Related papers: Cyclic Negations and Four-valuedness
Quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, a system of qubits, representing a…
We investigate the cyclic proof theory of extensions of Peano Arithmetic by (finitely iterated) inductive definitions. Such theories are essential to proof theoretic analyses of certain `impredicative' theories; moreover, our cyclic systems…
We review known factorization results in quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, the QR factorization. We prove there is a Schur factorization for commuting…
We give a purely model-theoretic characterization of the semantics of logic programs with negation-as-failure allowed in clause bodies. In our semantics the meaning of a program is, as in the classical case, the unique minimum model in a…
Some comments are made on the matrices which serve as the basis of a quaternionic algebra. We show that these matrices are related with the quaternionic action of the imaginary units from the left and from the right.
The notion of non-deterministic logical matrix (where connectives are interpreted as multi-functions) preserves many good properties of traditional semantics based on logical matrices (where connectives are interpreted as functions) whilst…
A cyclic order may be thought of informally as a way to seat people around a table, perhaps for a game of chance or for dinner. Given a set of agents such as $\{A,B,C\}$, we can formalize this by defining a cyclic order as a permutation or…
We present two deductively equivalent calculi for non-deterministic many-valued logics. One is defined by axioms and the other - by rules of inference. The two calculi are obtained from the truth tables of the logic under consideration in a…
The term quantum logic has different connotations for different people, having been considered as everything from a metaphysical attack on classical reasoning to an exercise in abstract algebra. Our aim here is to give a uniform…
Circular (or cyclic) proofs have received increasing attention in recent years, and have been proposed as an alternative setting for studying (co)inductive reasoning. In particular, now several type systems based on circular reasoning have…
Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear…
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational…
Propositional logics in general, considered as a set of sentences, can be undecidable even if they have "nice" representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already…
In this short paper we will discuss the similarities and differences between two semantic approaches to modal logics - non-deterministic semantics and restricted non-deterministic semantics. Generally speaking, both kinds of semantics are…
Given a first-order sentence, a model-checking computation tests whether the sentence holds true in a given finite structure. Data provenance extracts from this computation an abstraction of the manner in which its result depends on the…
Causal reasoning is essential to science, yet quantum theory challenges it. Quantum correlations violating Bell inequalities defy satisfactory causal explanations within the framework of classical causal models. What is more, a theory…
The paper is a contribution both to the theoretical foundations and to the actual construction of efficient automatizable proof procedures for non-classical logics. We focus here on the case of finite-valued logics, and exhibit: (i) a…
The paper describes an extension of well-founded semantics for logic programs with two types of negation. In this extension information about preferences between rules can be expressed in the logical language and derived dynamically. This…
With the increasing empirical success of distributional models of compositional semantics, it is timely to consider the types of textual logic that such models are capable of capturing. In this paper, we address shortcomings in the ability…
Negation is an important perspective of knowledge representation. Existing negation methods are mainly applied in probability theory, evidence theory and complex evidence theory. As a generalization of evidence theory, random permutation…