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A resurgence of scholarly work on proof of God’s existence is noticeable over the past decade, with considerable emphasis on attempts to provide ‘analytic proof’ based on the meanings and logic of various identity statements which constitute premises of the syllogisms of the ‘proof’. Most recently perhaps, Emmanuel Rutten’s ‘modal-epistemic proof’ has drawn serious academic attention. Like other ‘analytic’ and strictly logical proofs of God’s existence, Rutten’s proof has been found flawed. In this paper I discuss the possibility of an ‘ethics-based’ identity proof of God’s existence. Such a proof, the first version of which, I believe, has been offered, indirectly, by Nikolai Lossky, utilizes the form and structure of the analytic proof, but fundamentally rests on the perception of moral values we associate with God and Godliness. The nature of the proof shifts the focus of the very attempt to ‘prove’ God’s existence from what I believe is an unreasonable standard, unattainable even in ‘proving’ the existence of the more mundane world, towards a more functional, practical and attainable standard. The proof proposed initially by Lossky, and in a more systematic form here, I believe, shows the indubitable existence of God in the sense of his moral presence in the lives of the faithful, at least with the same degree of certainty as the presence or ‘existence’ of anything else that can be epistemically proven in principle. Tokom poslednje decenije uočljiva je intenziviran rad na izvođenju dokaza o postojanju Boga, sa posebnim naglaskom na takozvane “analitičke dokaze”, koji su zasnovani na značenjima i logici različitih iskaza o identitetu, koji predstavljaju premise samog silogizma “dokaza”. Možda akademski najuticajniji skorašnji analitički dokaz o postojanju Boga izložio je Emanuel Ruten u formi svog “modalno-epistemičkog dokaza”. Kao i za ostale analitčke i strogo logičke dokaze postojanja Boga, i za Rutenov je utvrđeno da je neispravan. Kroz kritiku Rutenovog dokaza, koju koristim kao uvod, ja u ovom tekstu rahzmatram mogućnost dokaza o postojanju Boga koji bi bio zasnovan na etičkim argumentima. Takav dokaz, Like other ‘analytic’ and strictly logical proofs of God’s existence, Rutten’s proof has been found flawed. In this paper I discuss the possibility of an ‘ethics-based’ identity proof of God’s existence. Such a proof, čiju je prvu verziju, po mom mišljenju, već izneo Nikolaj Loski, koristi formu i strukturu analitičkih dokaza, ali se fundamentalno oslanja na doživljaj moralnih vrednosti koje povezujemo sa Bogom ili božanstvenošću. “Etički” dokaz pomera naglasak samog rada na izvođenju dokaza o postojanju Boga sa jednog standarda za koji smatram da je nerazuman i koji se ne može dostići ni kada se “dokazuje” postojanje mnogo manje kontroverznih ontoloških kategorija, kao što su različite kategorije svakodnevnog, “običnog” sveta. Istovremeno, etički dokaz pomera naglasak dokazivanja ka jednom funkcionalnom, praktičnom i dostižnom standardu dokazivanja. Ovaj dokaz, i u formi u kojoj ga je izveo Loski, a i u sistematičnijoj formi u kojoj ga ovde izlažem, pokazuje nesumnjivo postojanje Boga u smislu moralnog prisustva Boga u životima verujućih ljudi. “Izvesnost” takvog dokaza nije ništa manja od izvesnosti bilo čega drugog što se uopšte može epistemički dokazivati.

As a human begin, it cannot be expected from us to remain indifferent to the events and happenings around us or not to think about their causes and effects. For that reason, people mostly seek to find the answers to the question “Why?” when they especially met unexpected or unpleasant something. Attribution theory deals with the question of how in

This research study investigated the change in students’ academic achievement with an achievement test designed for the implementations of theorem proofs carried out with the help of mathematical models. For that purpose, the study was carried out with 45 second-grade students studying in the Department of Primary Education Mathematics Teaching in 2014-2015 academic year. One-group pre-test post-test model, a weak experimental design, was used in the study. The data of the study were carried via Mathematical Knowledge Test. Descriptive analysis and t-test were used for the data analysis. The analysis results of the data revealed that proofs of theorem carried out with the help of mathematical models increased students’ academic achievement. Researchers are suggested that this study should be implemented not only with theorem proofs but also with the visualisation of any concept that is difficult to understand and with problem solutions.

In questo articolo è presentato un percorso didattico svolto in modalità flipped classroom in una scuola media. Tale modalità didattica, che può essere riassunta nell’espressione “teoria a casa e compiti a scuola”, può essere considerata innovativa in quanto introduce nuove tecnologie in aula, promuovendone un utilizzo consapevole. Il percorso, incentrato sul teorema di Pitagora, ha suscitato interesse negli studenti. Sono presentate tutte le lezioni svolte sull’argomento: l’attività introduttiva, la formulazione del teorema, le sue diverse dimostrazioni, l’implicazione inversa del teorema, esercizi vari e la verifica. Rispetto ad un’introduzione più tradizionale, si è riscontrato da parte degli allievi un gradimento maggiore e un apprendimento più radicato. / This article presents a didactic experience carried out in flipped classroom mode in a lower secondary school. This teaching method, which can be summarized as “theory at home and homework at school”, can be considered as innovative since it allows introducing new technologies in the classroom, promoting a conscious use of them. The experience, focused on the Pythagorean theorem, thrilled the students. The article reports on all the lessons on this theme: the introductory activity, the formulation of the theorem, its different proofs, the inverse implication of the theorem, various exercises, and the test. Compared to a more traditional introduction, the students were more motivated and their learning appeared to be more rooted.

SummaryIn his book The Nature of Mathematical Knowledge and in a series of articles, Philip Kitcher attacks the traditional conception of a priori mathematical knowledge. The reliabilism he develops as an alternative situates all our knowledge within a psychological framework. However, in Frege’s Epistemology he claims that Frege’s conception of a priori knowledge is compatible with a psychological account. Kitcher attributes to Frege a traditional concept of proof, according to which mathematical and logical proofs are psychological activities. I shall argue that Kitcher’s interpretation conflicts with Frege’s anti-psychologistic injunction against confusing reasons with causes. Moreover, the psychological explanation obscures one of the most interesting features of a priori knowledge.

The importance of user benefits in transport projects assessments is well-known by transport planners and economists. Generally they have the greatest impact on the result of cost-benefit analysis. It is common practice to adopt the consumer surplus measure for calculating transport user benefits. Normally the well-known “Rule of a Half”, as a practical approximation for the integral of the demand curve, is used to determine the change of consumer surplus. Changes in travel demand and consumer surplus are influenced by all modeled changing variables, which primarily comprise the generalized costs. However, travel demand models with multiple constraints additionally contain variable “shadow prices”, added to the generalized costs. Such models are often used for travel demand modeling of large-scale areas. The most discussed and well-known model in the field of transport modeling is the doubly constrained gravity model. Beside this model with inelastic constraints, there are also more flexible models with elastic constraints. In this paper we enter into the question of whether applying the Rule of a Half with regard only to the generalized costs and neglecting the shadow prices is valid in the case of travel demand models with multiple constraints. For this purpose, a theoretical analysis in this paper provides a mathematical proof.

In an earlier article (s. J Gen Philos Sci 40:341-355, 2009), I have rejected an interpretation of Aristotle's syllogistic which (since Patzig) is predominant in the literature on Aristotle, but wrong in my view. According to this interpretation, the distinguishing feature of perfect syllogisms is their being evident. Theodor Ebert has attempted to defend this interpretation by means of objections (s. J Gen Philos Sci 40:357-365, 2009) which I will try to refute in part [1] of the following article. I want to show that (1) according to Aristotle's Prior Analytics perfect and imperfect syllogisms do not differ by their being evident, but by the reason for their being evident, (2) Aristotle uses the same words to denote proofs of the validity of perfect and imperfect syllogisms (aEuroapodeixis", "deiknusthai" etc.), (3) accordingly, Aristotle defines perfect syllogisms not as being evident, but as "requiring nothing beyond the things taken in order to make the necessity evident", i.e. as not "requiring one or more things that are necessary because of the terms assumed, but that have not been taken among the propositions" (APr. I. 1), (4) the proofs by which the validity of perfect assertoric syllogisms can be shown according to APr. I. 4 are based on the Dictum de omni et nullo, (5) the fact that Aristotle describes these proofs only in rough outlines corresponds to the fact that his proofs of the validity of other fundamental rules are likewise produced in rough outlines, e.g. his proof of the validity of conversio simplex in APr. I. 2, which usually has been misunderstood (also by Ebert): (6) Aristotle does not prove the convertibility of E-sentences by presupposing the convertibility of I-sentences; only the reverse is true.