly simplified, '' ..., was introduced by , who also proved that, for theories of Peano arithmetic and any stronger theories with effective axiomatizations, cointerpretability is equivalent to \Sigma_1-conservativity. See also * Cotolerance * Interpretability logic. * Tolerance (in logic) References *. *. Math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpretability Logic
Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities. Main contributors to the field are Alessandro Berarducci, Petr Hájek, Konstantin Ignatiev, Giorgi Japaridze, Franco Montagna, Vladimir Shavrukov, Rineke Verbrugge, Albert Visser, and Domenico Zambella. Examples Logic ILM The language of ILM extends that of classical propositional logic by adding the unary modal operator \Box and the binary modal operator \triangleright (as always, \Diamond p is defined as \neg \Box\neg p). The arithmetical interpretation of \Box p is “p is provable in Peano arithmetic (PA)”, and p \triangleright q is understood as “PA+q is interpretable in PA+p”. Axiom schemata: 1. All classical tautologies 2. \Box(p \rightarrow q) \rightarrow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Studia Logica
''Studia Logica'' (full name: Studia Logica, An International Journal for Symbolic Logic), is a scienific journal publishing papers employing formal tools from Mathematics and Logic. The scope of papers published in Studia Logica covers all scientific disciplines; the key criterion for published papers is not their topic but their method: they are required to contain significant and original results concerning formal systems and their properties. The journal offers papers on topics in general logic and on applications of logic to methodology of science, linguistics, philosophy, and other branches of knowledge. The journal is published by the Institute of Philosophy and Sociology of the Polish Academy of Sciences and Springer publications. History The name Studia Logica appeared for the first time in 1934, but only one volume (edited by Jan Łukasiewicz) has been published that time. It had been published continuously since December 1953 in changing frequency by the Polish Acade ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
