50 research outputs found
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
The variety generated by all the ordinal sums of perfect MV-chains
We present the logic BL_Chang, an axiomatic extension of BL (see P. H\'ajek -
Metamathematics of fuzzy logic - 1998, Kluwer) whose corresponding algebras
form the smallest variety containing all the ordinal sums of perfect MV-chains.
We will analyze this logic and the corresponding algebraic semantics in the
propositional and in the first-order case. As we will see, moreover, the
variety of BL_Chang-algebras will be strictly connected to the one generated by
Chang's MV-algebra (that is, the variety generated by all the perfect
MV-algebras): we will also give some new results concerning these last
structures and their logic.Comment: This is a revised version of the previous paper: the modifications
concern essentially the presentation. The scientific content is substantially
unchanged. The major variations are: Definition 2.7 has been improved.
Section 3.1 has been made more compact. A new reference, [Bus04], has been
added. There is some minor modification in Section 3.
First-order Goedel logics
First-order Goedel logics are a family of infinite-valued logics where the
sets of truth values V are closed subsets of [0, 1] containing both 0 and 1.
Different such sets V in general determine different Goedel logics G_V (sets of
those formulas which evaluate to 1 in every interpretation into V). It is shown
that G_V is axiomatizable iff V is finite, V is uncountable with 0 isolated in
V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for
each of these cases are given. The r.e. prenex, negation-free, and existential
fragments of all first-order Goedel logics are also characterized.Comment: 37 page
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
First-order Nilpotent Minimum Logics: first steps
Following the lines of the analysis done in [BPZ07, BCF07] for first-order
G\"odel logics, we present an analogous investigation for Nilpotent Minimum
logic NM. We study decidability and reciprocal inclusion of various sets of
first-order tautologies of some subalgebras of the standard Nilpotent Minimum
algebra. We establish a connection between the validity in an NM-chain of
certain first-order formulas and its order type. Furthermore, we analyze
axiomatizability, undecidability and the monadic fragments.Comment: In this version of the paper the presentation has been improved. The
introduction section has been rewritten, and many modifications have been
done to improve the readability; moreover, numerous references have been
added. Concerning the technical side, some proofs has been shortened or made
more clear, but the mathematical content is substantially the same of the
previous versio
Vagueness and Formal Fuzzy Logic: Some Criticisms
In the common man reasoning the presence of vague predicates is pervasive and under the name âfuzzy logic in narrow senseâ or âformal fuzzy logicâ there are a series of attempts to formalize such a kind of phenomenon. This paper is devoted to discussing the limits of these attempts both from a technical point of view and with respect the original and principal task: to define a mathematical model of the vagueness. For example, one argues that, since vagueness is necessarily connected with the intuition of the continuum, we have to look at the order-based topology of the interval [0,1] and not at the discrete topology of the set {0,1}. In accordance, in switching from classical logic to a logic for the vague predicates, we cannot avoid the use of the basic notions of real analysis as, for example, the ones of âapproximationâ, âconvergenceâ, âcontinuityâ. In accordance, instead of defining the compactness of the logical consequence operator and of the deduction operator in terms of finiteness, we have to define it in terms of continuity. Also, the effectiveness of the deduction apparatus has to be defined by using the tools of constructive real analysis and not the one of recursive arithmetic. This means that decidability and semi-decidability have to be defined by involving effective limit processes and not by finite steps stopping processes
Fidel Semantics for Propositional and First-Order Version of the Logic of CGâ3
Paraconsistent extensions of 3-valued Gödel logic are studied as tools for knowledge representation and nonmonotonic reasoning. Particularly, Osorio and his collaborators showed that some of these logics can be used to express interesting nonmonotonic semantics. CGâ3 is one of these 3-valued logics. In this paper, we introduce Fidel semantics for a certain calculus of CGâ3 by means of Fidel structures, named CGâ3-structures. These structures are constructed from enriched Boolean algebras with a special family of sets. Moreover, we also show that the most basic CGâ3-structures coincide with da CostaâAlvesâ bi-valuation semantics; this connection is displayed through a Representation Theorem for CGâ3-structures. By contrast, we show that for other paraconsistent logics that allow us to present semantics through Fidel structures, this connection is not held. Finally, Fidel semantics for the first-order version of the logic of CGâ3 are presented by means of adapting algebraic tools
Fuzzy Logic
Fuzzy Logic is becoming an essential method of solving problems in all domains. It gives tremendous impact on the design of autonomous intelligent systems. The purpose of this book is to introduce Hybrid Algorithms, Techniques, and Implementations of Fuzzy Logic. The book consists of thirteen chapters highlighting models and principles of fuzzy logic and issues on its techniques and implementations. The intended readers of this book are engineers, researchers, and graduate students interested in fuzzy logic systems