126 research outputs found
Implicit Commitment in a General Setting
G\"odel's Incompleteness Theorems suggest that no single formal system can
capture the entirety of one's mathematical beliefs, while pointing at a
hierarchy of systems of increasing logical strength that make progressively
more explicit those \emph{implicit} assumptions. This notion of \emph{implicit
commitment} motivates directly or indirectly several research programmes in
logic and the foundations of mathematics; yet there hasn't been a direct
logical analysis of the notion of implicit commitment itself. In a recent
paper, \L elyk and Nicolai carried out an initial assessment of this project by
studying necessary conditions for implicit commitments; from seemingly weak
assumptions on implicit commitments of an arithmetical system , it can be
derived that a uniform reflection principle for -- stating that all
numerical instances of theorems of are true -- must be contained in 's
implicit commitments. This study gave rise to unexplored research avenues and
open questions. This paper addresses the main ones. We generalize this basic
framework for implicit commitments along two dimensions: in terms of iterations
of the basic implicit commitment operator, and via a study of implicit
commitments of theories in arbitrary first-order languages, not only couched in
an arithmetical language
Factor Varieties and Symbolic Computation
We propose an algebraization of classical and non-classical logics, based on factor varieties and decomposition operators. In particular, we provide a new method for determining whether a propositional formula is a tautology or a contradiction. This method can be autom-atized by defining a term rewriting system that enjoys confluence and strong normalization. This also suggests an original notion of logical gate and circuit, where propositional variables becomes logical gates and logical operations are implemented by substitution. Concerning formulas with quantifiers, we present a simple algorithm based on factor varieties for reducing first-order classical logic to equational logic. We achieve a completeness result for first-order classical logic without requiring any additional structure
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