G(3)' AS THE LOGIC OF MODAL 3-VALUED HEYTING ALGEBRAS

Autor(es)

Coniglio, Marcelo Esteban ^{[1, 2]} ; Figallo-Orellano, Aldo ^{[2, 3]} ; Hernandez-Tello, Alejandro ^[4] ; Perez-Gaspar, Miguel ^[5]

Afiliação do(s) autor(es)

^[1] Univ Estadual Campinas, UNICAMP, Inst Philosophy & Humanities IFCH, Campinas, SP - Brazil

^[2] Univ Estadual Campinas, Ctr Log Epistemol & Hist Sci CLE, UNICAMP, Campinas, SP - Brazil

^[3] Univ Nacl Sur UNS, Dept Matemat, Buenos Aires, DF - Argentina

^[4] Univ Tecnol Mixteca UTM, Inst Fis & Matemat, Oaxaca, Oaxaca - Mexico

^[5] Univ Nacl Autonoma Mexico, Fac Ingn, Mexico City, DF - Mexico

• Tipo de documento: Artigo Científico
• Fonte: JOURNAL OF APPLIED LOGICS-IFCOLOG JOURNAL OF LOGICS AND THEIR APPLICATIONS; v. 9, n. 1, p. 175-197, JAN 2022.
• Citações Web of Science: 0

Resumo

In 2001, W. Carnielli and Marcos considered a 3-valued logic in order to prove that the schema phi V (phi -> psi) is not a theorem of da Costa's logic C-omega. In 2006, this logic was studied (and baptized) as G(3)' by Osorio et al. as a tool to define semantics of logic programming. It is known that the truth-tables of G(3)' have the same expressive power than the one of Lukasiewicz 3-valued logic as well as the one of Godel 3-valued logic G(3). From this, the three logics coincide up-to language, taking into acccount that 1 is the only designated truth-value in these logics. From the algebraic point of view, Canals-Frau and Figallo have studied the 3-valued modal implicative semilattices, where the modal operator is the wellknown Moisil-Monteiro-Baaz A operator, and the supremum is definable from this. We prove that the subvariety obtained from this by adding a bottom element Delta is term-equivalent to the variety generated by the 3-valued algebra of G(3)' . The algebras of that variety are called G(3)'-algebras. From this result, we obtain the equations which axiomatize the variety of G(3)'-algebras. Moreover, we prove that this variety is semisimple, and the 3-element and the 2-element chains are the unique simple algebras of the variety. Finally an extension of G(3)' to first-order languages is presented, with an algebraic semantics based on complete G(3)'-algebras. The corresponding soundness and completeness theorems are obtained. (AU)

• Processo FAPESP: 16/21928-0 - Semânticas não-determinísticas para as lógicas da inconsistência formal
• Beneficiário: Aldo Figallo Orellano
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado