Hilbert-style formalism for two-dimensional notions of consequence
The present work proposes a two-dimensional Hilbert-style deductive formalism (H-formalism) for B-consequence relations, a class of two-dimensional logics that generalize the usual (Tarskian, one-dimensional) notions of logic. We argue that the two-dimensional environment is appropriate to the study...
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| Formato: | Dissertação |
| Idioma: | pt_BR |
| Publicado em: |
Universidade Federal do Rio Grande do Norte
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| Endereço do item: | https://repositorio.ufrn.br/handle/123456789/46792 |
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| Resumo: | The present work proposes a two-dimensional Hilbert-style deductive formalism (H-formalism) for B-consequence relations, a class of two-dimensional
logics that generalize the usual (Tarskian, one-dimensional) notions of logic.
We argue that the two-dimensional environment is appropriate to the study
of bilateralism in logic, by allowing the primitive judgments of assertion
and denial (or, as we prefer, the cognitive attitudes of acceptance and rejection) to act on independent but interacting dimensions in determining
what-follows-from-what. In this perspective, our proposed formalism constitutes an inferential apparatus for reasoning over bilateralist judgments. After
a thorough description of the inner workings of the proposed proof formalism,
which is inspired by the one-dimensional symmetrical Hilbert-style systems,
we provide a proof-search algorithm for finite analytic systems that runs in
at most exponential time, in general, and in polynomial time when only rules
having at most one formula in the succedent are present in the concerned
system. We delve then into the area of two-dimensional non-deterministic
semantics via matrix structures containing two sets of distinguished truthvalues, one qualifying some truth-values as accepted and the other as rejected,
constituting a semantical path for bilateralism in the two-dimensional environment. We present an algorithm for producing analytic two-dimensional
Hilbert-style systems for sufficiently expressive two-dimensional matrices, as
well as some streamlining procedures that allow to considerably reduce the
size and complexity of the resulting calculi. For finite matrices, we should
point out that the procedure results in finite systems. In the end, as a case
study, we investigate the logic of formal inconsistency called mCi with respect to its axiomatizability in terms of Hilbert-style systems. We prove that there is no finite one-dimensional Hilbert-style axiomatization for this logic,
but that it inhabits a two-dimensional consequence relation that is finitely
axiomatizable by a finite two-dimensional Hilbert-style system. The existence
of such system follows directly from the proposed axiomatization procedure,
in view of the sufficiently expressive 5-valued non-deterministic bidimensional
semantics available for the mentioned two-dimensional consequence relation. |
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