One-variable fragments of first-order logics

The one-variable fragment of a first-order logic may be viewed as an "S5-like" modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases -- notably, the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and intuitionistic logic -- but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion is given in this paper for the one-variable fragment of a semantically-defined first-order logic -- spanning families of intermediate, substructural, many-valued, and modal logics -- to admit a natural axiomatization. More precisely, such an axiomatization is obtained for the one-variable fragment of any first-order logic based on a variety of algebraic structures with a lattice reduct that has the superamalgamation property, building on a generalized version of a functional representation theorem for monadic Heyting algebras due to Bezhanishvili and Harding. An alternative proof-theoretic strategy for obtaining such axiomatization results is also developed for first-order substructural logics that have a cut-free sequent calculus and admit a certain interpolation property.Comment: arXiv admin note: text overlap with arXiv:2209.0856

Constructive Fuzzy Logics

We generalise Kripke’s semantics for Intuitionistic logic to Hajek’s BL and consider the constructive subsystems of GBLewf and Intuitionistic Affine logic or ALi. The genesis of our semantics is the Poset Product construction for GBL-algebras elucidated in a series of papers by Peter Jipsen, Simone Bova, and Franco Montagna. We present natural deduction systems for all of these systems and corresponding deduction theorems for these same. We present the algebraic semantics for each of the logics under consideration, demonstrate their soundness and completeness with respect to these algebraic semantics. We also show how the classical Kripke semantics for Intuitionistic logic can be recast in terms of Poset Products. We then proceed to the main results, showing how a very natural generalisation of the Kripke semantics holds for each of GBLewf , ALi and Hajek’s BL based on the embedding results of Jipsen and Montagna and the decidability results of Bova and Montagna. We demonstrate soundness and completeness of the logics under our semantics in each case, with the exception of ALi, whose robust completeness with respect to the intended models (relational models with frames valued in involutive pocrims) we leave as an open problem for the ambitious reader

One-Variable Fragments of First-Order Many-Valued Logics

In this thesis we study one-variable fragments of first-order logics. Such a one-variable fragment consists of those first-order formulas that contain only unary predicates and a single variable. These fragments can be viewed from a modal perspective by replacing the universal and existential quantifier with a box and diamond modality, respectively, and the unary predicates with corresponding propositional variables. Under this correspondence, the one-variable fragment of first-order classical logic famously corresponds to the modal logic S5. This thesis explores some such correspondences between first-order and modal logics. Firstly, we study first-order intuitionistic logics based on linear intuitionistic Kripke frames. We show that their one-variable fragments correspond to particular modal Gödel logics, defined over many-valued S5-Kripke frames. For a large class of these logics, we prove the validity problem to be decidable, even co-NP-complete. Secondly, we investigate the one-variable fragment of first-order Abelian logic, i.e., the first-order logic based on the ordered additive group of the reals. We provide two completeness results with respect to Hilbert-style axiomatizations: one for the one-variable fragment, and one for the one-variable fragment that does not contain any lattice connectives. Both these fragments are proved to be decidable. Finally, we launch a much broader algebraic investigation into one-variable fragments. We turn to the setting of first-order substructural logics (with the rule of exchange). Inspired by work on, among others, monadic Boolean algebras and monadic Heyting algebras, we define monadic commutative pointed residuated lattices as a first (algebraic) investigation into one-variable fragments of this large class of first-order logics. We prove a number of properties for these newly defined algebras, including a characterization in terms of relatively complete subalgebras as well as a characterization of their congruences

From Quantum Metalanguage to the Logic of Qubits

The main aim of this thesis is to look for a logical deductive calculus (we will adopt sequent calculus, originally introduced in Gentzen, 1935), which could describe quantum information and its properties. More precisely, we intended to describe in logical terms the formation of the qubit (the unit of quantum information) which is a particular linear superposition of the two classical bits 0 and 1. To do so, we had to introduce the new connective "quantum superposition", in the logic of one qubit, Lq, as the classical conjunction cannot describe this quantum link.Comment: 138 pages, PhD thesis in Mathematic

Weakening Relation Algebras and FL\u3csup\u3e2\u3c/sup\u3e-algebras

FL2-algebras are lattice-ordered algebras with two sets of residuated operators. The classes RA of relation algebras and GBI of generalized bunched implication algebras are subvarieties of FL2-algebras. We prove that the congruences of FL2-algebras are determined by the congruence class of the respective identity elements, and we characterize the subsets that correspond to this congruence class. For involutive GBI-algebras the characterization simplifies to a form similar to relation algebras. For a positive idempotent element p in a relation algebra A, the double division conucleus image p/A/p is an (abstract) weakening relation algebra, and all representable weakening relation algebras (RWkRAs) are obtained in this way from representable relation algebras (RRAs). The class S(dRA) of subalgebras of {p/A/p∶ A ϵ RA; 1 ≤ p2 = p ϵ A} is a discriminator variety of cyclic involutive GBI-algebras that includes RA. We investigate S(dRA) to find additional identities that are valid in all RWkRAs. A representable weakening relation algebra is determined by a chain if and only if it satisfies 0 ≤ 1, and we prove that the identity 1 ≤ 0 holds only in trivial members of S(dRA).https://digitalcommons.chapman.edu/scs_books/1050/thumbnail.jp

On the expressive power of Łukasiewicz square operator

The aim of the paper is to analyze the expressive power of the square operator of Łukasiewicz logic: ∗x=x⊙x⁠, where ⊙ is the strong Łukasiewicz conjunction. In particular, we aim at understanding and characterizing those cases in which the square operator is enough to construct a finite MV-chain from a finite totally ordered set endowed with an involutive negation. The first of our main results shows that, indeed, the whole structure of MV-chain can be reconstructed from the involution and the Łukasiewicz square operator if and only if the obtained structure has only trivial subalgebras and, equivalently, if and only if the cardinality of the starting chain is of the form n+1 where n belongs to a class of prime numbers that we fully characterize. Secondly, we axiomatize the algebraizable matrix logic whose semantics is given by the variety generated by a finite totally ordered set endowed with an involutive negation and Łukasiewicz square operator. Finally, we propose an alternative way to account for Łukasiewicz square operator on involutive Gödel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equation

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic

Fuzzy Description Logics with General Concept Inclusions

Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived