Provability, Computability and Reflection.

We introduce the notion of an alphabetic trace of a cut-free intuitionistic prepositional proof and show that it serves to characterize the equality of arrows in cartesian closed categories. We also show that alphabetic traces improve on the notion of the generality of proofs proposed in the literature. The main theorem of the paper yields a new and considerably simpler solution of the coherence problem for cartesian closed categories than those in [11, 14].

In this paper, we approach the problem of the relationship between Greek mathematics and Eleatic philosophy from a new perspective, which leads us to a reappraisal of Szabó’s hypothesis about the origin of mathematics out of Eleatic philosophy. We claim that Parmenidean philosophy, particularly its semantic core, has possibly been shaped by reflexion on the Pythagoreans’ mathematical practice, particularly in arithmetic. Furthermore, Pythagorean arithmetic originates not from another domain outside mathematics but from counting, i.e., it has its roots in man’s (...) practical activity. This interpretation restores the historically inverse relationship between mathematics and philosophy, refuting the attribution of mathematics’ origin to a field outside mathematics, for which Szabó’s hypothesis has been criticised. Moreover, Parmenidean theory of truth is viewed not as a defective predecessor of Aristotle’s classical theory of truth that needs to be remedied but as a semantic conception coordinated with the mathematics of Parmenides’ times. (shrink)

A recent analysis by de Barros and Suppes of experimentally realizable GHZ correlations supports the conclusion that these correlations cannot be explained by introducing local hidden variables. We show, nevertheless, that their analysis does not exclude local hidden variable models in which the inefficiency in the experiment is an effect not only of random errors in the detector equipment, but is also the manifestation of a pre-set, hidden property of the particles ("prism models"). Indeed, we present an explicit prism model (...) for the GHZ scenario; that is, a local hidden variable model entirely compatible with recent GHZ experiments. (shrink)

In 2001, Le Bars proved that there exists an existential monadic second-order sentence such that the probability that it is true on [Formula: see text] does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for [Formula: see text], and give new examples of sentences with fewer variables without convergence.

Some theorists, worried about liberalism’s potential as a foundation for public health ethics, suggest that republicanism provides a better background of justification for public health policies, interventions, etc. In this article, this suggestion is put to the test, and it is argued that (i) contemporary (civic) republicanism and liberalism are not nearly as opposed as it is sometimes suggested, and that (ii) the kind of republicanism which one leading scholar in the field, Bruce Jennings, as an alternative to liberalism, does (...) not reflect the contemporary understanding of republicanism as held by, e.g. Phillip Pettit et al. (shrink)

Let T 1 , T 2 be countable first-order theories, M i ⊨ T i , and ������ any regular ultrafilter on λ ≥ $\aleph_{0}$ . A longstanding open problem of Keisler asks when T 2 is more complex than T 1 , as measured by the fact that for any such λ, ������, if the ultrapower (M 2 ) λ /������ realizes all types over sets of size ≤ λ, then so must the ultrapower (M 1 ) λ /������. (...) In this paper, building on the author's prior work [12] [13] [14], we show that the relative complexity of first-order theories in Keisler's sense is reflected in the relative graph-theoretic complexity of sequences of hypergraphs associated to formulas of the theory. After reviewing prior work on Keisler's order, we present the new construction in the context of ultrapowers, give various applications to the open question of the unstable classification, and investigate the interaction between theories and regularizing sets. We show that there is a minimum unstable theory, a minimum TP 2 theory, and that maximality is implied by the density of certain graph edges (between components arising from Szemerédi-regular decompositions) remaining bounded away from 0, 1. We also introduce and discuss flexible ultrafilters, a relevant class of regular ultrafilters which reflect the sensitivity of certain unstable (non low) theories to the sizes of regularizing sets, and prove that any ultrafilter which saturates the minimal TP 2 theory is flexible. (shrink)

The Birkhoff-Maltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasivariety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are related to the Birkhoff-Maltsev problem are also considered, including ones that stem from the theory of propositional logics.

For a first-order formula φ(x; y) we introduce and study the characteristic sequence ⟨P n : n < ω⟩ of hypergraphs defined by P n (y₁…., y n ):= $(\exists x)\bigwedge _{i\leq n}\varphi (x;y_{i})$ . We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theoretic properties of φ and vice versa. The main results are a characterization of NIP and of simplicity in terms of persistence of configurations in the characteristic sequence. Specifically, we show that (...) some tree properties are detected by the presence of certain combinatorial configurations in the characteristic sequence while other properties such as instability and the independence property manifest themselves in the persistence of complicated configurations under localization. (shrink)

According to the standard realistic interpretation of classical electrodynamics, the electromagnetic field is conceived as a real physical entity existing in space and time. The problem we address in this paper is how to understand this spatiotemporal existence, that is, how to describe the persistence of a field-like physical entity like electromagnetic field. First, we provide a formal description of the notion of persistence: we derive an “equation of persistence” constituting a necessary condition that the spatiotemporal distributions of the fundamental (...) attributes of a persisting physical entity must satisfy. We then prove a theorem according to which the vast majority of the solutions of Maxwell’s equations, describing possible spatiotemporal distributions of the fundamental attributes of the electromagnetic field, violate the equation of persistence. Finally, we discuss the consequences of this result for the ontology of the electromagnetic field. (shrink)

In spite of his post-World War II works on international law, which seems more purely juridical, Hans Kelsen continues to put forward in his vast body of work an implicit – and sometimes even explicit – juridical objectivism and pacifism. Especially before and during the second World War he makes – by means of many moral-political writings – an effort for a more effective assurance of international peace. The fact that Kelsen regards the law as the pre-eminent means to achieve (...) the end of peace, is – in view of the tradition running from Hobbes to Kant – still not very innovating. But the primary and central role he assigns to the judiciary in order to guarantee international peace is on the other hand really original. To bring out Kelsen’s juridical objectivism and pacifism, I restrict myself in this article to his plea before and during the second World War for the judicial primacy in the international legal order. In that plea, which is spread over many writings and founded by several arguments, I distinguish however three main convincing arguments – namely an evolutionary, a technical and a moral-political argument – for the establishment of an international court with compulsory jurisdiction as the ‘guardian of international peace’. The fact that Kelsen’s original plea is still relevant to our times, is already evident from the post-World War II UN-system of peace-enforcement, which still functions little satisfactorily and of which the existing plans of reform go no further than strengthening the position of the Security Council, the current hardly effective ‘maintainer of world peace’. (shrink)

ABSTRACTThere are two distinct types of legal wrongdoing: civil and criminal. This article demonstrates in three ways that Immanuel Kant's Universal Principle of Right, properly interpreted, offers a plausible and resilient account of this important distinction. First, Kant's principle correctly identifies attempted crimes as crimes themselves even when they do not violate the rights of any individual. Second, it justifies our treatment of reckless endangerment as a crime by distinguishing it from ordinary negligence, traditionally thought to be only civilly wrong. (...) Third, it explains and justifies differences between the features of traditional criminal punishments and the features of civil remedies. Moreover, the Universal Principle of Right yields a Kantian standard for criminal wrongdoing that is compelling enough to inform future philosophical inquiries into the nature and limits of the state's criminal lawmaking authority. (shrink)