Abstract
In this paper we continue the investigation carried out in Albuquerque et al. (2018) on assertional logics and their relation with the Frege hierarchy, through the notions of relative point-regularity and relative congruence orderability. We provide new characterizations for the classes of logics within the Frege hierarchy under the underlying assumption of assertionality. In particular, an assertional logic \({\mathcal {S}}\) is fully Fregean if and only if the class \({\textsf{Alg}}{\mathcal {S}}\) is congruence orderable. Moreover, an assertional logic \({\mathcal {S}}\) is protoalgebraic if and only the class \({\textsf{Alg}}{\mathcal {S}}\) is point-regular. Finally, we introduce a new notion of relative strong congruence orderability and prove that the class \({\textsf{Alg}}{\mathcal {S}}\) satisfies this property if and only if the intrinsic variety \(\mathbb {V}({\mathcal {S}})\) is congruence orderable. As a consequence, we prove a sufficient condition for the variety problem in AAL.
Second reader: Daniele Mundici
Hugo Albuquerque is Independent Researcher.
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Notes
- 1.
- 2.
- 3.
We denote the operators of takin isomorphic copies, homomorphic images and subdirect products by the usual symbols \(\mathbb {I}\), \(\mathbb {H}\) and \(\mathbb {P}_\textrm{S}\), respectively.
- 4.
This definition does not need the assumption of \({\textsf{K}}\) being closed under \(\mathbb{I}\mathbb{P}_\textrm{S}\), only pointedness.
- 5.
Basically, a more general notion than that of Definition 9.2: Whenever we have
$$\begin{aligned} \Gamma \vdash _{\mathcal {S}}\varphi \Leftrightarrow \boldsymbol{\tau }^{\boldsymbol{Fm}}(\Gamma ) \models _{\textsf{K}}\boldsymbol{\tau }^{\boldsymbol{Fm}}(\varphi ) , \end{aligned}$$for some set of equations \(\boldsymbol{\tau }(x) \subseteq \textrm{Eq}\) and some class of algebras \({\textsf{K}}\), we call the class \({\textsf{K}}\) a \(\boldsymbol{\tau }\)-algebraic semantics for \({\mathcal {S}}\).
- 6.
\({\textsf{WH}_{\scriptscriptstyle (\textrm{N})}}\) is the subvariety of weakly Heyting algebras axiomatized by the defining equations of \(\textsf{WH}\) plus the equation \(x \wedge \square x \approx x\).
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Albuquerque, H., Jansana, R. (2024). Assertional Logics and the Frege Hierarchy. In: Malinowski, J., Palczewski, R. (eds) Janusz Czelakowski on Logical Consequence. Outstanding Contributions to Logic, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-031-44490-6_9
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