Since Darwin, the idea of psychological continuity between humans and other animals has dominated theory and research in investigating the minds of other species. Indeed, the field of comparative psychology was founded on two assumptions. First, it was assumed that introspection could provide humans with reliable knowledge about the causal connection between specific mental states and specific behaviors. Second, it was assumed that in those cases in which other species exhibited behaviors similar to our own, similar psychological causes were at (...) work. In this paper. we show how this argument by analogy is flowed with respect to the case of second-order mental states. As a test case, we focus on the question of how other species conceive of visual attention, and in particular whether chimpanzees interpret seeing as a mentalistic event involving internal states of perception, attention, and belief. We conclude that chimpanzees do not reason about seeing in this manner, and indeed, there is considerable reason to suppose that they do not harbor representations of mental states in general. We propose a reinterpretation model in which the majority of the rich social behaviors that humans and other primates share in common emerged long before the human lineage evolved the psychological means of interpreting those behaviors in mentalistic terms. Although humans, chimpanzees, and most other species may be said to possess mental states, humans alone may have evolved a cognitive specialization for reasoning about such states. (shrink)

American Academy of Pediatrics (AAP) and American College of Medical Genetics (ACMG) recently provided two recommendations about predictive genetic testing of children. The Clinical Sequencing Exploratory Research Consortium's Pediatrics Working Group compared these recommendations, focusing on operational and ethical issues specific to decision making for children. Content analysis of the statements addresses two issues: (1) how these recommendations characterize and analyze locus of decision making, as well as the risks and benefits of testing, and (2) whether the guidelines conflict (...) or come to different but compatible conclusions because they consider different testing scenarios. These statements differ in ethically significant ways. AAP/ACMG analyzes risks and benefits using best interests of the child and recommends that, absent ameliorative interventions available during childhood, clinicians should generally decline to order testing. Parents authorize focused tests. ACMG analyzes risks and benefits using the interests of the child and other family members and recommends that sequencing results be examined for additional variants that can lead to ameliorative interventions, regardless of age, which laboratories should report to clinicians who should contextualize the results. Parents must accept additional analysis. The ethical arguments in these statements appear to be in tension with each other. (shrink)

Most donated organs in the United States come from brain dead donors, while a small percentage come from patients who die in “controlled,” or expected, circumstances, typically after the family or surrogate makes a decision to withdraw life support. The number of organs available for transplant could be substantially if donations were permitted in “uncontrolled” circumstances–that is, from people who die unexpectedly, often outside the hospital. According to projections from the Institute of Medicine, establishing programs permitting “uncontrolled donation after circulatory (...) determination of death,” or uDCDD, throughout the United States has the potential to provide 22,000 more donation opportunities annually. In contrast, U.S. controlled donation after circulatory determination of death, or cDCDD, cases have increased progressively over the past decade from 87 to 848 donors, but currently account for only 10.6 percent of all deceased donors. Following the IOM recommendations, several projects exploring the feasibility of uDCDD were funded by the federal government, including a grant from the Health Resources and Services Administration that supported a pilot project in New York City in which the authors of this article participated.A key feature of our protocol, and indeed of many uDCDD protocols, is the initiation of preservation methods such as chest compressions and extracorporeal membrane oxygenation shortly after death in order to perfuse and preserve the donor's organs. Critics of uDCDD argue that the means of determining death deviates from generally ascribed principles. They assert that reinstituting circulation in order to preserve organs has the effect of “undoing” the prior determination of death. The result is that cDCDD is widely accepted and practiced routinely even though it only marginally increases the number of organs available for transplantation, and uDCDD is widely considered unacceptable despite being ethically embraced and proven to significantly increase organ donation opportunities in other countries. This article explores the evolution of this counterintuitive state of affairs and calls for a policy that, in line with the IOM report, allows for both cDCDD and uDCDD protocols. (shrink)

The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is (...) best explained by an example. Complete separable metric spaces are essentially countable because, although the spaces may be uncountable, they can be understood in terms of a countable basis. Simpson gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. Reverse mathematics is less suited to theorems of abstract functional analysis, abstract set theory, universal algebra, or general topology.Section 2 introduces the major subsystems of second order arithmetic used in reverse mathematics: RCA0, WKL0, ACA0, ATR0 and – CA0. Sections 3 through 7 consider various theorems of ordered group theory from the perspective of reverse mathematics. Among the results considered are theorems on ordered quotient groups, groups and semigroup conditions which imply orderability, the orderability of free groups, Hölder's Theorem, Mal'tsev's classification of the order types of countable ordered groups. (shrink)

We study theorems of ordered groups from the perspective of reverse mathematics. We show that suffices to prove Hölder's Theorem and give equivalences of both (the orderability of torsion free nilpotent groups and direct products, the classical semigroup conditions for orderability) and (the existence of induced partial orders in quotient groups, the existence of the center, and the existence of the strong divisible closure).

A totally ordered group G is essentially periodic if for every definable non-trivial convex subgroup H of G every definable subset of G is equal to a finite union of cosets of subgroups of G on some interval containing an end segment of H; it is coset-minimal if all definable subsets are equal to a finite union of cosets, intersected with intervals. We study definable sets and functions in such groups, and relate them to the quasi-o-minimal groups introduced (...) in Belegradek et al. . Main results: An essentially periodic group G is abelian; if G is discrete, then definable functions in one variable are ultimately piecewise linear. A group such that every model elementarily equivalent to it is coset-minimal is quasi-o-minimal , and its definable functions in one variable are piecewise linear. (shrink)

It is known that the spaces of orders on orderable computable fields can represent all Π10 classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent Π10 classes in even a weak manner. Next, we consider presentations of ordered abelian groups, and we show that there is a computable ordered abelian group for which no computable presentation admits a computable set of representatives for its Archimedean classes.

Let n be a positive integer and FAℓ be the free abelian lattice-ordered group on n generators. We prove that FAℓ and FAℓ do not satisfy the same first-order sentences in the language if m≠n. We also show that is decidable iff n{1,2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.

Dp-minimality is a common generalization of weak minimality and weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal (Fact 2.2), but there are dp-minimal densely ordered groups that are not weakly o-minimal. We introduce the even more general notion of inp-minimality and prove that in an inp-minimal densely ordered group, every definable unary function is a union of finitely many continuous locally monotonic functions (Theorem 3.2).

We introduce a deductive system Bal which models the logic of balance of opposing forces or of balance between conflicting evidence or influences. ‘‘Truth values’’ are interpreted as deviations from a state of equilibrium, so in this sense, the theorems of Bal are to be interpreted as balanced statements, for which reason there is only one distinguished truth value, namely the one that represents equilibrium. The main results are that the system Bal is algebraizable in the sense of [5] and (...) its equivalent algebraic semantics BAL is definitionally equivalent to the variety of abelian lattice ordered groups, that is, the categories of the algebras in BAL and of ℓ–groups are isomorphic (see [10], Ch.4, 4). We also prove the deduction theorem for Bal and we study different kinds of semantic consequence associated to Bal. Finally, we prove the co-NP-completeness of the tautology problem of Bal. (shrink)

We study o-minimal expansions of Archimedean totally ordered groups. We first prove that any such expansion must be elementarily embeddable via a unique (provided some nonzero element is 0-definable) elementary embedding into a unique o-minimal expansion of the additive ordered group of real numbers R. We then show that a definable function in an o-minimal expansion of R enjoys good differentiability properties and use this to prove that an Archimedean real closed field is definable in any nonsemilinear (...) expansion of R. Combining these results, we obtain several restrictions on possible o-minimal expansions of arbitrary Archimedean ordered groups and in particular of the rational ordered group. (shrink)

Robinson and Zakon gave necessary and sufficient conditions for an abelian ordered group to satisfy the same first‐order sentences as an archimedean abelian ordered group (i.e., which embeds in the group of real numbers). The present paper generalizes their work to obtain similar results for infinite subgroups of the group of unimodular complex numbers. Furthermore, the groups which satisfy the same first‐order sentences as ultraproducts of finite cyclic groups are characterized.

The following two assertions are equivalent for an o-minimal expansion of an ordered group $$\mathcal M=(M,<,+,0,\ldots )$$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function $$f:A \rightarrow M$$ defined on a definable closed subset of $$M^n$$ has a definable continuous extension $$F:M^n \rightarrow M$$.

The main objective of this paper is the following two results. There exists a computable bi-orderable group that does not have a computable bi-ordering; there exists a bi-orderable, two-generated computably presented solvable group with undecidable word problem. Both of the groups can be found among two-generated solvable groups of derived length $3$. [a]nswers a question posed by Downey and Kurtz; answers a question posed by Bludov and Glass in Kourovka Notebook.One of the technical tools used to obtain the (...) main results is a computational extension of an embedding theorem of B. Neumann that was studied by the author earlier. In this paper we also compliment that result and derive new corollaries that might be of independent interest. (shrink)

We prove that there are 2 χ 0 pairwise non elementarily equivalent existentially closed ordered groups, which solve the main open problem in this area (cf. [3, 10]). A simple direct proof is given of the weaker fact that the theory of ordered groups has no model companion; the case of the ordered division rings over a field k is also investigated. Our main result uses constructible sets and can be put in an abstract general framework. Comparison (...) with the standard methods which use forcing (cf. [4]) is sketched. (shrink)

We study the groups Gal L and Gal KP, and the associated equivalence relations EL and EKP, attached to a first order theory T. An example is given where EL≠ EKP. It is proved that EKP is the composition of EL and the closure of EL. Other examples are given showing this is best possible.

We establish, generalizing Di Nola and Lettieri’s categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various (...) results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang’s MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element. (shrink)

Special groups [M. Dickmann, F. Miraglia, Special Groups : Boolean-Theoretic Methods in the Theory of Quadratic Forms, Memoirs Amer. Math. Soc., vol. 689, Amer. Math. Soc., Providence, RI, 2000] are a first-order axiomatization of the theory of quadratic forms. In Section 2 we investigate reduced special groups which are a lattice under their natural representation partial order ; we show that this lattice property is preserved under most of the standard constructions on RSGs; in particular finite RSGs and RSGs of (...) finite chain length are lattice ordered. We prove that the lattice property fails for the RSGs of function fields of real algebraic varieties over a uniquely ordered field dense in its real closure, unless their stability index is 1 . We show that Open Problem 1 has a positive answer for the RSG of the field Q . In the final section we explore the meaning of Open Problem 1 for formally real fields, in terms of their orders and real valuations; we introduce the notion of “parameter-rank” of a positive-primitive first-order formula of the language for special groups. (shrink)

Given a group , G⊆Mm, definable in a first-order structure equation image equipped with a dimension function and a topology satisfying certain natural conditions, we find a large open definable subset V⊆G and define a new topology τ on G with which becomes a topological group. Moreover, τ restricted to V coincides with the topology of V inherited from Mm. Likewise we topologize transitive group actions and fields definable in equation image. These results require a series of (...) preparatory facts concerning dimension functions, some of which might be of independent interest. (shrink)

A simplified definition of point local clocks and the relationship between an inertial reference frame and a class of such clocks, at rest with respect to each other, are used for an algebraic determination of the geometry of Minkowski's space-time on the set of point events. The group of all automorphisms that preserve the time ordering induced by the set of all equivalent local clocks is shown to be generated by the inhomogeneous orthochronous Lorentz group and dilatations, consistently (...) with a well-known result of E. C. Zeeman. The existence of a universal limit velocity and of an invariant clock cone through any event is among the implications of our proof, thus effectively exploiting a suggestion due to F. Severi. (shrink)

Let M = 〈M, +, <, 0, {λ}λ∈D〉 be an ordered vector space over an ordered division ring D, and G = 〈G, ⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to the t-topology, then it is definably isomorphic to a 'definable quotient group' U/L, for some convex V-definable subgroup U of 〈Mⁿ, +〉 and a lattice L of rank n. As two consequences, (...) we derive Pillay's conjecture for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L. (shrink)

Those who invoke the notion of moral responsibility in ethical discourse seem to be faced with a dilemma. Apparently, they either have to violate the “control principle” which says that nobody can be held responsible for what is beyond one's control. Or they have to concede that in many cases there is a “responsibility void” which means that nobody is responsible. The first option seems unjustifiable. The second renders the concept of moral responsibility useless. This dilemma may be taken to (...) suggest that thinking about moral issues in terms of responsibility is an unproductive way of doing ethics. In our paper we offer a solution which, we hope, can rehabilitate responsibility as a moral concept. It combines order ethics (which is a kind of ethics that primarily focuses on the institutional structure of society) with a recently developed account of group agency. (shrink)

We show that the 'tail' of a doubly homogeneous chain of countable cofinality can be recognized in the quotient of its automorphism group by the subgroup consisting of those elements whose support is bounded above. This extends the authors' earlier result establishing this for the rationals and reals. We deduce that any group is isomorphic to the outer automorphism group of some simple lattice-ordered group.

In 1978, Yu. F. Borisov presented an axiom system using a few basic assumptions and four explicit axioms, the fourth being a formulation of the relativity principle; and he demonstrated that this axiom system had (up to choice of units) only two models: a relativistic one in which worldview transformations are Poincaré transformations and a classical one in which they are Galilean. In this paper, we reformulate Borisov’s original four axioms within an intuitively simple, but strictly formal, first-order logic framework, (...) and convert his basic background assumptions into explicit axioms. Instead of assuming that the structure of physical quantities is the field of real numbers, we assume only that they form an ordered field. This allows us to investigate how Borisov’s theorem depends on the structure of quantities. We demonstrate (as our main contribution) how to construct Euclidean, Galilean, and Poincaré models of Borisov’s axiom system over every non-Archimedean field. We also demonstrate the existence of an infinite descending chain of models and transformation groups in each of these three cases, something that is not possible over Archimedean fields. As an application, we note that there is a model of Borisov’s axioms that satisfies the relativity principle, and in which the worldview transformations are Euclidean isometries. Over the field of reals it is easy to eliminate this model using natural axioms concerning time’s arrow and the absence of instantaneous motion. In the case of non-Archimedean fields, however, the Euclidean isometries appear intrinsically as worldview transformations in models of Borisov’s axioms and neither the assumption of time’s arrow, nor the rejection of instantaneous motion, can eliminate them. (shrink)

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with ‘solubility’ replaced by ‘nilpotence’ and ‘perfectness’, among others, are false.These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini’s theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

It is shown that there is a formula σ(g) in the first-order language of group theory with the following property: for every finite group G, the largest soluble normal subgroup of G consists precisely of the elements g of G such that σ(g) holds.

In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, a so-called pseudo-order. The prime example is the order of the constructive real numbers. We examine two kinds of constructive completions of pseudo-orders: order completions of pseudo-orders and Cauchy completions of ordered groups and fields. It is shown how these can be predicatively defined in type theory, also when the underlying set is non-discrete. Provable choice principles, in particular a generalisation (...) of dependent choice, are used for showing set-representability of cuts. (shrink)

We consider R-torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. In the first part of the paper [4] we compared the theory T of all R-torsionfree RG-modules and the theory T0 of RG-lattices , and we realized that they are almost always different. Now we compare their behaviour with respect to decidability, when RG-lattices are of finite, or wild representation type.

In this paper we study elimination of imaginaries in some classes of pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma /\Delta $ for each definable convex subgroup $\Delta $, and sorts for the quotient groups $\Gamma /(\Delta + \ell \Gamma )$ where $\Delta $ is a definable convex subgroup and $\ell \in \mathbb (...) {N}_{\geq 2}$. We refer to these sorts as the quotient sorts. For the dp-minimal case we obtain a complete elimination of imaginaries if we also add constants to distinguish the cosets of $\ell \Gamma $ in $\Gamma $, where $\ell \in \mathbb {N}_{\geq 2}$. (shrink)

In [GK], Gurevich and Kokorin proved that any two non-trivial ordered abelian groups (o-groups, for short) satisfy the same existential sentences. Let nowG, H be non-trivialo-groups with a commono-subgroupG 0. We determine whetherG andH are existentially equivalent overG 0. As a corollary, we obtain algebraic criteria for deciding, whether ano-subgroupG is existentially closed in ano-groupH. Corresponding results are proved foro-groups in which congruences are regarded as atomic relations.

We show that if H is an effectively completely decomposable computable torsion-free abelian group, then there is a computable copy G of H such that G has computable orders but not orders of every degree.

In his testamentary letter to Auguste Chevalier, Évariste Galois states that, in modern terminology, the smallest simple group has order 60. No proof of this statement survives in his papers, and it has been suggested that a proof would have been impossible using the methods available at the time. We argue that this assertion is unduly pessimistic. Moreover, one fragmentary document, dismissed as a triviality and misunderstood, looks suspiciously like cryptic notes related to this result. We give an elementary (...) proof of Galois’s statement, explain why it is likely that he would have been aware of the methods involved, and discuss the potential relevance of the fragment. (shrink)

Scholars of ethnomethodologically informed discourse studies are often sceptical of the use of interview data such as focus group data. Some scholars quite simply reject interview data with reference to a general preference for so-called naturally occurring data. Other scholars acknowledge that interview data can be of some use if the distinction between natural and contrived data is given up and replaced with a distinction between interview data as topic or as resource. In greater detail, such scholars argue that (...) interview data are perfectly adequate if the researcher wants to study the topic of interview interaction, but inadequate as data for studying phenomena that go beyond the phenomenon of interview interaction. Neither of these more and less sceptical positions are, on the face of it, surprising due to the ethnomethodological commitment to study social order as accomplished in situ, not as something that pre-exists or goes beyond the situated interaction. This article, however, challenges not only the first, but also the second position and suggests that it is, after all, possible to do committedly ethnomethodological studies of focus group data that demonstrate how members of a focus group setting accomplish certain rational orders, and, significantly, how they do so by utilizing certain available resources. (shrink)

We consider R-torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. In the first part of the paper [4] we compared the theory T of all R-torsionfree RG-modules and the theory T0 of RG-lattices , and we realized that they are almost always different. Now we compare their behaviour with respect to decidability, when RG-lattices are of finite, or wild representation type.