We show that Intuitionistic Open Induction iop is not closed under the rule DNS(∃ - 1 ). This is established by constructing a Kripke model of iop + $\neg L_y(2y > x)$ , where $L_y(2y > x)$ is universally quantified on x. On the other hand, we prove that iop is equivalent with the intuitionistic theory axiomatized by PA - plus the scheme of weak ¬¬LNP for open formulas, where universal quantification on the parameters precedes double negation. We also show (...) that for any open formula φ(y) having only y free, $(PA^-)^i \vdash L_y\varphi(y)$ . We observe that the theories iop, i∀ 1 and iΠ 1 are closed under Friedman's translation by negated formulas and so under VR and IP. We include some remarks on the classical worlds in Kripke models of iop. (shrink)

In this paper we are concerned with definably, with or without parameters, complete expansions of ordered fields, i. e. those with no definable gaps. We present several axiomatizations, like being definably connected, in each of the two cases. As a corollary, when parameters are allowed, expansions of ordered fields are o-minimal if and only if all their definable subsets are finite disjoint unions of definably connected subsets. We pay attention to how simply a definable gap in an expansion is so. (...) Next we prove that over parametrically definably complete expansions of ordered fields, all one-to-one definable continuous functions are monotone and open. Moreover, in both parameter and parameter-free cases again, definably complete expansions of ordered fields satisfy definable versions of the Heine-Borel and Extreme Value theorems and also Bounded Intersection Property for definable families of closed bounded subsets. (shrink)

In "Intuitionistic validity in -normal Kripke structures," Buss asked whether every intuitionistic theory is, for some classical theory , that of all -normal Kripke structures for which he gave an r.e. axiomatization. In the language of arithmetic and denote PA plus Open Induction or Open LNP, and are their intuitionistic deductive closures. We show is recursively axiomatizable and , while . If proves PEM but not totality of a classically provably total Diophantine function of , then and so . A (...) result due to Wehmeier then implies . We prove is not -conservative over . If , then is not closed under MR or Friedman's translation, so range (). Both and are closed under the negative translation. (shrink)

The administration of pro re nata medications is the responsibility of the nurse. However, ethical uncertainties often happen due to the inability of incapacitated patients to collaborate with the nurse in the process of decision making for pro re nata medication administration. There is a lack of integrative knowledge and insufficient understanding regarding ethical considerations surrounding the administration of pro re nata medications to incapacitated patients. Therefore, they have been discussed in this paper and practical strategies to avoid unethical practices (...) have been suggested. The complicated caring situation surrounding the administration of pro re nata medications is intertwined with ethical issues affecting the consideration of the patient's wishes and interventions that override them. The patient's right of autonomy and treatment refusal, surrogacy role, paternalism, and coercion are the main ethos of ethical pro re nata medication administration. Education and training can help nurses avoid legal and ethical issues in pro re nata medicines management and improve the quality and safety of healthcare. Empirical research is needed to improve our understanding of this phenomenon in the multidisciplinary environment of medicines management. (shrink)

Number facts are commonly assumed to be verbally stored in an associative multiplication fact retrieval network. Prominent evidence for this assumption comes from so-called operand-related errors (e.g. 4 × 6 = 28). However, little is known about the development of this network in children and its relation to verbal and non-verbal memories. In a longitudinal design, we explored elementary school children from grades 3 and 4 in a multiplication verification task with the operand-related and -unrelated distractors. We examined the contribution (...) of multiplicative fact retrieval by verbal and visuo-spatial short-term and working memory. Children in grade 4 showed smaller reaction times in all conditions. However, there was no significant difference in errors between grades. Contribution of verbal and visuo-spatial working memory also changed with grade. Multiplication correlated with verbal working memory and performance in grade 3 but with visuo-spatial working memory and performance in grade 4. We suggest that the relation to verbal working memory in grade 3 indicates primary linguistic learning of and access to multiplication in grade 3 which is probably based on verbal repetition of the multiplication table heavily practiced in grades 2 and 3. However, the relation to visuo-spatial semantic working memory in grade 4 suggests that there is a shift from verbal to visual and semantic learning in grade 4. This shifting may be induced because later in elementary school, multiplication problems are rather carried out via more written, i.e., visual tasks, which also involve executive functions. More generally, the current data indicates that mathematical development is not generally characterized by a steady progress in performance; rather verbal and non-verbal memory contributions of performance shift over time, probably due to different learning contents. (shrink)

In Western literatures, "conflict" is a general term that refers to discord between two or more entities. In Islamic jurisprudence, however, in addition to the term "conflict", there is another term which is called tazāhum. The two terms, however, have different definitions. Conflict between two concepts, for instance, indicates that one is right and the other is wrong, while tazāhum does not necessarily have to be between right and wrong, and may appear between two equally right concepts. Moreover, conflict exists (...) on a legislative level, while tazāhum is a matter of obedience and adherence, meaning that in practice, both sides cannot continue to coexist. Conflict of interest is a known term in Western literatures, and according to D.F. Thompson, it refers to a situation where professional judgment regarding a primary interest is improperly and unjustifiably influenced by a secondary interest. Taking into account Thompson's definition and the distinction between "conflict" and "tazāhum", the English term "conflict of interest" translates to "tazāhum of interest" in Islamic jurisprudence as it refers to a person's action without reflecting right or wrong, and simply concerns priority of one interest over another. The resolution to tazāhum in Islamic jurisprudence lies in two principles: the principle of significance and the principle of choice. For instance, in case of conflict or tazāhum between the interests of patient and physician, the patient's interest should be the main concern based on the principle of significance. Although Western literatures propose methods such as disclosure or prohibition in order to resolve conflict of interest, the foundation for these solutions seems to have been the principle of significance. (shrink)

The purpose of this article is to discuss the ethical issues impacting the phenomenon of patient safety and to present implications for nursing management. Previous knowledge of this perspective is fragmented. In this discussion, the main drivers are identified and formulated in ‘the ethical imperative’ of patient safety. Underlying values and principles are considered, with the aim of increasing their visibility for nurse managers’ decision-making. The contradictory nature of individual and utilitarian safety is identified as a challenge in nurse management (...) practice, together with the context of shared responsibility and identification of future challenges. As a conclusion, nurse managers play a strategic role in patient safety. Their role is to incorporate ethical values of patient safety into decision-making at all levels in an organization, and also to encourage clinical nurses to consider values in the provision of care to patients. Patient safety that is sensitive to ethics provides sustainable practice where the humanity and dignity of all stakeholders are respected. (shrink)

We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using these axiomatizations, we prove interpolation theorems for basic propositional logic.

We study the relations of being substructure and elementary substructure between Kripke models of intuitionistic predicate logic with the same arbitrary frame. We prove analogues of Tarski's test and Löwenheim-Skolem's theorems as determined by our definitions. The relations between corresponding worlds of two Kripke models [MATHEMATICAL SCRIPT CAPITAL K] ⪯ [MATHEMATICAL SCRIPT CAPITAL K]′ are studied.

Background Care is the basis of the nursing profession and nurse’s caring behavior is one of the important factors in patient satisfaction. On the other hand, psychological empowerment can improve the provision of care services, and leaders have a significant impact on the behavior of followers. This study determined the correlation between ethical leadership, psychological empowerment, and caring behavior from nurses’ perspective. Methods This cross-sectional descriptive study was conducted in 2019. A total of 200 nurses were selected by stratified random (...) sampling. Data were collected with ethical leadership, caring behavior and psychological empowerment questionnaires and analyzed with descriptive statistics and inferential statistics using SPSS20. Results The findings showed a significant correlation between ethical leadership, psychological empowerment and caring behavior of nurses. Also, there was a significant correlation between psychological empowerment and nurses’ caring behavior. The regression analysis showed that ethical leadership and psychological empowerment are a predictor of caring behavior of nurses. Conclusion Applying an ethical leadership style by nursing managers can lead to nurses’ improved psychological empowerment and caring behaviors and ultimately, patient care quality. Thus, considering guidelines at the team and organizational levels and programs such as providing workshops and periodic counseling to cultivate ethical nurse leadership can lead to better caring behavior. (shrink)

This volume is a collection of essays in honour of Professor Mohammad Ardeshir. It examines topics which, in one way or another, are connected to the various aspects of his multidisciplinary research interests. Based on this criterion, the book is divided into three general categories. The first category includes papers on non-classical logics, including intuitionistic logic, constructive logic, basic logic, and substructural logic. The second category is made up of papers discussing issues in the contemporary philosophy of mathematics and logic. (...) The third category contains papers on Avicenna’s logic and philosophy. Mohammad Ardeshir is a full professor of mathematical logic at the Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, where he has taught generations of students for around a quarter century. Mohammad Ardeshir is known in the first place for his prominent works in basic logic and constructive mathematics. His areas of interest are however much broader and include topics in intuitionistic philosophy of mathematics and Arabic philosophy of logic and mathematics. In addition to numerous research articles in leading international journals, Ardeshir is the author of a highly praised Persian textbook in mathematical logic. Partly through his writings and translations, the school of mathematical intuitionism was introduced to the Iranian academic community. (shrink)

In this paper we prove a bounded translation of intuitionistic propositional logic into basic propositional logic. Our new theorem, compared with the translation theorem in [1], has the advantage that it gives an effective bound on the translation, depending on the complexity of formulas.

In this paper, we show that the implication fragment of classical propositional logic is finitary for unification with parameters.

Dynamic topological logic is a multi-modal logic that was introduced for reasoning about dynamic topological systems, i.e. structures of the form $\langle{\mathfrak{X}, f}\rangle $, where $\mathfrak{X}$ is a topological space and $f$ is a continuous function on it. The problem of finding a complete and natural axiomatization for this logic in the original tri-modal language has been open for more than one decade. In this paper, we give a natural axiomatization of $\textsf{DTL}$ and prove its strong completeness with respect to (...) the class of all dynamic topological systems. Our proof system is infinitary in the sense that it contains an infinitary derivation rule with countably many premises and one conclusion. It should be remarked that $\textsf{DTL}$ is semantically non-compact, so no finitary proof system for this logic could be strongly complete. Moreover, we provide an infinitary axiomatic system for the logic ${\textsf{DTL}}_{\mathcal{A}}$, i.e. the $\textsf{DTL}$ of Alexandrov spaces, and show that it is strongly complete with respect to the class of all dynamical systems based on Alexandrov spaces. (shrink)

As pressure to publish increases in the academic nursing world, journal submission numbers and rejection rates are soaring. The review process is crucial to journals in publishing high quality, cutting‐edge knowledge development, and to authors in preparing their papers to a high quality to enable the nursing world to benefit from developments in knowledge that affect nursing practice and patient outcomes and the development of the discipline. This paper does not intend to contribute to the debate regarding the ethics of (...) reviewing, but rather seeks to explore notions of how the quality of the reviewing process can be enhanced to benefit authors, the reviewers, and the state of nursing knowledge. Furthermore, a call is made to editors to devise strategies for aiding reviewers to attain higher validity and reliability within the reviewing process by establishing clear standards and expectations and to ensure published work is judged against industry norms for quality. (shrink)

Recent years have seen new general notions of contextuality emerge. Most of these employ context-independent symbols to represent random variables in different contexts. As an example, the operational theory of (Spekkens in Phys Rev A 71(5):52108, 2005) treats an observable being measured in two different contexts identically. Non-contextuality in this approach is the impossibility of drawing ontological distinctions between identical elements of the operational theory. However, a recent collection of work seeks to exploit context-dependent symbols of random variables to interpret (...) contextuality (Kujala et al. in Phys Rev Lett 115(15):150401, 2015; Dzhafarov and Kujala in Phys Scr T163:014009, 2014). This approach associates contextuality with the possibility of imposing a particular joint distribution on random variables recorded under different experimental contexts. This paper compares these two different treatments of random variables and highlights the limitations of the context-dependent approach as a physical theory. (shrink)

We introduce a Gentzen-style sequent calculus axiomatization for Basic Predicate Calculus. Our new axiomatization is an improvement of the previous axiomatizations, in the sense that it has the subformula property. In this system the cut rule is eliminated.

For a function $f$ with domain $[X]^{n}$, where $X\subseteq\mathbb{N}$, we say that $H\subseteq X$ is canonical for $f$ if there is a $\upsilon\subseteq n$ such that for any $x_{0},\ldots,x_{n-1}$ and $y_{0},\ldots,y_{n-1}$ in $H$, $f=f$ iff $x_{i}=y_{i}$ for all $i\in\upsilon$. The canonical Ramsey theorem is the statement that for any $n\in\mathbb{N}$, if $f:[\mathbb{N}]^{n}\rightarrow\mathbb{N}$, then there is an infinite $H\subseteq\mathbb{N}$ canonical for $f$. This paper is concerned with a model-theoretic study of a finite version of the canonical Ramsey theorem with a largeness (...) condition and also a version of the Kanamori–McAloon principle. As a consequence, we produce new indicators for cuts satisfying $\operatorname{PA}$. (shrink)

There are several ways for defining the notion submodel for Kripke models of intuitionistic first‐order logic. In our approach a Kripke model A is a submodel of a Kripke model B if they have the same frame and for each two corresponding worlds Aα and Bα of them, Aα is a subset of Bα and forcing of atomic formulas with parameters in the smaller one, in A and B, are the same. In this case, B is called an extension of (...) A. We characterize theories that are preserved under taking submodels and also those that are preserved under taking extensions as universal and existential theories, respectively. We also study the notion elementary submodel defined in the same style and give some results concerning this notion. In particular, we prove that the relation between each two corresponding worlds of finite Kripke models A ≤ B is elementary extension (in the classical sense) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

For a classical theory T, ℋ(T) denotes the intuitionistic theory of T-normal (i.e. locally T) Kripke structures. S. Buss has asked for a characterization of the theories in the range of ℋ and raised the particular question of whether HA is an ℋ-theory. We show that Ti∈ range(ℋ) iff Ti = ℋ(T). As a corollary, no fragment of HA extending iΠ1 belongs to the range of ℋ. A. Visser has already proved that HA is not in the range of H (...) by different methods. We provide more examples of theories not in the range of ℋ. We show PA-normality of once-branching Kripke models of HA + MP, where it is not known whether the same holds if MP is dropped. (shrink)

This paper proves some independence results for weak fragments of Heyting arithmetic by using Kripke models. We present a necessary condition for linear Kripke models of arithmetical theories which are closed under the negative translation and use it to show that the union of the worlds in any linear Kripke model of HA satisfies PA. We construct a two-node PA-normal Kripke structure which does not force iΣ2. We prove i∀1 ⊬ i∃1, i∃1 ⊬ i∀1, iΠ2 ⊬ iΣ2 and iΣ2 ⊬ (...) iΠ2. We use Smorynski's operation Σ′ to show HA ⊬ lΠ1. (shrink)

In the first section of this paper we show that i Π1 ≡ W⌝⌝lΠ1 and that a Kripke model which decides bounded formulas forces iΠ1 if and only if the union of the worlds in any path in it satisflies IΠ1. In particular, the union of the worlds in any path of a Kripke model of HA models IΠ1. In the second section of the paper, we show that for equivalence of forcing and satisfaction of Πm-formulas in a linear Kripke (...) model deciding Δ0-formulas it is necessary and sufficient that the model be Σm-elementary. This implies that if a linear Kripke model forces PEMprenex, then it forces PEM. We also show that, for each n ≥ 1, iΦn does not prove ℋ(IΠn's are Burr's fragments of HA. (shrink)

In this paper we first introduce a new neighbourhood semantics for propositional intuitionistic logic. We then naturally extend this semantics to first-order intuitionistic logic. We also study bisimulation between neighbourhood models and prove some of their basic properties for both propositional and first-order intuitionistic logic.

We give a corrected proof of the main result in the paper [2] mentioned in the title.

We give a corrected proof of the main result in the paper [2] mentioned in the title.

In this paper we define a suitable version of the notion of homomorphism for Kripke models of intuitionistic first-order logic and characterize theories that are preserved under images and also those that are preserved under inverse images of homomorphisms. Moreover, we define a notion of union of chain for Kripke models and define a class of formulas that is preserved in unions of chains. We also define similar classes of formulas and investigate their behavior in Kripke models. An application to (...) intuitionistic first-order arithmetic is also given. (shrink)

We construct ω-framed Kripke models of i∀1 and iΠ1 non of whose worlds satisfies ∀x∃y(x=2y∨x=2y+1) and ∀x,y∃zExp(x, y, z) respectively. This will enable us to show that i∀1 does not prove ¬¬∀x∃y(x=2y∨x=2y+1) and iΠ1 does not prove ¬¬∀x, y∃zExp(x, y, z). Therefore, i∀1⊬¬¬lop and iΠ1⊬¬¬iΣ1. We also prove that HA⊬lΣ1 and present some remarks about iΠ2.

IPV is the intuitionistic theory axiomatized by Cook's equational theory PV plus PIND on NP-formulas. Two extensions of IPV were introduced by Buss and by Cook and Urquhart by adding PIND for formulas of the form A ∨ B, respectively ¬¬A, where A is NP and x is not free in B. Cook and Urquhart posed the question of whether these extensions are proper. We show that in each of the two cases the extension is proper unless the polynomial hierarchy (...) collapses. (shrink)

It is shown that the feasibly constructive arithmetic theory IPV does not prove LMIN, unless the polynomial hierarchy CPV-provably collapses. It is proved that PV plus LMIN intuitionistically proves PIND. It is observed that PV + PIND does not intuitionistically prove NPB, a scheme which states that the extended Frege systems are not polynomially bounded.

In this paper we prove conservation theorems for theories of classical first-order arithmetic over their intuitionistic version. We also prove generalized conservation results for intuitionistic theories when certain weak forms of the principle of excluded middle are added to them. Members of two families of subsystems of Heyting arithmetic and Buss-Harnik’s theories of intuitionistic bounded arithmetic are the intuitionistic theories we consider. For the first group, we use a method described by Leivant based on the negative translation combined with a (...) variant of Friedman’s translation. For the second group, we use Avigad’s forcing method. (shrink)

In this paper we naturally define when a theory has bounded quantifier elimination, or is bounded model complete. We give several equivalent conditions for a theory to have each of these properties. These results provide simple proofs for some known results in the model theory of the bounded arithmetic theories like CPV and PV1. We use the mentioned results to obtain some independence results in the context of intuitionistic bounded arithmetic. We show that, if the intuitionistic theory of polynomial induction (...) on strict $\Pi_2^{b}$ formulas proves decidability of $\Sigma_1^b$ formulas, then P=NP. We also prove that, if the mentioned intuitionistic theory proves ${\rm LMIN}(\Sigma_{1}^{b})$ , then P=NP. (shrink)

Objective: Electrical stimulation potentially delineates epileptogenic cortex through induction of typical seizures. Although frequently employed, its value for epilepsy surgery remains controversial. Similarly, ES is used to identify symptomatogenic zones, but with greater success and a long-standing evidence base. Recent work points to new seizure symptoms such as ictal central apnea that may enhance presurgical hypotheses. The aims of this review are 2-fold: to determine the value of ES-induced seizures in epilepsy surgery and to analyze current evidence on ICA as (...) a new surrogate of symptomatogenic cortex.Methods: Three databases were searched for ESIS. Investigators independently selected studies according to pre-specified criteria. Studies reporting postoperative outcome in patients with ESIS were included in a meta-analysis. For ES-induced apnea, a thorough search was performed and reference list searching was employed.Results: Of 6,314 articles identified for ESIS, 25 were considered eligible to be reviewed in full text. Fourteen studies were included in the qualitative synthesis ; six studies were included in the meta-analysis. The meta-analysis showed that favorable outcome is associated with ESIS prior to surgery. In addition, the overall estimation of the occurrence of favorable outcome among cases with ESIS is 68.13%. On the other hand, recent studies have shown that stimulation of exclusively mesial temporal lobe structures elicits central apnea and represents symptomatogenic anatomic substrates of ICA. This is in variance with traditional teaching that mesial temporal ES is non-symptomatogenic.Conclusions: ES is a tool highly likely to aid in the delineation of the epileptogenic zone, since ESIS is associated with favorable postoperative outcomes. There is an urgent need for prospective evaluation of this technique, including effective stimulation parameters and surgical outcomes, that will provide knowledge base for practice. In addition, ES-induced apnea studies suggest that ICA, especially when it is the first or only clinical sign, is an important semiological feature in localizing the symptomatogenic zone to mesial temporal lobe structures, which must be considered in SEEG explorations where this is planned, and in surgical resection strategies. (shrink)

The current study was a cross-sectional research and aimed to investigate the factor structure, internal consistency, and validities of the Persian version of the Teasing Questionnaire-Revised. Forward and backward translations of the TQ-R were performed; face and content validities were determined based on comments by a sample of psychology students and specialists. Using the cluster sampling method, 290 participants were recruited, and 201 valid data were analyzed. The factor structure was assessed by confirmatory and exploratory factor analysis. The result of (...) the confirmatory factor analysis did not confirm the proposed three, four, and five-factor models. EFA revealed four factors with 23 items, explaining 52.03% of the total variance. The internal consistency of the Persian version of Teasing Questionnaire 23 was in the excellent range, and its expected associations with external correlates supported the measure’s convergent validity. The findings indicated that the Persian version of the TQ-R has sound psychometric properties and can be used as a valid and reliable tool in research and clinical outcomes. (shrink)

In this paper, we study bounded versions of some model-theoretic notions and results. We apply these results to the context of models of bounded arithmetic theories as well as some related complexity questions. As an example, we show that if the theory \(\rm S_2 ^1(PV)\) has bounded model companion then \(\rm NP=coNP\). We also study bounded versions of some other related notions such as Stone topology.

We define a class of first-order formulas $\mathsf{P}^{\ast }$ which exactly contains formulas $\varphi$ such that satisfaction of $\varphi$ in any classical structure attached to a node of a Kripke model of intuitionistic predicate logic deciding atomic formulas implies its forcing in that node. We also define a class of $\mathsf{E}$-formulas with the property that their forcing coincides with their classical satisfiability in Kripke models which decide atomic formulas. We also prove that any formula with this property is an $\mathsf{E}$-formula. (...) Kripke models of intuitionistic arithmetical theories usually have this property. As a consequence, we prove a new conservativity result for Peano arithmetic over Heyting arithmetic. (shrink)

In order to build high-quality concrete, it is imperative to know the raw materials in advance. It is possible to accurately predict the quality of concrete and the amount of raw materials used using machine learning-enhanced methods. An automated process based on machine learning strategies is proposed in this paper for predicting the compressive strength of concrete. Fusion-learning-based optimization is used in the proposed approach to generate a strong learner by pooling support vector regression models. The SVR technique proposes an (...) optimization method for finding the kernel radial basis function parameters based on improving the innovative gunner algorithm. As a result of AIG's diverse solutions, local optima are effectively avoided. Therefore, the novelty of our research is that, in solving the uncertainty of predicted outputs based on integrated models, we use fusion-learning-based optimization to improve regression discrimination. We also collected a standard dataset to analyze the proposed algorithm, and subsequently, the dataset was designed from concrete laboratory tests on 244 samples, seven features, and three outputs. Different regression intensities are determined by correlation analysis of responses. Regression fusion is sufficiently accurate to estimate the number of desired outcomes examined based on the appropriate input data sample. The best quality concrete can be achieved with an error rate of less than 5%. (shrink)

When humans are required to perform two or more tasks concurrently, their performance declines as the tasks get closer together in time. Here, we investigated the mechanisms of this cognitive performance decline using a dual-task paradigm in a simulated driving environment, and using drift-diffusion modeling, examined if the two tasks are processed in a serial or a parallel manner. Participants performed a lane change task, along with an image discrimination task. We systematically varied the time difference between the onset of (...) the two tasks and measured its effect on the amount of dual-task interference. Results showed that the reaction times of the two tasks in the dual-task condition were higher than those in the single-task condition. SOA influenced the RTs of both tasks when they were presented second and the RTs of the image discrimination task when it was presented first. Results of drift-diffusion modeling indicated that dual-task performance affects both the rate of evidence accumulation and the delays outside the evidence accumulation period. These results suggest that a hybrid model containing features of both parallel and serial processing best accounts for the results. Next, manipulating the predictability of the order of the two tasks, we showed that in unpredictable conditions, the order of the response to the two tasks changes, causing attenuation in the effect of SOA. Together, our findings suggest higher-level executive functions are involved in managing the resources and controlling the processing of the tasks during dual-task performance in naturalistic settings. (shrink)

We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A built up of atoms p1,..., pn, BPC⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vdash}$$\end{document}A if and only if for all arithmetical sentences B1,..., Bn, BA⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vdash}$$\end{document}A. The technique used in our proof can easily be applied to some known extensions of BA.