SAGGI DI VITTORIO MARCHIS, UMBERTO BOTTAZZINI, FABIO TOSCANO, MASSIMO NEGROTTI, GIUSEPPE O. LONGO, STEFANO A. E. LEONI, ALBERTO CASADEI, SILVANO TAGLIAGAMBE.

This paper discusses an earlier emendation to fr. 54 GRF Funaioli from Varro's De bibliothecis and argues that, while the text et citro refers to cedar oil, it should not be emended to et cedro. A comparison with a passage from Pliny the Elder (HN 13.86) is used to support the view presented in the article.

By a comparison between Maxwell's electrodynamics classically interpreted (MT) and relativistic electrodynamics (RED), this paper discusses whether the “asymmetries” in MT mentioned by A. Einstein in his 1905 relativity paper are only of a conceptual nature or rather involve specific empirical claims. It is shown that in fact MT predicts strongly asymmetric behaviour for very simple interactions, and an analysis is made of the extent of the “symmetry” achieved by means of relativistic postulates. A “low” velocity experiment is suggested which (...) could provide another test of the accuracy of RED with respect to MT. (shrink)

In preference aggregation a set of individuals express preferences over a set of alternatives, and these preferences have to be aggregated into a collective preference. When preferences are represented as orders, aggregation procedures are called social welfare functions. Classical results in social choice theory state that it is impossible to aggregate the preferences of a set of individuals under different natural sets of axiomatic conditions. We define a first-order language for social welfare functions and we give a complete axiomatisation for (...) this class, without having the number of individuals or alternatives specified in the language. We are able to express classical axiomatic requirements in our first-order language, giving formal axioms for three classical theorems of preference aggregation by Arrow, by Sen, and by Kirman and Sondermann. We explore to what extent such theorems can be formally derived from our axiomatisations, obtaining positive results for Sen’s Theorem and the Kirman-Sondermann Theorem. For the case of Arrow’s Theorem, which does not apply in the case of infinite societies, we have to resort to fixing the number of individuals with an additional axiom. In the long run, we hope that our approach to formalisation can serve as the basis for a fully automated proof of classical and new theorems in social choice theory. (shrink)

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the mathematical (...) practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences. (shrink)

This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic |$FL_{ew}$| (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. |$FL_{ew}$|-algebras) that includes both Heyting and Nelson algebras (...) and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics. (shrink)

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite logical (...) matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

Serendipities is a careful unraveling of the fabulous and the false, a brilliant exposition of how unanticipated truths often spring from false ideas. From Leibniz's belief that the I Ching illustrated the principles of calculus to Marco Polo's mistaking a rhinoceros for a unicorn, Umberto Eco offers a dazzling tour of intellectual history, illuminating the ways in which we project the familiar onto the strange to make sense of the world. Uncovering layers of mistakes that have shaped human history, (...) Eco offers with wit and clarity such instances as Columbus's voyage to the New World, the fictions that grew around the Rosicrucians and Knights Templar, and the linguistic endeavors to recreate the language of Babel, to show how serendipities can evolve out of mistakes. With erudition, anecdotes, and scholarly rigor, this new collection of essays is sure to entertain and enlighten any reader with a passion for the curious history of languages and ideas. (shrink)

In this paper we re-assess the philosophical foundation of Exactly True Logic ($$\mathcal {ET\!L}$$ ET L ), a competing variant of First Degree Entailment ($$\mathcal {FDE}$$ FDE ). In order to do this, we first rebut an argument against it. As the argument appears in an interview with Nuel Belnap himself, one of the fathers of $$\mathcal {FDE}$$ FDE, we believe its provenance to be such that it needs to be taken seriously. We submit, however, that the argument ultimately fails, (...) and that $$\mathcal {ET\!L}$$ ET L cannot easily be dismissed. We then proceed to give an overview of the research that was inspired by this logic over the last decade, thus providing further motivation for the study of $$\mathcal {ET\!L}$$ ET L and, more generally, of $$\mathcal {FDE}$$ FDE -related logics that result from semantical analyses alternative to Belnap’s canonical one. We focus, in particular, on philosophical questions that these developments raise. (shrink)

In a companion volume to his "History of Beauty," the renowned philosopher and cultural critic analyzes our attraction to the gruesome, horrific, and repellant in visual culture and the arts, drawing on abundant examples of painting and sculpture, ranging from antiquity to the works of Bosch, Goya, and others.

We experience our own body through both touch and vision. We further see that others’ bodies are similar to our own body, but we have no direct experience of touch on others’ bodies. Therefore, relations between vision and touch are important for the sense of self and for mental representation of one’s own body. For example, seeing the hand improves tactile acuity on the hand, compared to seeing a non-hand object. While several studies have demonstrated this visual enhancement of touch (...) effect, its relation to the ‘bodily self’, or mental representation of one’s own body remains unclear. We examined whether VET is an effect of seeing a hand, or of seeing my hand, using the rubber hand illusion. In this illusion, a prosthetic hand which is brushed synchronously—but not asynchronously—with one’s own hand is felt to actually be one’s hand. Thus, we manipulated whether or not participants felt like they were looking directly at their hand, while holding the actual stimulus they viewed constant. Tactile acuity was measured by having participants judge the orientation of square-wave gratings. Two characteristic effects of VET were observed: cross-modal enhancement from seeing the hand was inversely related to overall tactile acuity, and participants near sensory threshold showed significant improvement following synchronous stroking, compared to asynchronous stroking or no stroking at all. These results demonstrate a clear functional relation between the bodily self and basic tactile perception. (shrink)

Recent research on algebraic models of _quasi-Nelson logic_ has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a _nucleus_. Among these various algebraic structures, for which we employ the umbrella term _intuitionistic modal algebras_, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations (...) arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of _weak implicative semilattices_, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus. (shrink)