If A and B are subsets of natural numbers we say that A is recursively equivalent to B (denoted A ≃ B) if there is a one-one partial recursive function which maps A onto B, and that A is recursively isomorphic to B (denoted A ≅ B) if there is a one-one total recursive function which maps A onto B and Ā (the complement of A) onto B#x00AF;.

We introduce the notion of constructive suprema and of constructively directed sets. The Axiom of Choice turns out to be equivalent to the postulate that every supremum is constructive, but also to the hypothesis that every directed set admits a function assigning to each finite subset an upper bound. The Axiom of Multiple Choice implies a simple set-theoretical induction principle , stating that any system of sets that is closed under unions of well-ordered subsystems and contains all finite subsets of (...) a given set must also contain that set itself. This is not provable without choice principles but equivalent to the statement that the existence of joins for constructively directed subsets of a poset follows from the existence of joins for nonempty well-ordered subsets. Moreover, we establish the equivalence of SIP with several other fundamental statements concerning inductivity, compactness, algebraic closure systems, and the exchange between chains and directed sets. (shrink)

J. N. Crossley [1] raised the question of whether the implication 2 + A = A ⇒ 1 + A = A is true for constructive order types (C.O.T.'s). Using an earlier definition of constructive order type, A. G. Hamilton [2] presented a counterexample. Hamilton left open the general question, however, since he pointed out that Crossley considers only orderings which can be embedded in a standard dense r.e. ordering by a partial recursive function, and that his counterexample (...) fails to meet this requirement. We resolve the question by finding a C.O.T. A which meets Crossley's requirement and such that 2 + A = A but 1 + A ≠ A. At the suggestion of A. B. Manaster and A. G. Hamilton we easily extend this construction to show that for any n ≧ 2, there is a C.O.T. A such that n + A = A but m + A ≠ A for 0 < m < n. Hence, Theorem 3 of [2] and all of its corollaries hold with the new definition of C.O.T. The construction is not difficult and requires no priority argument. The techniques are similar to those developed in [3], but no outside results are needed here. (shrink)

We define and investigate constructibility in higher order arithmetics. In particular we get an interpretation ofn-order arithmetic inn-order arithmetic without the scheme of choice such that ∈ and the property “to be a well-ordering” are absolute in it and such that this interpretation is minimal among such interpretations.

We re-express a previous general result in a way which seems easier to remember, using the terminology of infinite games. We show how this can be applied to construct recursive linear orderings, showing, for example, that if there is a ▵ 0 2β + 1 linear ordering of type τ, then there is a recursive ordering of type ω β · τ.

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)

In a first part, I defend that formal semantics can be used as a guide to ontological commitment. Thus, if one endorses an ontological view \(O\) and wants to interpret a formal language \(L\) , a thorough understanding of the relation between semantics and ontology will help us to construct a semantics for \(L\) in such a way that its ontological commitment will be in perfect accordance with \(O\) . Basically, that is what I call constructing formal semantics from an (...) ontological perspective. In the rest of the paper, I develop rigorously and put into practice such a method, especially concerning the interpretation of second-order quantification. I will define the notion of ontological framework: it is a set-theoretical structure from which one can construct semantics whose ontological commitments correspond exactly to a given ontological view. I will define five ontological frameworks corresponding respectively to: (i) predicate nominalism, (ii) resemblance nominalism, (iii) armstrongian realism, (iv) platonic realism, and (v) tropism. From those different frameworks, I will construct different semantics for first-order and second-order languages. Notably I will present different kinds of nominalist semantics for second-order languages, showing thus that we can perfectly quantify over properties and relations while being ontologically committed only to individuals. I will show in what extent those semantics differ from each other; it will make clear how the disagreements between the ontological views extend from ontology to logic, and thus why endorsing an ontological view should have an impact on the kind of logic one should use. (shrink)

Partial order optimality theory is a conservative generalization of classical optimality theory that makes possible the modeling of free variation and quantitative regularities without any numerical parameters. Solving the ranking problem for PoOT has so far remained an outstanding problem: allowing for free variation, given a finite set of input/output pairs, i.e., a dataset, \ that a speaker S knows to be part of some language L, how can S learn the set of all grammars G under some constraint (...) set C compatible with \?. Here, allowing for free variation, given the set of all PoOT grammars GPoOT over a constraint set C, for an arbitrary \, I provide set-theoretic means for constructing the actual set G compatible with \. Specifically, I determine the set of all STRICT ORDERS of C that are compatible with \. As every strict total order is a strict order, our solution is applicable in both PoOT and COT, showing that the ranking problem in COT is a special instance of a more general one in PoOT. (shrink)

Euclid’s definition of a point as “that which has no part” has been a major source of controversy in relation to the epistemological and ontological presuppositions of classical geometry, from the medieval and modern disputes on indivisibilism to the full development of point-free geometries in the 20th century. Such theories stem from the general idea that all talk of points as putative lower-dimensional entities must and can be recovered in terms of suitable higher-order constructs involving only extended regions (or (...) bodies). Here I focus on what is arguably the first thorough proposal of this sort, Whitehead’s theory of “extensive abstraction”, offering a critical reconstruction of the theory through its successive installments: from the purely mereological version of ‘La théorie relationniste de l’espace’ (1916) to the refined versions presented in An Enquiry Concerning the Principles of Natural Knowledge (1919) and in The Concept of Nature (1920) to the last, mereotopological version of Process and Reality (1929). (shrink)

Adaptationists explain the evolution of religion from the cooperative effects of religious commitments, but which cooperation problem does religion evolve to solve? I focus on a class of symmetrical coordination problems for which there are two pure Nash equilibriums: (1) ALL COOPERATE, which is efficient but relies on full cooperation; (2) ALL DEFECT, which is inefficient but pays regardless of what others choose. Formal and experimental studies reveal that for such risky coordination problems, only the defection equilibrium is evolutionarily stable. (...) The following makes sense of otherwise puzzling properties of religious cognition and cultures as features of cooperative designs that evolve to stabilise such risky exchange. The model is interesting because it explains lingering puzzles in the data on religion, and better integrates evolutionary theories of religion with recent, well-motivated models of cooperative niche construction. (shrink)

We give a construction assigning classes of Boolean algebras to first-order theories; several classes of Boolean algebras considered previously in the literature can be thus obtained. In particular it turns out that the class of semigroup algebras can be defined in this way, in fact by a Horn theory, and it is the largest class of Boolean algebras defined by a Horn theory.

This article examines how employees form their perceptions of managerial responsibility in a concrete organizational setting. Drawing on negotiated order theory, it shows that these perceptions are the result of complex processes of social construction and negotiation, rather than the application of predetermined ethics models or norms. Employees’ perceptions appear to be unstable; they are subject to constant alterations, fluctuating with the organizational circumstances, and are likely to create considerable organizational perturbations, especially when managers make complex and ambiguous decisions. (...) This process is illustrated through an ethnographic study that analyzed the evolution of employee perceptions during a three-year crisis—one that led managers to repeatedly postpone salary payments to save jobs. The process approach adopted by the study highlights important dynamics that traditional business ethics approaches overlook, such as the fragility of the construct of managerial responsibility, which cannot be coherent unless it is constantly renegotiated among an organization’s various employee groups. (shrink)

For a long time, international relations scholars have adopted a narrow view of what is global order, who are its makers and managers, and what means they employ to realize their goals. Amitav Acharya argues that the nature and scope of agency in the global order - who creates it and how - needs to be redefined and broadened. Order is built not by material power alone, but also by ideas and norms. While the West designed the (...) post-war order, the non-Western countries were not passive. They contested and redefined Western ideas and norms, and contributed new ones of their own making. This book examines such acts of agency, especially the redefinitions of sovereignty and security, shaping contemporary world politics. With the decline of Western dominance, ideas and agency from the Rest may make it possible to imagine and build a truly global order. (shrink)

This article praises the development of second order cybernetics by von Foerster, Maturana, and Varela as an important step in deepening our understanding of the bio-psychological foundation of the dynamics of information, cognition, and communication. Luhmann's development of the theory into the realm of social communication is seen as a necessary and important move. The triple autopoietic differentiation between biological, psychological, and social-communicative autopoiesis and the introduction of a technical concept of meaning is central. Finally, the paper shows that (...) second order cybernetics lacks explicit and ontological concepts of emotion, meaning, and a concept of signs. C. S. Peirce's theory is introduced for this purpose. It is then shown, through Varela's development of Spencer Brown's ‘Laws of Form’ from a dual to a dynamic triadic categorical structure, that both theories are triadic and second order, and therefore can be fruitfully fused to a Cybersemiotics. (shrink)

Methodical doubt is usually associated with Descartes. However, it is with Francis Bacon that its function and scope are first recognized – as a preliminary stage in the attainment of knowledge, and as an epistemological tool (a rule) for achieving true knowledge. In this paper, I follow the various steps of construction and use of Baconian doubt as it appears in the first book of the New Organon. I will argue that Bacon - in distancing himself from traditional scepticism – (...) will come to conceive methodical doubt not only as a procedure for renovating knowledge, but more specifically as a prescriptive condition for identifying what 'mind' is required to pursue the aim of renovating knowledge. (shrink)

The constructs of motivation (or needs, motives, etc.) to explain higher-order behavior have burgeoned in psychology. In this article, we critically evaluate such high-level motivation constructs that many researchers define as causal determinants of behavior. We identify a fundamental issue with this predominant view of motivation, which we called the black-box problem. Specifically, high-level motivation constructs have been considered as causally instigating a wide range of higher-order behavior, but this does not explain what they actually are or how (...) behavioral tendencies are generated. The black box problem inevitably makes the construct ill-defined and jeopardizes its theoretical status. To address the problem, we discuss the importance of mental computational processes underlying motivated behavior. Critically, from this perspective, motivation is not a unitary construct that causes a wide range of higher-order behavior --- it is an emergent property that people construe through the regularities of subjective experiences and behavior. The proposed perspective opens new avenues for future theoretical development, i.e., the examination of how motivated behavior is realized through mental computational processes. (shrink)

We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke semantics, and prove that, over Zermelo–Fraenkel set theory, the completeness of such semantics is equivalent to the Boolean prime ideal theorem. Using a result of McCarty, we conclude that the completeness of Kripke semantics is equivalent, over intuitionistic Zermelo–Fraenkel set theory, to the Law of Excluded Middle plus BPI. Along the way, we also prove the equivalence, over ZF, (...) between BPI and the completeness theorem for Kripke semantics for both first-order and propositional theories. (shrink)

Certain extensions of Nelson's constructive logic N with strong negation have recently become important in arti.cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's .rst-order logic as a basis for de.ning nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for .rst-order here-and-there logics, (...) and their minimal strong negation extensions, for both constant and varying domains. We choose the constant domain version, which we denote by QNc5, as a basis for de.ning a .rst-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of QNc5, including Skolem forms and Herbrand theorems and Interpolation, and show that the .rst-oder version of equilibrium logic can be used as a foundation for answer set inference. (shrink)

Higher-order realists about properties express their view that there are properties with the help of higher-order rather than first-order quantifiers. They claim two types of advantages for this way of formulating property realism. First, certain gridlocked debates about the nature of properties, such as the immanentism versus transcendentalism dispute, are taken to be dissolved. Second, a further such debate, the tropes versus universals dispute, is taken to be resolved. In this paper I first argue that higher-order (...) realism does not in fact resolve the tropes versus universals dispute. In a constructive spirit, I then develop higher-order realism in a way that leads to a dissolution, rather than a resolution, of this dispute too. (shrink)

We show that various fragments of the intuitionistic/constructive theory of the reals are decidable.

Niche construction theory highlights the active role of organisms in modifying their environment. A subset of these modifications is the developmental niche, which concerns ecological, epistemic, social and symbolic legacies inherited by organisms as resources that scaffold their developmental processes. Since in this theory development is a situated process that takes place in a culturally structured environment, we may reasonably ask if implicit cultural biases may, in some cases, be responsible for maladaptive developmental niches. In this paper we wish to (...) argue for an affirmative answer. In order to do so, we first propose to conceptualize implicit bias as embodied perceptual habits, and then proceed to show that these habits are at least partially responsible for maladaptive developmental niches of children with Down syndrome and autism. With this framework we thus hope to bring together two fields of research that haven’t been explicitly connected: implicit bias and niche construction theory. Linking these two theories may bring benefits both to implicit bias researchers, who can extend this concept to characterize other sets of processes as biased, as well as to niche construction theorists, who will have a useful theoretical tool to diagnose maladaptive features of niches brought about by sociocultural biases. (shrink)

A key idea in both Frege's development of arithmetic in theGrundlagen[7] and Zermelo's 1904 proof [10] of the well-ordering theorem is that of a “type reducing” correspondence between second-level and first-level entities. In Frege's construction, the correspondence obtains betweenconceptandnumber, in Zermelo's (through the axiom of choice), betweensetandmember. In this paper, a formulation is given and a detailed investigation undertaken of a system ℱ of many-sorted first-order logic (first outlined in the Appendix to [6]) in which this notion of type (...) reducing correspondence is accorded a central role and which enables Frege's and Zermelo's constructions to be presented in such a way as to reveal their essential similarity. By adapting Bourbaki's version of Zermelo's proof of the well-ordering theorem, we show that, within ℱ, any correspondencecbetween second-level entities (here calledconcepts) and first-level ones (here calledobjects) induces a well-ordering relationW(c) in a canonical manner. We shall see that, whencis the “Fregean” correspondence between concepts and cardinal numbers,W(c) is (the well-ordering of) the ordinalω+ 1, and whencis a “Zermelian” choice function on concepts,W(c) is a well-ordering of the universal concept embracing all objects.In ℱ an important role is played by the notion ofextensionof a concept. To each conceptXwe assume there is assigned an objecte(X) in such a way that, for any conceptsX, Ysatisfying a certain predicateE, we havee(X) =e(Y) iff the same objects fall underXandY. (shrink)

As an object which is new, complex and potentially challenging some of the foundations of the Social Order (SO), we sought to study the influence of the Political Orientation (PO) and Social Representations (SR) of the Social Order (Staerklé et al., 2007). Qui a droit à quoi? Représentations et légitimation de l’ordre social. PUG) on the socio-representational construction of the Universal Basic Income (UBI) and stances towards this measure (attitude and estimated fair amount). Data were collected via Facebook (...) from 264 French general population participants (Mage = 37.8; SD = 12.7; 174 women). Our Structural Equation Model revealed that the PO would be associated with the stances towards the UBI, mediated by the different ways of representing it and the representations of the SO. Results show two differents socio-representational logics in relation to being right-wing and one in relation to being left-wing. This leads to contrasting stances towards the UBI and seems to reflect different desires to maintain or change the SO. (shrink)