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In this paper we introduce a modal theory iHσ which is sound and complete for arithmetical Σ1-interpretations in HA, in other words, we will show that iHσ is the Σ1-provability logic of HA. Moreover we will show that iHσ is decidable. As a by-product of these results, we show that HA+□⊥ has de Jongh property. © 2018 Elsevier B.V  

For the Heyting Arithmetic HA,HA is defined [14, 15] as the theory {A | HA-A}, where is called the box translation of A (Definition 2.4). We characterize the Σ1-provability logic of HA as a modal theory (Definition 3.17). © 2019 The Association for Symbolic Logic  

We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A(p1,..., pn) built up of atoms p1,..., pn, BPC(Formula presented.)A(p1,..., pn) if and only if for all arithmetical sentences B1,..., Bn, BA(Formula presented.)A(B1,..., Bn). The technique used in our proof can easily be applied to some known extensions of BA  

Let L denote a first-order logic in a language that contains infinitely many constant symbols and also containing intuitionistic logic IQC. By ¬¬L, we mean the associated logic axiomatized by the double negation of the universal closure of the axioms of L plus IQC. We shall show that if L is strongly complete for a class of Kripke models K, then ¬¬L is strongly complete for the class of Kripke models that are ultimately in K  

We show that the provability logic of. PA,. GL and the truth provability logic, i.e. the provability logic of. PA relative to the standard model N, GLS are reducible to their. Σ Σ1-provability logics,. GLV and. GLSV, respectively, by only propositional substitutions  

We give a counterexample to the claim that every provably total function of Basic Arithmetic is a polynomially bounded primitive recursive function. © 2019 by University of Notre Dame  

We give a counterexample to the claim that every provably total function of Basic Arithmetic is a polynomially bounded primitive recursive function. © 2019 by University of Notre Dame  

In this study, a cryogenic distillation column has been designed and simulated via a computer code based on the theta method of convergence. The required thermodynamic properties are determined from the enhanced predictive Peng-Robinson (E-PPR 78) equation of state which has a good accuracy in predicting the corresponding thermodynamic properties of natural gas components. The combined code of distillation column/equation of state has been verified with that of another study. In the present study, the results are achieved by the constant molar over-flow and inclusion of energy equations assumptions. In order to have more accuracy in the results, the energy equations were considered in the... 

In logical literature, the phrase 'basic logic' refers to at least three different logical systems. The first one, basic propositional logic, BPL was introduced by Albert Visser in 1981. This logic is a subintuitionistic logic that can be obtained from intuitionistic logic by weakening of modus ponens. The second logical system with the name 'basic logic', is the system B that was introduced by G. Sambin and G. Battilotti in 1997. The goal of this logical system is to provide a common foundation for all usual non-modal logics. The third one is called 'basic logic' by P. Hajek in the field of Fuzzy Logic. We show that the two systems BPL and B do not have a direct relationship (i.e. none of... 

Let A be a subset of the constructive real line. What are the necessary and sufficient conditions for the set A such that A is continuously separated from other reals, i.e., there exists a continuous function f with f -1(0) = A? In this paper, we study the notions of closed sets and closure maps in constructive reverse mathematics  

In the context of intuitionistic analysis, we consider the set F consisting of all continuous functions φ{symbol} from [0, 1] to R such that φ{symbol} (0) = 0 and φ{symbol} (1) = 1, and the set I0 consisting of φ{symbol}'s in F where there exists x ∈ [0, 1] such that φ{symbol} (x) = frac(1, 2). It is well-known that there are weak counterexamples to the intermediate value theorem, and with Brouwer's continuity principle we have I0 ≠ F. However, there exists no satisfying answer to I0¬ ¬ =? F. We try to answer to this question by reducing it to a schema (which we call ED) about intuitionistic decidability that asserts "there exists an intuitionistically enumerable set that is not... 

A bounded monotone sequence of reals without a limit is called a Specker sequence. In Russian constructive analysis, Church's Thesis permits the existence of a Specker sequence. In intuitionistic mathematics, Brouwer's Continuity Principle implies it is false that every bounded monotone sequence of real numbers has a limit. We claim that the existence of Specker sequences crucially depends on the properties of intuitionistic decidable sets. We propose a schema (which we call ED) about intuitionistic decidability that asserts "there exists an intuitionistic enumerable set that is not intuitionistic decidable" and show that the existence of a Specker sequence is equivalent to ED. We show that... 

A latarre is a lattice with an arrow. its axiomatization looks natural. Latarres have a nontrivial theory which permits many constructions of latarres. Latarres appear as an end result of a series of generalizations of better known structures. These include Boolean algebras and Heyting algebras. Latarres need not have a distributive lattice. © 2017 Springer Science+Business Media B.V  

In classical analysis, every compact subset of ℝ is Lebesgue measurable, but it is not true in constructive analysis. In this paper, we prove that the statement 'every compact set K in a locally compact space X is integrable with respect to a positive measure μ' is equivalent to LPO, over Bishop's constructive analysis. We also prove that the existence of a compact subset of ℝ which is not Lebesgue integrable is equivalent to the schema ED, which asserts that 'there exists an intuitionistically enumerable subset of ℕ which is not intuitionistically decidable'. Moreover, classically, every open subset of ℝ is Lebesgue measurable, but it is not true constructively. We show that Lebesgue... 

The schema ED asserts that 'there exists an intuitionistically enumerable subset of N which is not intuitionistically decidable.' In this article, we prove that in the presence of Markov's Principle over Bishop's constructive analysis, ¬ED is equivalent to the principle of open induction on [0,1], via Specker sequences. © 2016. Oxford University Press. All rights reserved  

A latarre is a lattice with an arrow. Its axiomatization looks natural. Latarres have a nontrivial theory which permits many constructions of latarres. Latarres appear as an end result of a series of generalizations of better known structures. These include Boolean algebras and Heyting algebras. Latarres need not have a distributive lattice. © 2017, Springer Science+Business Media B.V  

We study Basic Arithmetic BA, which is the basic logic BQC equivalent of Heyting Arithmetic HA over intuitionistic logic IQC, and of Peano Arithmetic PA over classical logic CQC. It turns out that The Friedman translation is applicable to BA. Using this translation, we prove that BA is closed under a restricted form of the Markov rule. Moreover, it is proved that all nodes of a finite Kripke model of BA are classical models of Ι∃1+, a fragment of PA with Induction restricted to the formulas made up of ∃, ∧ and/or ∨. We also study an interesting extension of BQC, called EBQC, which is the extension by the axiom schema ⊤ → →. We show that this extension behaves very like to IQC, and the... 

We give answers to some questions raised by L. Åqvist in [6] and [7]. The question raised in [7] which we will answer positively is about the representability of Åqvist's system G in a hierarchy of alethic modal logics Hm, m = 1, 2, .... On the other hand, the questions raised in [6] are about the completeness of some monadic alethic denotic logics with respect to their Kripke semantics. © Copyright 2006 Oxford University Press  

We prove that every infinite rooted narrow tree Kripke model of HA is locally PA  

We introduce a new aspect of the notion of obligation inspired from the Islamic legal system. We then construct a dynamic deontic logic to model this notion of obligation. A semantics for this logic is introduced, and then its soundness and completeness theorems with respect to this semantics are proved. © 2022, Springer Nature Switzerland AG