Related papers: Refining the arithmetical hierarchy of classical p…

This paper investigates the proof-theoretic foundations of double negation introduction (DNI) and double negation elimination (DNE) in classical logic. By examining both sequent calculus and natural deduction, it is shown that these rules…

We present a calculus providing a Curry-Howard correspondence to classical logic represented in the sequent calculus with explicit structural rules, namely weakening and contraction. These structural rules introduce explicit erasure and…

In the context of natural deduction for propositional classical logic, with classicality given by the inference rule reductio ad absurdum, we investigate the De Morgan translation of disjunction in terms of negation and conjunction. Once…

The logic of constant domains is intuitionistic logic extended with the so-called forall-shift axiom, a classically valid statement which implies the excluded middle over decidable formulas. Surprisingly, this logic is constructive and so…

Axiomatizing mathematical structures is a goal of Mathematical Logic. Axiomatizability of the theories of some structures have turned out to be quite difficult and challenging, and some remain open. However axiomatization of some…

Akama et al. systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification…

This paper explores the connection between two central results in the proof theory of classical logic: Gentzen's cut-elimination for the sequent calculus and Herbrands "fundamental theorem". Starting from Miller's expansion-tree-proofs, a…

The model theory of a first-order logic called N^4 is introduced. N^4 does not eliminate double negations, as classical logic does, but instead reduces fourfold negations. N^4 is very close to classical logic: N^4 has two truth values;…

Almost from the inception of Hilbert's program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various…

The general aim of this article is to study the negation fragment of classical logic within the framework of contemporary (Abstract) Algebraic Logic. More precisely, we shall find the three classes of algebras that are canonically…

The classical fields with fractional derivatives are investigated by using the fractional Lagrangian formulation.The fractional Euler-Lagrange equations were obtained and two examples were studied.

We give some new refinements and reverses Young inequalities in both additive-type and multiplicative-type for two positive numbers/operators. We show our advantages by comparing with known results. A few applications are also given. Some…

In this paper we will see deductive systems for classical propositional and predicate logic in the calculus of structures. Like sequent systems, they have a cut rule which is admissible. In addition, they enjoy a top-down symmetry and some…

We reconstruct finite-dimensional quantum theory with superselection rules, which can describe hybrid quantum-classical systems, from four purely operational postulates: symmetric sharpness, complete mixing, filtering, and local equality.…

We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question. For such theories it was already known that proofs strongly…

This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of `maximal formula', `segment' and `maximal segment' suitable to the system, and gives…

Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schr\"odinger…

We investigate the properties of arithmetic differentiation, an attempt to adapt the notion of differentiation to the integers by preserving the Leibniz rule, (ab)' = a'b + ab'. This has proved to be a very rich topic with many different…

The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition.…

An investigation of classical fields with fractional derivatives is presented using the fractional Hamiltonian formulation. The fractional Hamilton's equations are obtained for two classical field examples. The formulation presented and the…