Community Research and Development Information Service - CORDIS


Adams — Result In Brief

Project ID: 299071
Funded under: FP7-PEOPLE
Country: Netherlands

A many-valued turn in logic

EU-funded scientists, seeking new interpretations to many-valued logic, have investigated new approaches using tools traditionally developed in intuitionistic and modal logic.
A many-valued turn in logic
Many-valued logics offer conceptual tools for the formal description of fuzzy, vague and uncertain information. In the last few years, the study of these logical systems has bloomed thanks to new research related to the most diverse areas of mathematics. In particular, many-valued algebras – the semantic counterpart of Łukasiewicz's infinite-valued logic – have attracted the attention of researchers.

In the EU-funded project ADAMS (A dual approach to many-valued semantics), researchers looked into dual categories of many-valued algebras. Generalisation of the classical dual adjunction between ideals of polynomials and affine varieties to many-valued algebras provided an adjunction of the latter with Tychonoff spaces.

The duality of finitely presented many-valued algebras and rational polyhedra was then obtained by specialising such an adjunction. The researchers identified which parts of the theory of many-valued algebras are needed to establish these results, with an eye on future applications to other algebraic varieties.

Before the end of the ADAMS project, the algebraic approach to canonical formulas opened the way to its application to non-classical logics. Canonical formulas have provided a uniform way to formulate extensions of intuitionistic logic as well as modal logics.

In this new approach, canonical formulas are built from a finite irreducible algebra by describing the behaviour of certain operations completely and that of others only partially. This tool can be extended to the wider realm of substructural logics using the finite embeddability property (FEP).

On the one hand, FEP is a key property for canonical formulas to work and, on the other, to establish links between substructural and many-valued logics. This line of research revealed a means to connect canonical formulas with dualities in non-classical logics.

The findings of ADAMS researchers' investigations into the relation of canonical formulas and dualities have been summarised in seven papers published in or under consideration at high-impact international journals.

Related information


Many-valued logic, intuitionistic, modal logic, many-valued algebras, canonical formulas
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