ABSTRACT. We start with an arbitrary but fixed positive integer n. If n is even, we divide it by 2 and get ⁿ/₂. If n is odd, we multiply n by 3 and add 1 to obtain the even number 3n + 1. Then we repeat the procedure indefinitely. Each sequence defined in this way is referred to as a Collatz sequence. The Collatz conjecture says that no matter what number n we start with, we will always eventually reach 1. For example, if one starts with 5, one gets

5, 3 · 5 + 1 = 16, 8, 4, 2, 1.

The number 7 yields the sequence:

7, 3 · 7 + 1 = 22, 11, 3 · 11 + 1 = 34, 17, 3 · 17 + 1 = 52, 26, 13, 3 · 13 + 1 = 40, 20, 10, 5, 3 · 5 + 1 = 16, 8, 4, 2, 1.

The Collatz conjecture is an old problem in number theory, named after Lothar Collatz, who proposed it in 1937.

In the talk we show that the Collatz conjecture is unprovable in the elementary Peano arithmetic PA. The proof refers to the general approach to first-order logic based on Rasiowa-Sikorski Lemma and the derived notion of a Rasiowa-Sikorski set. The central idea consists in constructing countable models from individual variables by means of appropriate Rasiowa-Sikorski sets. This idea is developed and applied to the language of Peano arithmetic.

Advanced methods borrowed from the contemporary mathematical logic are applied. Such a ``logical'' methodology is a useful addition to the dominant approach to number theory based on analytical methods.

ABSTRACT. It is common in various non-classical logics, especially in relevant logics, to characterize negation semantically via the operation known as Routley star. This operation works well within relational semantic frameworks based on prime theories. We study this operation in the context of ``information-based'' semantics for which it is characteristic that sets of formulas supported by individual information states are theories that do not have to be prime. We show that, somewhat surprisingly, the incorporation of Routley star into the information-based semantics does not lead to a collapse or a trivialization of the whole semantic system. On the contrary, it leads to a technically elegant though quite restricted semantic framework that determines a particular logic. We study some basic properties of this semantics. For example, we show that within this framework double negation law is valid only in involutive linear frames. We characterize axiomatically the logic of all linear frames and show that the logic of involutive linear frames coincides with a system that Mike Dunn coined Kalman logic. This logic is the fragment of the ``semi-relevant'' logic known as R-mingle, for the language restricted to conjunction, disjunction and negation. Finally, we characterize by a deductive system the logic of all information frames equipped with Routley star.

ABSTRACT. In this paper I show, with a rich and systematized diet of examples, that many contra-classical logics can be presented as variants of \textbf{FDE}, obtained by modifying at least one of the truth or falsity conditions of some connective. Then I argue that using Dunn semantics provides a clear understanding of the source of contra-classicality, namely, connectives that have either the classical truth or the classical falsity condition of another connective. This requires a fine-grained analysis of the sorts of modifications that can be made to an evaluation condition, analysis which I offer here as well.

ABSTRACT. The introduction of paraconsistency in labelled transition systems provides a sound and smooth way to address some critical is- sues in quantum computing, as, for example, qubit decoherence. This paper characterises a modal, intuitionistic logic able to express and ver- ify properties of such systems. Notions of simulation and bisimulation in this setting are discussed and shown to be preserved by the logic.

ABSTRACT. Many-sorted logic is not only a natural logic for reasoning about more than one sort of objects, but also a unifying logic, the best target logic in translation issues, due to its efficient proof theory, flexibility, naturalness and versatility to adapt to reasoning about a variety of objects.

Translations can be seen as the path to completeness in three possible stages: abstract completeness at the first stage, weak and strong completeness at the other two. The case of second-order logic can act as the source of inspiration for translations of other logical systems. The idea is to include in the many-sorted structure obtained by direct conversion of the logic being translated a domain for sets (and relations); namely, for all sets and relations defined in the original logic with their own formulas. And so, we are somehow shifting to second-order logic. In these structures, the Comprehension Schema restricted to formulas which are translations is always true. The idea of restricting CS to obtain new logics is taken from Henkin [1].

References:

[1] Leon Henkin (1953): Banishing the rule of substitution for functional variables. Journal of Symbolic Logic 18(3), pp. 201–208, doi:10.2307/2267403.

ABSTRACT. There is a common agreement that each connexive logic should satisfy the Aristotles and Boethian Theses (AB). However, the sole AB theses don't guarantee any common content or other form of "connexions" as they are true in binary matrix {0,1} with distinguished value of 1, with classical material implication and negation defined as ~1 = ~0 = 1. Similarly, AB are true in a binary matrix with classical negation and implication defined as x ⇒ y = 1 iff x = y.

Counterexamples above show that the sole AB theses are very weak and should be strengthen in some way. We can eliminate the first counterexample by assuming that negation behaves in a classical way. We did it in [6] and [7]. It brings us to the notion of Boolean connexive logic. Boolean algebras are defined in terms of ∧, ∨, ¬. In Boolean connexive logic these connectives behave in classical way. It made us to refer to George Boole here.

We can eliminate a second counterexample and consider connexive logics with material implication. It hardly know lead to promising research area. Anyway we could call such logics, a material connexive logics. Obviously we could eliminate both the counterexamples. It would bring us to connexive logics which are neither Boolean not material.

In this lecture we concetrate on Boolean connexive logics. We indicate a numer of possible desiderata strengthening Aristotle and Boethius' theses. We investigate them by means of relating semantics.

A relating semantics has its origins in Epstein and Walton papers. However in its full generality it has been proposed by Jarmuek and Kaczkowski in [4]. It is extensively studied among others in [3], [5], [10]. The general idea of relating semantics is based on the relating relation R. The fact that ARB – sentences A and B relate with respect to R – could be interpreted in many ways depending on the motivations of a given logical system. For example, A and B could be related causally, they could be related in the sense of time order, they could have a common content. Generally, two sentences are related because they have something in common.

In Boolean connexive logics only implication depends on a relating relation R. The truth conditions for the propositional letters and ∧, ∨, and ¬ remain standard, i.e. as in classical logic. Only implication → has an intensional nature, it depends on a relation R: 〈v,R〉 ⊨ A → B iff [〈v,R〉 ⊭ A or 〈v,R〉 ⊨ B] & ARB.

By a minimal Boolean connexive logic we mean the least set of sentences containing all classical tautologies expressed by means of ∧, ∨, ¬, (A1), (A2), (B1), (B2), (A → B) ⊃ (A ⊃ B) and closed under substitutions and modus ponens with respect to ⊃. ⊃ denotes material implication. Let JT denote a class of all relations R such that R is (a1), (a2), (b1), (b2). Let JT^¬ denote a class of all relations from JT satisfying (c1), where:

• R is (a1) iff for all A ∈ For_CF, AR̃~A;
• R is (a2) iff for all A ∈ For_CF, ~AR̃A;
• R is (b1) iff for all A,B ∈ For_CF, if ARB, then AR̃~B and (A →^w B)R~(A →^w ~B);
• R is (b2) iff for all A,B ∈ For_CF, if ARB, then AR̃~B and (A →^w ~B)R~(A →^w B);
• R is (c1) iff for all A,B ∈ For_CF, if ARB then ~AR~B.

In [6] we characterized Boolean connexive logics by means of relating semantics. Then Mateusz Klonowski (forthcoming) made this result stronger. Klonowski proved that the class JT determines minimal Boolean connexive logics. The class JT^¬ determines the least Boolean connexive logics satisfying the following two axioms: (A → B) ⊃ (¬¬A → ¬¬B) (A → B) ⊃ ((¬A → ¬B) ∨ (¬A ∧ B)).

In [2] Estrada-González and Ramírez-Cámara consider a number of properties added to AB. They used terms hyper-connexive and totally connexive logics. Malinowski and Nicolás-Francisco in [9] analyzed it in terms of relating semantics for Boolean connexive logcs. In particular they show that Minimal Boolean Connexive Logic (or alternatively the logic determined by JT) is Abelardian, strongly consistent, Kapsner strong and antiparadox. They also construct examples showing that it is not simplificative, neither conjunction-idempotent nor strongly inconsistent logics. Malinowski and Nicolás-Francisco also describe in terms of relating semantics an interrelation between hyper-connexive logics and JT logic.

In the last part we remind less known Barbershop paradox published by Lewis Carrol in 1894 [1], see also [8], [10]. We show that if interpreted by means of material implication paradox reduces to simple paralogism. Then we interpret it by means of connexive implication. We develop relating semantics for Boolean connexive logics to eludicate Barbershop paradox.

References:

[1] Lewis Carroll (1894): A logical paradox. Mind 3(11), pp. 436–438, doi:10.1093/mind/III.11.436.

[2] Luis Estrada-González & Elisngela Ramírez-Cámara (2016): A Comparison of Connexive Logics. The If- CoLog Journal of Logics and their Applications 3(3), pp. 341–355.

[3] Tomasz Jarmużek (2021): Relating semantics as fine-grained semantics for intensional propositional logics. In Alessandro Giordani & Jacek Malinowski, editors: Logic in High Definition. Trends in Logical Semantics, Trends in Logic 56, Springer, Cham, pp. 13–30, doi:10.1007/978-3-030-53487-5_2.

[4] Tomasz Jarmużek & Bartosz J. Kaczkowski (2014): On some logic with a relation imposed on formulae: Tableau system F. Bulletin of the Section of Logic 43(1/2), pp. 53–72.

[5] Tomasz Jarmużek & Mateusz Klonowski (2021): Some intensional logics defined by relating semantics and tableau systems. In Alessandro Giordani & Jacek Malinowski, editors: Logic in High Definition. Trends in Logical Semantics, Trends in Logic 56, Springer, Cham, pp. 31–48, doi:10.1007/978-3-030-53487-5_3.

[6] Tomasz Jarmużek & Jacek Malinowski (2019): Boolean Connexive Logics: Semantics and tableau approach. Logic and Logical Philosophy 28(3), pp. 427–448, doi:10.12775/LLP.2019.003.

[7] Tomasz Jarmużek & Jacek Malinowski (2019): Modal Boolean Connexive Logics: Semantics and tableau approach. Bulletin of the Section of Logic 48(3), pp. 213–243, doi:10.18778/0138-0680.48.3.05.

[8] Jacek Malinowski (2019): Barbershop Paradox and Connexive Implication. Ruch Filozoficzny (Philosophical Movement) 75(2), pp. 109–115, doi:10.12775/RF.2019.023.

[9] Jacek Malinowski & Ricardo Arturo Nicolás-Francisco: Relating semantics for hyper-connexive and totally connexive logics. Forthcoming.

[10] Jacek Malinowski & Rafał Palczewski (2021): Relating Semantics for Connexive Logic. In Alessandro Giordani & Jacek Malinowski, editors: Logic in High Definition. Trends in Logical Semantics, Trends in Logic 56, Springer, Cham, pp. 49–65, doi:10.1007/978-3-030-53487-5_4.

ABSTRACT. With the emerging applications that involve complex distributed systems branching-time specifications are specifically important as they reflect dynamic and nondeterministic nature of such applications. We describe the expressive power of a simple yet powerful branching-time specification framework which has been developed as part of clausal resolution for branching-time temporal logics. We show the encoding of Buchi Tree Automata in the language of the normal thus representing, syntactically, tree automata in a high-level way.

ABSTRACT. We reconsider the idea of superconnexivity, an idea that has not received much attention so far. We inspect more closely the problems with the proposal that are responsible for this disregard. However, we also suggest a slight modification of the idea that has a much better chance of delivering the desired results, which we call super-bot-connexivity.

ABSTRACT. This paper formulates a bilateral account of harmony that is an alternative to one proposed by Francez. It builds on an account of harmony for unilateral logic proposed by Kürbis and the observation that reading the rules for the connectives of bilateral logic bottom up gives the grounds and consequences of formulas with the opposite speech act. I formulate a process of inversion which allows the determination of assertive elimination rules from assertive introduction rules, and rejective elimination rules from rejective introduction rules, and conversely. It corresponds to Francez's notion of vertical harmony. I also formulate a process of conversion, which allows the determination of rejective introduction rules from assertive elimination rules and conversely, and the determination of assertive introduction rules from rejective elimination rules and conversely. It corresponds to Francez's notion of horizontal harmony.

ABSTRACT. In "Aristotle's thesis in consistent and inconsistent logics", Chris Mortensen introduced a connexive logic commonly known as `\textbf{M3V}'. \textbf{M3V} is obtained by adding a special conditional to González-Asenjo/Priest's \textbf{LP}. Among its most notable features, besides its being connexive, \textbf{M3V} is negation-inconsistent and it validates the negation of every conditional. But Mortensen has also studied and applied extensively other non-connexive logics, for example, \emph{closed set logic}, \textbf{CSL}, and a variant of Sette's logic, identified and called \textbf{P$^2$} by Marcos in "A problem of da Costa".

In this paper, we analyze and compare systematically the connexive variants of \textbf{CSL} and \textbf{P$^2$}, obtained by adding the \textbf{M3V} conditional to them. Our main observations are two. First, that the inconsistency of \textbf{M3V} is exacerbated in the connexive variant of closed set logic, while it is attenuated in the connexive variant of the Sette-like \textbf{P$^2$}. Second, that the \textbf{M3V} conditional is, unlike other conditionals, "connexively stable", meaning that it remains connexive when combined with the main paraconsistent negations.