Artículos con la etiqueta ‘Historia de la Lógica’

Restructuring Logic

Por • 12 mar, 2014 • Category: Filosofía

The outline of a programme for restructuring mathematical logic. We explain what we mean by «restructuring» and carry out exemplary parts of the programme

A Galois connection between classical and intuitionistic logics

Por • 11 dic, 2013 • Category: Ciencia y tecnología

In a 1985 commentary to his collected works, Kolmogoroff remarked that his 1932 paper «was written in hope that with time, the logic of solution of problems will become a permanent part of the logical curriculum. Creation of a unified logical apparatus dealing with objects of two types – propositions and problems – was intended.» We construct such a formal system QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC, and sheds new light on the basics of intuitionism:
1) The only new connectives ? and ! of QHC induce a Galois connection (i.e., a pair of adjoint functors) between the Lindenbaum algebras of QH and QC.
2) Kolmogoroff’s double negation translation of QC into QH extends to an interpretation of QHC in QH that is the identity on QH.
3) Goedel’s provability translation of QH into the classical modal logic QS4 extends to an interpretation of QHC in QS4, which is identified with a fragment of QHC.
Some models of QHC are constructed, including a sheaf-valued model inspired by dependent type theory, which appears to be of interest even as a model of QH (not to be confused with the well-known open-set-valued sheaf models of QH), since it can be seen as a rather accurate formalization of the BHK interpretation of intuitionistism.
The paper is addressed to a general mathematical audience and includes a rather unconventional introduction to intuitionistic logic, featuring (a) a motivation via Hilbert’s 24th Problem and Lafont’s observation that in classical logic, any two proofs of a given theorem are «homotopic»; (b) a derivation of Tarski topological models of QH via a model in «Venn diagrams» of a classical first-order theory extracted from the clauses of the BHK interpretation.

Rational Lukasiewicz logic and DMV-algebras

Por • 4 dic, 2012 • Category: Filosofía

In this paper we present some results on the variety of divisible MV-algebras. Any free divisible MV-algebra is an algebra of continuous piecewise linear functions with rational coefficients. Correspondingly, Rational {\L}ukasiewicz logic is defined and its tautology problem is shown to be co-NP-complete.

How Peircean was the «‘Fregean’ Revolution» in Logic?

Por • 24 ene, 2012 • Category: Crítica

The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912-1986) and Donald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848-1925) in his Begriffsschrift of 1879, and with Bertrand Russell (1872-1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806-1871), George Boole (1815-1864), Charles Sanders Peirce (1838-1914), and Ernst Schr\»oder (1841-1902) as belonging to the Aristotelian tradition. The «Booleans» are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise.

Truth and the liar paradox

Por • 8 ene, 2012 • Category: Filosofía

We analyze the informal notion of truth and conclude that it can be formalized in essentially two distinct ways: constructively, in terms of provability, or classically, as a hierarchy of concepts which satisfy Tarski’s biconditional in limited settings. This leads to a complete resolution of the liar paradox.