"new foundations must give new math"

from "The Basic Picture. Structures for constructive topology"

by Giovanni Sambin, Oxford University Press, forthcoming


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Under the patronage of

Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Italy

University of Padova, Italy

AILA (Italian Association of Logic and Applications), whose first president was Giovanni Sambin (from 1987 to 1994).

Abstracts » Macedonio

Damiano Macedonio

Sambin's Principle of Reflection and Relational Semantics

Basic Logic has been introduced with the aim of finding a structure in the space of the logics. Classical, intuitionistic and non-modal linear logics, are all obtained as extensions in a uniform way.

The logical constants and the connectives are introduced by following the foundational principle of reflection: every logical entity is bound to a meta-linguistic link by a definitional equation, whose solution provides the syntactic inference rules.

This principle, along with the definitional equations, is essential to provide a notion of model for the logic itself: the relational monoids. We prove soundness and refined completeness results for this class of models. In particular the completeness result allows a semantical proof of cut-elimination.

Basic Logic can be extended in two directions: one direction is the addition of structural rules, the other is the move to intuitionistic logic, thus obtaining Intuitionistic Linear Logic and Intuitionistic Logic. The notion of model, in turn, can be extended to these logics, and in each case the soundness and refined completeness results are retained. These newly found models can be related to more traditional models of the logics so discovered: Kripke models for Intuitionistic Logic and Phase Spaces for Linear Logic. Finally, combining the two extensions leads us to the Logic of Bunched Implications.