236 research outputs found
Rejection in Łukasiewicz's and Słupecki's Sense
The idea of rejection originated by Aristotle. The notion of rejection
was introduced into formal logic by Łukasiewicz [20]. He applied it to
complete syntactic characterization of deductive systems using an axiomatic
method of rejection of propositions [22, 23]. The paper gives not only genesis,
but also development and generalization of the notion of rejection. It also
emphasizes the methodological approach to biaspectual axiomatic method of
characterization of deductive systems as acceptance (asserted) systems and
rejection (refutation) systems, introduced by Łukasiewicz and developed by
his student Słupecki, the pioneers of the method, which becomes relevant in
modern approaches to logic
Interval-valued algebras and fuzzy logics
In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter
Refutation Systems : An Overview and Some Applications to Philosophical Logics
Refutation systems are systems of formal, syntactic derivations, designed to derive the non-valid formulas or logical consequences of a given logic. Here we provide an overview with comprehensive references on the historical development of the theory of refutation systems and discuss some of their applications to philosophical logics
EF4, EF4-M and EF4-Ł: A companion to BN4 and two modal four-valued systems without strong Łukasiewicz-type modal paradoxes
The logic BN4 was defined by R.T. Brady as a four-valued extension of Routley and Meyer’s basic logic B. The system EF4 is defined as a companion to BN4 to represent the four-valued system of (relevant) implication. The system Ł was defined by J. Łukasiewicz and it is a four-valued modal logic that validates what is known as strong Łukasiewicz-type modal paradoxes. The systems EF4-M and EF4-Ł are defined as alternatives to Ł without modal paradoxes. This paper aims to define a Belnap-Dunn semantics for EF4, EF4-M and EF4-Ł. It is shown that EF4, EF4-M and EF4-Ł are strongly sound and complete w.r.t. their respective semantics and that EF4-M and EF4-Ł are free from strong Łukasiewicz-type modal paradoxes
Zero-one laws with respect to models of provability logic and two Grzegorczyk logics
It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems
Dyck algebras, interval temporal logic and posets of intervals
We investigate a natural Heyting algebra structure on the set of Dyck paths
of the same length. We provide a geometrical description of the operations of
pseudocomplement and relative pseudocomplement, as well as of regular elements.
We also find a logic-theoretic interpretation of such Heyting algebras, which
we call Dyck algebras, by showing that they are the algebraic counterpart of a
certain fragment of a classical interval temporal logic (also known as
Halpern-Shoham logic). Finally, we propose a generalization of our approach,
suggesting a similar study of the Heyting algebra arising from the poset of
intervals of a finite poset using Birkh\"off duality. In order to illustrate
this, we show how several combinatorial parameters of Dyck paths can be
expressed in terms of the Heyting algebra structure of Dyck algebras together
with a certain total order on the set of atoms of each Dyck algebra.Comment: 17 pages, 3 figure
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