http://hdl.handle.net/11089/41010 2026-04-06T20:18:35Z 2026-04-06T20:18:35Z Hao, Yunge Tourlakis, George http://hdl.handle.net/11089/41068 2022-03-11T02:41:26Z 2021-08-23T00:00:00Z

An Arithmetically Complete Predicate Modal Logic Hao, Yunge; Tourlakis, George This paper investigates a first-order extension of GL called \(\textup{ML}^3\). We outline briefly the history that led to \(\textup{ML}^3\), its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness (with respect to finite reverse well-founded Kripke models) is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that \(\textup{ML}^3\) is arithmetically complete. As expanded below, \(\textup{ML}^3\) is a first-order modal logic that along with its built-in ability to simulate general classical first-order provability―"\(\Box\)" simulating the the informal classical "\(\vdash\)"―is also arithmetically complete in the Solovay sense.

2021-08-23T00:00:00Z Sayed Ahmed, Tarek http://hdl.handle.net/11089/41067 2022-03-11T02:41:30Z 2021-07-21T00:00:00Z

On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results Sayed Ahmed, Tarek Fix a finite ordinal \(n\geq 3\) and let \(\alpha\) be an arbitrary ordinal. Let \(\mathsf{CA}_n\) denote the class of cylindric algebras of dimension \(n\) and \(\sf RA\) denote the class of relation algebras. Let \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for the class of polyadic (equality) algebras of dimension \(\alpha\). We reprove that the class \(\mathsf{CRCA}_n\) of completely representable \(\mathsf{CA}_n\)s, and the class \(\sf CRRA\) of completely representable \(\mathsf{RA}\)s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \(\sf V\) between polyadic algebras of dimension \(n\) and diagonal free \(\mathsf{CA}_n\)s. We show that that the class of completely and strongly representable algebras in \(\sf V\) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \(\sf CRRA\) is not closed under \(\equiv_{\infty,\omega}\). In contrast, we show that given \(\alpha\geq \omega\), and an atomic \(\mathfrak{A}\in \mathsf{PEA}_{\alpha}\), then for any \(n/p>

2021-07-21T00:00:00Z Inoue, Takao http://hdl.handle.net/11089/41066 2022-03-11T02:41:28Z 2021-11-09T00:00:00Z

A Sound Interpretation of Leśniewski's Epsilon in Modal Logic KTB Inoue, Takao In this paper, we shall show that the following translation \(I^M\) from the propositional fragment \(\bf L_1\) of Leśniewski's ontology to modal logic \(\bf KTB\) is sound: for any formula \(\phi\) and \(\psi\) of \(\bf L_1\), it is defined as(M1) \(I^M(\phi \vee \psi) = I^M(\phi) \vee I^M(\psi)\),(M2) \(I^M(\neg \phi) = \neg I^M(\phi)\),(M3) \(I^M(\epsilon ab) = \Diamond p_a \supset p_a . \wedge . \Box p_a \supset \Box p_b .\wedge . \Diamond p_b \supset p_a\),where \(p_a\) and \(p_b\) are propositional variables corresponding to the name variables \(a\) and \(b\), respectively. In the last, we shall give some comments including some open problems and my conjectures.

2021-11-09T00:00:00Z Szmuc, Damian E. http://hdl.handle.net/11089/41065 2022-03-11T02:41:31Z 2021-05-27T00:00:00Z

The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework) Szmuc, Damian E. We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined in terms of a \(p\)-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints.

2021-05-27T00:00:00Z