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Completeness and Herbrand Theorems for Nominal Logic
 Journal of Symbolic Logic
, 2006
"... Nominal logic is a variant of firstorder logic in which abstract syntax with names and binding is formalized in terms of two basic operations: nameswapping and freshness. It relies on two important principles: equivariance (validity is preserved by nameswapping), and fresh name generation (&qu ..."
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Nominal logic is a variant of firstorder logic in which abstract syntax with names and binding is formalized in terms of two basic operations: nameswapping and freshness. It relies on two important principles: equivariance (validity is preserved by nameswapping), and fresh name generation ("new" or fresh names can always be chosen).
The evolution of types and logic in the 20th century: A journey through Frege, Russell and . . .
 ILLC ALUMNI EVENT, AMSTERDAM 2004
, 2004
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Languages et Modèles pour la Formalisation et l’Automation de la
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Linguas y Modeles por el Formalisatione y el Automation del Matemáticas y el Informatica Fairouz Kamareddine (Universidad de HeriotWatt, Edimbourgo, RU)
"... • In less than a half a century, computers have revolutionised the way we all live. • Google, Wikipedia, and other information and search engines have changed the way we store and exchange information. • Computerisation also enables excellent collaborations between different disciplines (think of Bi ..."
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• In less than a half a century, computers have revolutionised the way we all live. • Google, Wikipedia, and other information and search engines have changed the way we store and exchange information. • Computerisation also enables excellent collaborations between different disciplines (think of BioInformatics) and enables new discoveries in different disciplines. • This computerisation of information is only at its beginning. We need a lot of investments in research methods that enable faster, correct, and efficient information storage and retrieval. • Information here means every aspect of information (mathematical, medical, social, educational, law, etc). • Calculators process numbers, computers process information. Brasilià, Novembre 2009 1The languages of Mathematics Usually, mathematicians ignore formal logic and write mathematics using a certain language style which we call Cml. Advantages of Cml: • Expressivity: We can express all sorts of notions. • Acceptability: Cml is accepted by most mathematicians. • Traditionality: Cml exists since very long and has been refined with the time.
From the Foundation of Mathematics to the Birth of Computation
, 2011
"... deduction/Logic was taken as a foundation for Mathematics, computation was also taken throughout as an essential tool in mathematics. • Our ancestors used sandy beaches to compute the circomference of a circle, and to work out approximations/values of numbers like π. • The word algorithm dates back ..."
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deduction/Logic was taken as a foundation for Mathematics, computation was also taken throughout as an essential tool in mathematics. • Our ancestors used sandy beaches to compute the circomference of a circle, and to work out approximations/values of numbers like π. • The word algorithm dates back centuries? Algorithms existed in anciant Egypt at the time of Hypatia. The word is named after AlKhawarizmi. • But even more impressively, the following important 20th century (un)computability result was known to Aristotle. • Assume a problem Π, – If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. HAPOC11: History and Philosophy of Computing 1 • But, this result was already known to Aristotle: • Assume a proposition Φ. – If you give me a proof of Φ, I can check whether this proof really proves Φ. – But, if you ask me to find a proof of Φ, I may go on forever trying but without success. • In fact, programs are proofs: – program = algorithm = computable function = λterm. – By the PAT principle: Proofs are λterms.
Parts of this talk are based on Kamareddine [2001]; Kamareddine et al. [2002]; Kamaredine and Nederpelt [2004], and on joint work with Maarek and Wells in Kamaredine et al. [2004b,a] University of LeipzigA Century of Complexity
, 1900
"... The formalisation and computerization of ..."
The word ’predicative ’ first appeared in Russell’s note On Some Difficulties in the Theory of Transfinite Numbers and Order Types [Rus07]. Paradoxes such
, 2008
"... as Russell’s Paradox show that we cannot form the class {x  φ(x)} for all propositional functions φ(x). Russell proposed we call φ(x) predicative if it defines a class and nonpredicative otherwise, but did not offer a criterion by which we could decide which propositional functions are which. A fi ..."
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as Russell’s Paradox show that we cannot form the class {x  φ(x)} for all propositional functions φ(x). Russell proposed we call φ(x) predicative if it defines a class and nonpredicative otherwise, but did not offer a criterion by which we could decide which propositional functions are which. A first such criterion was offered by Poincare ́ in the third part of his paper Les Mathématiques et la Logique [Poi06]. He proposed the vicious circle principle: “The definitions which ought to be regarded as nonpredicative are those which contain a vicious circle. ” (p. 1063) He indicated by example what he meant by ‘vicious circle’: in both the Richard paradox and the BuraliForti paradox, we define an aggregate E, and make use of E within its own definition. Poincare ́ proposed that definitions involving such a ‘vicious circle ’ are illegitimate: “A definition containing a vicious circle defines nothing. ” [Poi06, p. 1065] Poincare ́ justified this as follows. For a definition to be legitimate, it must be possible to substitute the definiens for the defined term. Recursive definitions