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Curry's Anticipation of the Types Used in Programming Languages
, 2003
"... This paper shows that H. B. Curry anticipated both the kind of data types used in computer programming languages and also the dependent function type. ..."
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This paper shows that H. B. Curry anticipated both the kind of data types used in computer programming languages and also the dependent function type.
Handbook of the History of Logic. Volume 6
"... ABSTRACT: Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory logic); 2. suggesting methods of proving fundamental metamathematical results (fixed p ..."
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ABSTRACT: Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory logic); 2. suggesting methods of proving fundamental metamathematical results (fixed point theorems, incompleteness, undecidability, undefinability); 3. applying inductive definability and generalized recursion; 4. introducing new semantical methods (e. g. revision theory, semiinductive definitions, which require nontrivial set theoretic results); 5. (partly) enhancing new axioms in set theory: the case of antifoundation AFA and the mathematics of circular phenomena; 6. suggesting the investigation of nonclassical logical systems, from contractionfree and manyvalued logics to systems with generalized quantifiers; 7. suggesting frameworks with flexible typing for the foundations of Mathematics and Computer Science; 8. applying forms of selfreferential truth and in Artificial Intelligence, Theoretical Linguistics, etc. Below we attempt to shed some light on the genesis of the issues 1–8 through the history of the paradoxes in the twentieth century, with a special emphasis on semantical aspects.
On the Role of Implication in Formal Logic
, 1998
"... Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with no ..."
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Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higherorder BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. 1991 AMS (MOS) Classification: 03B40, 03F05, 03B20 Key words: Implication, negation, combinatory logic, lambda calculus, comprehension principle, normalization, cutelimination...
On the Role of Implication in Formal Logic
"... Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with no ..."
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Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higherorder BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. 1991 AMS (MOS) Classification: 03B40, 03F05, 03B20 Key words: Implication, negation, combinatory logic, lambda calculus, comprehension principle, normalization, cutelimination...
LambdaCalculus and Functional Programming tions.
"... The lambdacalculus is a formalism for representing funcBy the second half of the nineteenth century, the concept of function as used in mathematics had reached the point at which the standard notation had become ambiguous. For example, consider the operator P defined on real functions as follows: ..."
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The lambdacalculus is a formalism for representing funcBy the second half of the nineteenth century, the concept of function as used in mathematics had reached the point at which the standard notation had become ambiguous. For example, consider the operator P defined on real functions as follows: ⎧f(x) – f(0) for x 0 P[f(x)] = ⎨ x ⎩f ′(0) for x = 0 What is P[f(x + 1)]? To see that this is ambiguous, let f(x) = x 2. Then if g(x) = f(x + 1), P[g(x)] = P[x 2 + 2x + 1] = x + 2. But if h(x) = P[f(x)] = x, then h(x + 1) = x + 1 P[g(x)]. This ambiguity has actually led to an error in the published literature; see the discussion in (Curry and Feys