caa a22 4500 469126612 CHVBK 20180323133152.0 cr unu---uuuuu 170328e19921101xx s 000 0 eng 10.1007/BF01277482 doi (NATIONALLICENCE)springer-10.1007/BF01277482 Pavlović Duško Department of Mathematics and Statistics, McGill University, Burnside Hall, H3A 2K6, Montreal, Quebec, Canada aut On the structure of paradoxes [Elektronische Daten] [Duško Pavlović] Summary: Paradox is a logical phenomenon. Usually, it is produced in type theory, on a type Ω of "truth values”. A formula Ψ (i.e., a term of type Ω) is presented, such that Ψ↔¬Ψ (with negation as a term¬∶Ω→Ω)-whereupon everything can be proved: In Sect. 1 we describe a general pattern which many constructions of the formula Ψ follow: for example, the well known arguments of Cantor, Russell, and Gödel. The structure uncovered behind these paradoxes is generalized in Sect. 2. This allows us to show that Reynolds' [R] construction of a typeA ≃℘℘A in polymorphic λ-calculus cannot be extended, as conjectured, to give a fixed point ofevery variable type derived from the exponentiation: for some (contravariant) types, such a fixed point causes a paradox. Pursueing the idea that $$\frac{{{\text{type theory}}}}{{{\text{categorical interpretation}}}} = \frac{{{\text{(propositional) logic}}}}{{{\text{Lindebaum algebra}}}}$$ the language of categories appears here as a natural medium for logical structures. It allows us to abstract from the specific predicates that appear in particular paradoxes, and to display the underlying constructions in "pure state”. The essential role of cartesian closed categories in this context has been pointed out in [L]. The paradoxes studied here remain within the limits of the cartesian closed structure of types, as sketched in this Lawvere's seminal paper — and do not depend on any logical operations on the type Ω. Our results can be translated in simply typed λ-calculus in a straightforward way (although some of them do become a bit messy). Springer-Verlag, 1992 Archive for Mathematical Logic Springer-Verlag 31/6(1992-11-01), 397-406 0933-5846 31:6<397 1992 31 153 https://doi.org/10.1007/BF01277482 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01277482 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Pavlović Duško Department of Mathematics and Statistics, McGill University, Burnside Hall, H3A 2K6, Montreal, Quebec, Canada aut NATIONALLICENCE 773 0- Archive for Mathematical Logic Springer-Verlag 31/6(1992-11-01), 397-406 0933-5846 31:6<397 1992 31 153 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer