Poset products as relational models

Abstract

We introduce a relational semantics based on poset products, and provide sufficient conditions guaranteeing its soundness and completeness for various substructural logics. We also demonstrate that our relational semantics unifies and generalizes two semantics already appearing in the literature: Aguzzoli, Bianchi, and Marra's temporal flow semantics for Hájek's basic logic, and Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Lukasiewicz logic. As a consequence of our general theory, we recover the soundness and completeness results of these prior studies in a uniform fashion, and extend them to infinitely-many other substructural logics.

Some Modal and Temporal Translations of Generalized Basic Logic

“…The translation results of this paper rely heavily on the poset product construction of Jipsen and Montagna (see [19,20]), which we now sketch. Our discussion of poset products is drawn mainly from [12], to which we refer the reader for a more detailed summary.…”

“…On the other hand, [12] generalizes the temporal flow semantics for basic logic [1], which is deployed in [2] to obtain a modal translation of Gödel-Dummett logic into an extension of Prior's tense logic [23]. Inspired by this work, the present study makes the modal connection from [12] explicit and offers modal and temporal translations of generalized basic logic into certain expanded Lukasiewicz logics. Like [2], these translations are directly analogous to the well known Gödel-McKinsey-Tarski translation of intuitionistic logic into the classical modal logic S4, and are connected to the broader theory of modal companions of superintuitionistic logics.…”

“…In this formulation, generalized basic logic is related to Lukasiewicz logic [8] in much the same way that intuitionistic logic is related to classical logic. For instance, generalized basic logic admits a Kripke-style relational semantics [12] in which worlds are valued in MV-algebra chains, mirroring the well known Kripke semantics for intuitionistic logic (in which worlds are valued in the 2-element Boolean algebra). It is evident from [12] that generalized basic logic may be viewed as a fragment of a modal Lukasiewicz logic, but the details of this modal connection are therein left implicit.…”

“…For instance, generalized basic logic admits a Kripke-style relational semantics [12] in which worlds are valued in MV-algebra chains, mirroring the well known Kripke semantics for intuitionistic logic (in which worlds are valued in the 2-element Boolean algebra). It is evident from [12] that generalized basic logic may be viewed as a fragment of a modal Lukasiewicz logic, but the details of this modal connection are therein left implicit. On the other hand, [12] generalizes the temporal flow semantics for basic logic [1], which is deployed in [2] to obtain a modal translation of Gödel-Dummett logic into an extension of Prior's tense logic [23].…”

“…It is evident from [12] that generalized basic logic may be viewed as a fragment of a modal Lukasiewicz logic, but the details of this modal connection are therein left implicit. On the other hand, [12] generalizes the temporal flow semantics for basic logic [1], which is deployed in [2] to obtain a modal translation of Gödel-Dummett logic into an extension of Prior's tense logic [23]. Inspired by this work, the present study makes the modal connection from [12] explicit and offers modal and temporal translations of generalized basic logic into certain expanded Lukasiewicz logics.…”