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## Banca de DEFESA: THIAGO NASCIMENTO DA SILVA

STUDENT : THIAGO NASCIMENTO DA SILVA
DATE: 18/02/2022
TIME: 08:00
TITLE:

Algebraic Semantics and Calculi for Nelson's logics

KEY WORDS:

Algebraisable logics, Substructural logics, Residuated lattices, Nelson’s logics, Algebraic logic.

PAGES: 140
BIG AREA: Ciências Exatas e da Terra
AREA: Ciência da Computação
SUMMARY:

The aim of this thesis is to study a family of logics, comprised of Nelson’s logic S, constructive logic with strong negation N3, quasi-Nelson logic QN and quasi-Nelson implicative logic QNI. This is done in two ways. The first is by means of an axiomatisation via a Hilbert Calculus and the second is by studying some of the properties of the corresponding quasi-variety of algebras. The main contribution of the thesis is to prove that these logics fit within the theory of algebraisable logics. Making use of this result, the following are also proven. Regarding S, we introduced its first semantics, axiomatised by means of a finite Hilbert-style calculus, as well as established a version of the deduction theorem for it. Regarding QN and QNI, we showed that both are algebraisable with respect to the class of quasi-Nelson algebras and quasi-Nelson implication algebras, respectively; we showed that they are non-self-extensional; we showed how to obtain from them, by axiomatic extensions, other well-known logics, such as the {->, ~}-fragment of intuitionistic propositional logic, the {->, ~}-fragment of Nelson’s constructive logic with strong negation and the classical logic; and finally, we made explicit the quaternary term that guarantees that both QN and QNI satisfy the deduction theorem. Regarding N3, we study the role of the Nelson identity ((φ -> (φ -> ψ))∧(~ ψ -> (~ ψ -> φ)) = φ -> ψ) in establishing order-theoretic properties for its algebraic semantics. Moreover, we have studied the ⟨^, v, ~, ¬, 0, 1⟩-subreducts of quasi-Nelson algebras, and by making use of their twist representation, proved that this object-level correspondence can be stated as a categorical equivalence. Lastly, it is worth noting that QN I is the {->, ~}-fragment of QN , so some results concerning QNI may be easily extended to QN.

BANKING MEMBERS:
Presidente - 1517271 - JOAO MARCOS DE ALMEIDA
Interno - 2251108 - UMBERTO RIVIECCIO
Externo à Instituição - FEY LIANG
Externo à Instituição - TOMMASO FLAMINIO
Externa à Instituição - MANUELA BUSANICHE