Testing definitional equivalence of theories via automorphism groups

Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections that preserve ultraproducts in the model categories up to isomorphism. Based on these results, we settle several conjectures of Barrett, Glymour and Halvorson

Vienna Circle and Logical Analysis of Relativity Theory

In this paper we present some of our school's results in the area of building up relativity theory (RT) as a hierarchy of theories in the sense of logic. We use plain first-order logic (FOL) as in the foundation of mathematics (FOM) and we build on experience gained in FOM. The main aims of our school are the following: We want to base the theory on simple, unambiguous axioms with clear meanings. It should be absolutely understandable for any reader what the axioms say and the reader can decide about each axiom whether he likes it. The theory should be built up from these axioms in a straightforward, logical manner. We want to provide an analysis of the logical structure of the theory. We investigate which axioms are needed for which predictions of RT. We want to make RT more transparent logically, easier to understand, easier to change, modular, and easier to teach. We want to obtain deeper understanding of RT. Our work can be considered as a case-study showing that the Vienna Circle's (VC) approach to doing science is workable and fruitful when performed with using the insights and tools of mathematical logic acquired since its formation years at the very time of the VC activity. We think that logical positivism was based on the insight and anticipation of what mathematical logic is capable when elaborated to some depth. Logical positivism, in great part represented by VC, influenced and took part in the birth of modern mathematical logic. The members of VC were brave forerunners and pioneers.Comment: 25 pages, 1 firgure

Comparing theories: the dynamics of changing vocabulary. A case-study in relativity theory

There are several first-order logic (FOL) axiomatizations of special relativity theory in the literature, all looking essentially different but claiming to axiomatize the same physical theory. In this paper, we elaborate a comparison, in the framework of mathematical logic, between these FOL theories for special relativity. For this comparison, we use a version of mathematical definability theory in which new entities can also be defined besides new relations over already available entities. In particular, we build an interpretation of the reference-frame oriented theory SpecRel into the observationally oriented Signalling theory of James Ax. This interpretation provides SpecRel with an operational/experimental semantics. Then we make precise, "quantitative" comparisons between these two theories via using the notion of definitional equivalence. This is an application of logic to the philosophy of science and physics in the spirit of Johan van Benthem's work.Comment: 27 pages, 8 figures. To appear in Springer Book series Trends in Logi

Solving non-uniqueness in agglomerative hierarchical clustering using multidendrograms

In agglomerative hierarchical clustering, pair-group methods suffer from a problem of non-uniqueness when two or more distances between different clusters coincide during the amalgamation process. The traditional approach for solving this drawback has been to take any arbitrary criterion in order to break ties between distances, which results in different hierarchical classifications depending on the criterion followed. In this article we propose a variable-group algorithm that consists in grouping more than two clusters at the same time when ties occur. We give a tree representation for the results of the algorithm, which we call a multidendrogram, as well as a generalization of the Lance and Williams' formula which enables the implementation of the algorithm in a recursive way.Comment: Free Software for Agglomerative Hierarchical Clustering using Multidendrograms available at http://deim.urv.cat/~sgomez/multidendrograms.ph

Chromatic number, clique subdivisions, and the conjectures of Haj\'os and Erd\H{o}s-Fajtlowicz

For a graph $G$, let $\chi(G)$ denote its chromatic number and $\sigma(G)$ denote the order of the largest clique subdivision in $G$. Let H(n) be the maximum of $\chi(G)/\sigma(G)$ over all $n$-vertex graphs $G$. A famous conjecture of Haj\'os from 1961 states that $\sigma(G) \geq \chi(G)$ for every graph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. This conjecture was disproved by Catlin in 1979. Erd\H{o}s and Fajtlowicz further showed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for some absolute constant $c>0$. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant $C$ such that $\chi(G)/\sigma(G) \leq Cn^{1/2}/\log n$ for all $n$-vertex graphs $G$. In this paper we prove the Erd\H{o}s-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on $n$ vertices with independence number $\alpha$.Comment: 14 page

17 ways to say yes:Toward nuanced tone of voice in AAC and speech technology

People with complex communication needs who use speech-generating devices have very little expressive control over their tone of voice. Despite its importance in human interaction, the issue of tone of voice remains all but absent from AAC research and development however. In this paper, we describe three interdisciplinary projects, past, present and future: The critical design collection Six Speaking Chairs has provoked deeper discussion and inspired a social model of tone of voice; the speculative concept Speech Hedge illustrates challenges and opportunities in designing more expressive user interfaces; the pilot project Tonetable could enable participatory research and seed a research network around tone of voice. We speculate that more radical interactions might expand frontiers of AAC and disrupt speech technology as a whole

Axiomatizing relativistic dynamics without conservation postulates

A part of relativistic dynamics (or mechanics) is axiomatized by simple and purely geometrical axioms formulated within first-order logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein's famous $E=mc^2$. The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and four-momentum is also investigated.Comment: 21 pages, 7 figure

Twin Paradox and the logical foundation of relativity theory

We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization SpecRel of special relativity from the literature. SpecRel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in SpecRel. As it turns out, this is practically equivalent to asking whether SpecRel is strong enough to "handle" (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to SpecRel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of SpecRel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that the Twin Paradox becomes provable in AccRel, but it is not provable without IND.Comment: 24 pages, 6 figure

Pressure-dependent hydrometra dimensions in hysteroscopy

AIM: To investigate the relation between intrauterine pressures and volumes for virtual-reality-based surgical training in hysteroscopy. MATERIAL AND METHODS: Ten fresh extirpated uteri were insufflated by commercial hysteroscopy pump and imaged by computer tomography (CT) under intrauterine air pressure in distension-collapse cycles between 0 , 20 (150 mmHg), and 0 kPa, performing a CT scan at every step at about 2.7 kPa (20 mmHg). RESULTS: An initial threshold pressure to distend the cavity was avoided by introducing the insufflation tube up to the fundus. The filling and release phases of seven uteri that were completely distended showed the typical characteristics of a hysteresis curve which is expected from a viscoelastic, nonlinear, anisotropic soft tissue organ like the uterus. In three cases tightening the extirpated uterus especially at the lateral resection lines caused significant problems that inhibited registration of a complete distension-collapse cycle. Interpolated volumes for complete distended cavities and extrapolated for incomplete data sets, derived from the digitally reconstructed three-dimensional (3D) geometries, ranged from 0.6 to 11.4 mL at 20 kPa. These values highly correlate with the uterine volume (not insufflated) considering different biometric data of the uteri and patient data. Linear (R (2) = 0.66) and quadratic least-squares fits (R (2) = 0.74) were used to derive the formulas y = 0.069x and y = 0.00037x (2) + 0.036x, where x is the uterine volume in mL (not insufflated) and y is the cavity volume in mL at 20 kPa intrauterine pressure. CONCLUSIONS: Our experimental hysteroscopical setup enabled us to reconstruct the changes in volumes of insufflated uteri under highly realistic conditions in 3D. The relation between intrauterine pressure and cavity volume in distension-collapse cycles describes a typical hysteresis curve