This paper argues for the reasonableness of an inclusive conception of default reasoning. The inclusive conception allows untriggered default rules to influence beliefs: Since a default “from , infer ” is a defeasible inference rule, it by default warrants a belief in the material implication , even if is not believed. Such inferences are not allowed in standard default logic of the Reiter tradition, but are reasonable by analogy to the Deduction Theorem for classical logic. Our main contribution is a formal framework for inclusive default reasoning. The framework has a solid philosophical foundation, it draws conclusions non-trivially different from non-inclusive frameworks, and it exhibits a host of benchmark properties deemed desirable in the literature—e.g., that extensions always exist and are consistent.

In this paper we show how to introduce a conditional to Kripke’s theory of truth that respects the deduction theorem for the consequence relation associated with the theory. To this effect we develop a novel supervaluational framework, called strong Kleene supervaluation, that we take to be a promising framework for handling the truth-conditions of non-monotone notion in the presence of semantic indeterminacy more generally.

Abstract The logico-algebraic study of Lewis’s hierarchy of variably strict conditional logics has been essentially unexplored, hindering our understanding of their mathematical foundations, and the connections with other logical systems. This work starts filling this gap by providing a logico-algebraic analysis of Lewis’s logics. We begin by introducing novel finite axiomatizations for Lewis’s logics on the syntactic side, distinguishing between global and local consequence relations on Lewisian sphere models on the semantical side, in parallel to the case of modal logic. As first main results, we prove the strong completeness of the calculi with respect to the corresponding semantical consequence on spheres, and a deduction theorem. We then demonstrate that the global calculi are strongly algebraizable in terms of a variety of Boolean algebras with a binary operator representing the counterfactual implication; in contrast, we show that the local ones are generally not algebraizable, although they can be characterized as the degree-preserving logic over the same algebraic models. This yields the strong completeness of all the logics with respect to the algebraic models.

We consider the long-time dynamics for the self-dual Chern–Simons–Schrödinger (CSS) equation within equivariant symmetry. The self-dual CSS equation is a self-dual L^{2} -critical equation having pseudoconformal invariance and solitons. In this paper, we show that any m -equivariant, m\geq1 , H^{3} finite-time blow-up solution to the self-dual CSS equation is a pseudoconformal blow-up solution. More precisely, such a solution decomposes into the sum of one modulated soliton, which contracts at the pseudoconformal rate \lambda(t)\sim T-t , and a radiation. Applying the pseudoconformal transform in reverse, we also obtain a refined soliton resolution theorem for m -equivariant, m\geq1 , sufficiently regular and localized solutions: any such solutions blow up in the pseudoconformal regime, scatter (to 0 ), or scatter to a modulated soliton with some fixed scale and phase. To our knowledge, this is the first result on the full classification of the dynamics of arbitrary smooth and spatially decaying solutions in the class of nonlinear Schrödinger equations which are not known to be completely integrable. Our analysis not only builds upon the previous works, especially the soliton resolution theorem by the author, Kwon, and Oh, but also refines all steps of the arguments typically employed in the forward construction of blow-up dynamics. The key feature of the proof is that we can identify the singular and regular parts of any H^{3} finite-time blow-up solutions, such that the evolution of the singular part is governed by a universal modulation dynamics while the regular part is kept H^{3} -bounded even up to the blow-up time. As a byproduct, we also observe that the asymptotic profile has a universal singular structure.

On trivalent logics, probabilistic weak deduction theorems, and a general import-export principle

Quillen's Resolution Theorem in algebraic K-theory provides a powerful computational tool for calculating K-groups of exact categories. At the level of K0, this result goes back to Grothendieck. In this article, we first establish an extriangulated version of Grothendieck's Resolution Theorem.Second, we use this Extriangulated Resolution Theorem to gain new insight into the index theory of triangulated categories. Indeed, we propose an index with respect to an extension-closed subcategory N of a triangulated category C and we prove an additivity formula with error term. Our index recovers the index with respect to a contravariantly finite, rigid subcategory X defined by Jørgensen and the second author, as well as an isomorphism between K0sp(X) and the Grothendieck group of a relative extriangulated structure CRX on C when X is n-cluster tilting. In addition, we generalize and enhance some results of Fedele. Our perspective allows us to remove certain restrictions and simplify some arguments.Third, as another application of our ExtriangulatedResolution Theorem, we show that if X is n-cluster tilting in an abelian category, then the index introduced by Reid gives an isomorphism K0(CRX)≅K0sp(X).

It is shown that both classical and intuitionistic propositional logics of an associative binary modality are undecidable. The proof is based on the deduction theorem for these logics.

ABSTRACT In this paper, I examine Ruth Barcan Marcus's early formal work on modal systems and the deduction theorem, both for the material and the strict conditional. Marcus proved that the deduction theorem for the material conditional does not hold for system S2 but holds for S4. This last result is at odds with the recent claim that without proper restrictions the deduction theorem fails also for S4. I explain where the contrast stems from. For the strict conditional, Marcus proved the deduction theorem for S4 though restricted to arguments with necessary premises. I discuss Marcus's result and analyze her philosophical position on the significance of the deduction theorem for modal systems designed to express the notion of deducibility.

Semiconic idempotent logic I: Structure and local deduction theorems

In this paper, we study resolving subcategories and singularity categories. First, if the left perpendicular category of a module [Formula: see text] over an Artin algebra [Formula: see text] is the additive closure of another module [Formula: see text], then the singularity category of [Formula: see text] and that of the endomorphism algebra [Formula: see text] of [Formula: see text] are closed related. This gives a categorical version of a recent result of Zhang ( [31, Theorem 2]). Second, we apply the resolution theorem for derived categories to elliptic curves, the monomorphism subcategory of a Gorenstein algebra and of a kind of Eilenberg–Moore category. As consequences, their singularity categories are equivalent, which explains why monomorphism categories are closely related to singularity categories.

Abstract Orthomodular logic is a weakening of quantum logic in the sense of Birkhoff and von Neumann. Orthomodular logic is shown to be a nonlinear noncommutative logic. Sequents are given a physically motivated semantics that is consistent with exactly one semantics for propositional formulas that use negation, conjunction, and implication. In particular, implication must be interpreted as the Sasaki arrow, which satisfies the deduction theorem in this logic. As an application, this deductive system is extended to two systems of predicate logic: the first is sound for Takeuti’s quantum set theory, and the second is sound for a variant of Weaver’s quantum logic.

The increasing significance of theorem proving-based formalization in mathematics and computer science highlights the necessity for formalizing foundational mathematical theories. In this work, we employ the Coq interactive theorem prover to methodically formalize the language, semantics, and syntax of propositional logic, a fundamental aspect of mathematical reasoning and proof construction. We construct four Hilbert-style axiom systems and a natural deduction system for propositional logic, and establish their equivalences through meticulous proofs. Moreover, we provide formal proofs for essential meta-theorems in propositional logic, including the Deduction Theorem, Soundness Theorem, Completeness Theorem, and Compactness Theorem. Importantly, we present an exhaustive formal proof of the Completeness Theorem in this paper. To bolster the proof of the Completeness Theorem, we also formalize concepts related to mappings and countability, and deliver a formal proof of the Cantor–Bernstein–Schröder theorem. Additionally, we devise automated Coq tactics explicitly designed for the propositional logic inference system delineated in this study, enabling the automatic verification of all tautologies, all internal theorems, and the majority of syntactic and semantic inferences within the system. This research contributes a versatile and reusable Coq library for propositional logic, presenting a solid foundation for numerous applications in mathematics, such as the accurate expression and verification of properties in software programs and digital circuits. This work holds particular importance in the domains of mathematical formalization, verification of software and hardware security, and in enhancing comprehension of the principles of logical reasoning.

Deduction Theorem in Congruential Modal Logics

Transfer matrix method for linear vibration analysis of flexible multibody systems

The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.

The identification of stiffness parameters of armoured vehicles is the prerequisite and key to the accurate calculation of armoured vehicle dynamics. A linear multi-rigid-flexible body dynamics model of the armoured vehicle system is developed by using transfer matrix method of multibody system. The overall transfer equation, overall transfer matrix, and characteristics equation of the system are deduced based on the automatic deduction theorem of overall transfer equation of multibody system. Combined with an adaptive genetic algorithm, the objective function is built using the eigenfrequencies obtained from the modal test and simulation to optimize the minimum value. In comparison with the ordinary genetic algorithm, the adaptive genetic algorithm has a faster convergence speed. Numerical simulation and experimental verification show the effectiveness and accuracy of the method. The paper provides an effective method for studying the identification of stiffness parameters of armoured vehicles.

Abstract In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication $\phi \rightarrow \psi $ with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in [24] since that article is about the property more correctly named ‘deductive interpolation’ (see Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent.

Philosophical issues often turn into logic. That is certainly true of Moore’s Paradox, which tends to appear and reappear in many philosophical contexts. There is no doubt that its study belongs to pragmatics rather than semantics or syntax. But it is also true that issues in pragmatics can often be studied fruitfully by attending to their projection, so to speak, onto the levels of semantics or syntax — just in the way that problems in spherical geometry are often illuminated by the study of projections onto a plane. To begin I will describe a potentially vast landscape of logics of a certain form, with some illustrations of how they appear naturally in response to some problems in philosophical logic. Then I will turn Moore’s Paradox into logic, within that landscape, and show how far it can be illuminated therein.

This article is a tribute to the scientific legacy of automated reasoning pioneer and JAR founder Lawrence T. (Larry) Wos. Larry’s main technical contributions were the set-of-support strategy for resolution theorem proving, and the demodulation and paramodulation inference rules for building equality into resolution. Starting from the original definitions of these concepts in Larry’s papers, this survey traces their evolution, unearthing the often forgotten trails that connect Larry’s original definitions to those that became standard in the field.

Here “group” means additive abelian group. A compact group G contains δ–subgroups, that is, compact totally disconnected subgroups Δ such that G/Δ is a torus. The canonical subgroup Δ(G) of G that is the sum of all δ–subgroups of G turns out to have striking properties. Lewis, Loth and Mader obtained a comprehensive description of Δ(G) when considering only finite dimensional connected groups, but even for these, new and improved results are obtained here. For a compact group G, we prove the following: Δ(G) contains tor(G), is a dense, zero-dimensional subgroup of G containing every closed totally disconnected subgroup of G, and G/Δ(G) is torsion-free and divisible; Δ(G) is a functorial subgroup of G, it determines G up to topological isomorphism, and it leads to a “canonical” resolution theorem for G. The subgroup Δ(G) appeared before in the literature as td(G) motivated by completely different considerations. We survey and extend earlier results. It is shown that td, as a functor, preserves proper exactness of short sequences of compact groups.