Existential Assertions Research Articles

Continuous unimodal maps in economic dynamics: On easily verifiable conditions for topological chaos

In this paper, we offer necessary and sufficient conditions for the presence of odd-period cycles and turbulence in a continuous unimodal interval map. The characterizations we present are original both to the economic and the mathematical literature, and go beyond existential assertions to easy verifiability. We apply our two theorems to six different canonical models in the literature on economic dynamics, all being grounded in the fact that their policy functions are given by continuous unimodal maps. An unintended outcome of the work presented here is to alert the economics profession to a richer conception of erratic and chaotic dynamics, one that goes considerably beyond the 1964-1975 Sharkovsky-Li-Yorke emphasis on three-period cycles and on uncountable scrambled sets.

The strange fate of the morphed ‘rump materialism’: a comment on the vagaries of social science as seen through Alexander Wendt's Quantum Mind and Social Science

AbstractIn putting Wendt's recent Quantum Mind in a larger context both of his own disciplinary engagement and some larger philosophical issues, I try to avoid a hasty dismissal, since the book seems at first blush to offer a ‘theory of everything’, or an uncritical acceptance, since the desire to know what makes the world hand together has always been part of the knowledge game. As to the first problem, I find it rather odd that Wendt spends little time in justifying his particular take on quantum theory, which is far from uncontroversial. Second, I attempt to understand why he has given up on the profession trying now to solve puzzles in the field by claiming that ‘quantum consciousness theory’ provides us with an ‘ace up the sleeve’. But the fact that wave collapse plays havoc with our traditional notions of cause, location, and mass, does not without further ado entitle us to claim that all or most problems in social science dealing with issues of validity and meaning of our concepts (rather than ‘truth/falsity’, as decided by making existential assertions) have been solved by quantum mechanics.

Epistemic Awareness of Doxastic Distinctions: Delineating Types of Beliefs in Belief-Formation

Doxastic distinctions help us define the basis and biases in belief–formation. Empirical and extra-empirical justification play an important role in determining doxastic distinctions. When we distinguish the different types of beliefs, we understand (1) that there are basically three kinds of beliefs, namely, verifiable, falsifiable, and unfalsifiable beliefs. Empirical justification provides the basis for establishing the veracity of verifiable and falsifiable beliefs. Extra-empirical justification provides the basis for establishing the veracity of unfalsifiable or irrefutable beliefs. (2) Verifiable or falsifiable beliefs that are reductive require the mandatory acceptance of their truth. However, unfalsifiable beliefs which are non-reductive require the volitional acceptance of their truth. Because there is both empirical and extra-empirical justification in belief-formation, we can accept beliefs with or without, against, or regardless of empirical evidence. Unfalsifiable beliefs do not mean that these beliefs are unjustifiable; it simply means that these beliefs are not empirically justifiable. Understanding the basis and biases of belief-formation is to be aware of how we come to know what we believe. As empirical basis allows us to hold beliefs based on either the principle of confirmation or falsification. Extra-empirical basis, namely phenomenological conjectures, allows us to hold beliefs founded on existential assertions. The intent of this paper is to present doxastic distinctions to help us understand the basis and biases associated with belief-formation. As long as epistemic claims are accepted, and their content is considered believable, the means used to arrive at those beliefs must also be respected. Every doxastic distinction not only provides the basis for belief-formation but also defines the role and limits of both empirical and extra-empirical justification in belief-formation. When we recognize the different types and different ways of justifying beliefs: we understand (1) why we can accept beliefs with or without evidence; against or regardless of empirical justification, and (2) know when to define beliefs as Plato argued as justified true belief and when to define beliefs as justified belief-decisions. Keywords: beliefs, justified true belief; justified belief-decisions, empirical and extra-empirical

¿PROFUNDIZACIÓN O CAÍDA DEL REALISMO JURÍDICO COMO TEORÍA DESCRIPTIVA DE NORMAS?

Algunas versiones del realismo jurídico pretenden compatibilizar la pretensión de que el derecho es un conjunto de normas con un fuerte compromiso con el empirismo. De conformidad con este último, el derecho no está constituido por entidades abstractas de ningún tipo sino por hechos empíricamente constatables. En vistas a llevar a cabo esta compatibilización, en varios trabajos Riccardo Guastini ha defendido una concepción de las proposiciones normativas, i.e. aserciones existenciales sobre normas jurídicas, como enunciados teóricos acerca del derecho vigente, necesariamente referentes a ciertos hechos. Se concibe así al derecho vigente como el conjunto de textos que son resultado de interpretaciones estables, consolidadas y dominantes que los jueces han llevado a cabo en sus decisiones en el ordenamiento de que se trate. Partiendo de esta versión del realismo jurídico, este trabajo procura sembrar algunas dudas. Primero, sobre este modo de concebir a las proposiciones normativas. Segundo, sobre el modo en que, en consecuencia, queda configurada la teoría del derecho. Tercero, y más en general, sobre la pretensión de compatibilizar la visión del derecho como conjunto de normas con la tesis empirista.

Propositions on Emergency

Predefinablessense: Things can be held or beheld, defined or dealt with, because they inhabit a sense-dimension. To see things is to see past them. The here and now are meaningful only in a distance and a span. The apparently immediate needs a medium in which to appear.excess: Things surpass their sense. They are more than the mediating dimension in which they are met. Their excess can elude definition, resist visibility, and solicit reinterpretation.Definitionsbeing: the sense of excess and the excess over sense. The concept of being indicates nonconceptuality: to grasp a thing as something that is (an entity) is to acknowledge that it surpasses the grasping. To recognize something as an entity is to experience it, within a sense-dimension, as exceeding that dimension.Being as such pertains to any entity that can be met in any way. Particular ways of being pertain to particular kinds of entities, with their characteristic senses and ways of exceeding sense.Scholium: As excess, being is often called existence. As sense, being is often called essence. The distinction is useful, but the experiences of existence and essence are normally interdependent.From a logical point of view, being appears as the separate functions of existential quantification and predication. But the capacity to make existential assertions (more broadly, to recognize excess) is tightly linked to the capacity to assign predicates (more broadly, to make sense of entities).emergency: a clash between sense and excess. An event in which excess challenges sense and resists being interpreted, so that sense has trouble getting a purchase on entities.The Emergency would be the primordial clash between sense and excess, the contest that generates the contenders as such, first establishing sense with excess as its counterpart.Unsettling emergencies are non-primordial emergencies, events in which some settled sense is unsettled by some excess.To emerge as emergent is to originate in a clash between sense and excess and call for an interpretation-to issue from an emergency and become an issue, hovering, unresolved and dissonant, demanding to be addressed.Scholium: What is most often called an emergency, the irruption of a threat that demands an immediate response, does not necessarily challenge sense, since the interpretation of the irruption may follow well-worn paths. But such an irruption always has the potential for trauma and surprise, so it can become an emergency as defined here.Emergencies as defined here are not necessarily threatening; they can bring pleasure and delight. Sense can be unsettled by an experience of wonder or rapture.Unsettling emergencies range from a passing confusion to a conceptual revolution to the shattering of a world. The degree to which an emergency can be understood within established sense-dimensions is a matter for judgment. Mature judgment develops through experiences of emergencies of various scopes.An emergent issue may hover in the background, even though it seems to have been resolved. The emergent is pacified when it takes the form of an answered question. It is dormant when it is not even recognized as a question.The Emergency can be defined in terms of sense and excess, as their origin; but it itself would be neither sense nor excess. To define it is not to understand it, nor to reduce it to the terms of the definition.Postulates 1. There are entities whose own being emerges for them. Call them we.1.1. Our excess over sense emerges for us. That we are here, that we exist and may not exist-regardless of how we may interpret our existence-has become an issue for us. We are faced with death as the ultimate excess, the extreme withdrawal from the sphere of sense. This issue can fade into the background as we take our existence for granted.1.2. The sense of our excess emerges for us. We are faced with the question of who we are, the demand to interpret ourselves as entities. …

Category Theory is a Contentful Theory

In (Linnebo & Pettigrew, 2011), some objections to category theory as an autonomous foundation are presented. The authors of that paper do a commendable job making clear several distinct senses of “autonomous” as it occurs in the phrase “autonomous foundation.” Unfortunately, the paper seems to treat the “categorist” perspective rather unfairly. Several infelicities of this sort were addressed by (McLarty, 2012). I wish in this note to address yet another apparent infelicity. The subject of this paper is the comments in (Linnebo & Pettigrew, 2011) concerning the contentfulness of William Lawvere’s axiomatic system CCAF. For details of this system itself the reader is encouraged to consult (Lawvere, 1966) or (Lawvere, 1963). No technical details from these expositions will be needed, however. The authors of (Linnebo & Pettigrew, 2011) are willing to admit “CCAF asserts the existence of certain categories and describes some of the functors between them.” However, as they correctly point out, this by itself is insufficient for CCAF to serve as a foundation of mathematics. As the authors point out, a necessary condition for CCAF to serve as a foundation of mathematics is that be contentful, and, “if it is to make a contentful assertion, we need to be able to identify its subject matter – namely, categories – independently of the theory.” The authors provide us an illustrative example of a non-contentful theory to help make this objection more clear. “[A lack of content] would be the problem, for instance, with the theory that consists of the following sentence: ‘the mome raths outgrabe’. The reason that this theory lacks content is that there is no way of identifying, independently of the theory, what mome raths are, or what it is to outgrabe. Without identifying the subject matter of the theory in such a way, the statements of the theory do not make contentful assertions.” (Linnebo & Pettigrew, 2011, p. 231) Presumably (and the authors are certainly correct here) a theory that runs afoul this objection (which I will call the Contentful Theory Objection [CTO]) cannot serve as an autonomous foundation for mathematics. Of course, a proponent of CCAF can respond to CTO by simply providing a CCAF-free identification of the subject matter of category theory. The problem, the authors hold, is that this cannot be accomplished without running afoul what they call the Logical Dependence Objection [LDO]: if any theory T is to provide an alternative foundation for mathematics to the foundation provided by set theory, it must not be the case that T “depend[s] logically on a prior theory of classes and functions in order to ground [its] existential assertions.” That is, if one cannot identify This work was supported by the National Science Foundation under Grant No. 00006595. A side note: CCAF is often assumed to abbreviate either “category of categories and functors” or “category of categories as foundation.” In Lawvere’s thesis, however, the system normally called CCAF appears under the heading “category of categories and adjoint functors”.

Primitive recursive selection functions for existential assertions over abstract algebras

We generalize to abstract many-sorted algebras the classical proof-theoretic result due to Parsons, Mints and Takeuti that an assertion ∀ x ∃ y P ( x , y ) (where P is Σ 1 0 ), provable in Peano arithmetic with Σ 1 0 induction, has a primitive recursive selection function. This involves a corresponding generalization to such algebras of the notion of primitive recursiveness. The main difficulty encountered in carrying out this generalization turns out to be the fact that equality over these algebras may not be computable, and hence atomic formulas in their signatures may not be decidable. The solution given here is to develop an appropriate concept of realizability of existential assertions over such algebras, generalized to realizability of sequents of existential assertions. In this way, the results can be seen to hold for classical proof systems. This investigation may give some insight into the relationship between specifiability and computability for data types such as the reals, where the atomic formulas, i.e., equations between terms of type real, are not computable.

Existential assertions and quantum levels on the tree of the situation calculus

In a seminal paper, Reiter introduced a variant of the situation calculus along with a set of its properties. To the best of our knowledge, one of these properties has remained unproved and ignored despite its relevance to the planning problem and the expressivity of the theories of actions. We state this property in a more general form and provide its proof. Intuitively, whenever a theory of actions entails that there exists a situation satisfying a first order formula (e.g., a goal), at least one such situation must be found within a predetermined distance from the initial situation. This distance is finite and the same in all the models of the theory, since it depends only on the theory and the formula at hand.

Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems

The tangential Hilbert 16th problem is to place an upper bound for the number of isolated ovals of algebraic level curves { H ( x , y ) = const } \{H(x,y)=\operatorname {const}\} over which the integral of a polynomial 1-form P ( x , y ) d x + Q ( x , y ) d y P(x,y)\,dx+Q(x,y)\,dy (the Abelian integral) may vanish, the answer to be given in terms of the degrees n = deg ⁡ H n=\deg H and d = max ( deg ⁡ P , deg ⁡ Q ) d=\max (\deg P,\deg Q) . We describe an algorithm producing this upper bound in the form of a primitive recursive (in fact, elementary) function of n n and d d for the particular case of hyperelliptic polynomials H ( x , y ) = y 2 + U ( x ) H(x,y)=y^2+U(x) under the additional assumption that all critical values of U U are real. This is the first general result on zeros of Abelian integrals that is completely constructive (i.e., contains no existential assertions of any kind). The paper is a research announcement preceding the forthcoming complete exposition. The main ingredients of the proof are explained and the differential algebraic generalization (that is the core result) is given.

A logic-based approach to program flow analysis

A formalism is presented for tracking assertions which hold universally, i.e., at the end of all the execution paths to a given program point, and assertions which hold existentially, i.e., at the end of some execution paths. In the formalism, the assertions which hold at a given execution path are uniformly defined by an entry environment which contains the assertions which hold when the execution of the program begins and an environment transformer for every program construct. The novel aspect of our formalism is that Horn clauses are used to specify the consistent environments and the meaning of program constructs. The best iterative algorithm (a notion defined by P. Cousot and R. Cousot) for tracking universal and existential assertions simultaneously is given. Conditions are presented under which the best iterative algorithm can be efficiently implemented. The formalism is applied to the pointer equality problem in Pascal. It is shown that universal pointer equalities may be used to reduce the number of superfluous existential equalities, and that existential equalities may be used to obtain more universal equalities. Recent empirical results indicate that tracking the combination of may and must equalities leads to substantial improvements in the result of the analysis. For programs without recursively defined records, the best iterative algorithm can be effectively implemented. These results apply to multiple levels of pointers and can be extended to handle possibly recursive procedures. However, for programs with recursively defined data types further approximations are necessary, e.g., by using a finite graph to model all the possible pointer equalities. For simplicity, this paper does not present an analysis algorithm for this case.

Conceptual Associations and Weak Existential Assertions

The purpose of this paper is to analyze the difference between two types of existential assertions, viz. strong existential assertions and weak existential assertions. This distinction is based on the concept of relevance as defined by Sperber and Wilson (1986). The strengh of an existential assertion can be influenced by different factors. In this article, we try to identify the factors which make an existential assertion weak. A particular attention is devoted to the conceptual factors, among which conceptual associations play a prominent role

Some relations between classical and constructive mathematics

This paper is devoted to the general question, which assertions φ have the property,if φ is provable classically, then φ is(*) provable constructively.More generally, we consider the question, what is the “constructive content” of a given classical proof? Our aim is to formulate rules in a form applicable to mathematical practice. Often a mathematician has the feeling that there will be no difficulty constructivizing a certain proof, only a number of routine details; although it can be quite laborious to set them all out. We believe that most such situations will come quite easily under the scope of the rules given here; the metamathematical machinery will then take care of the details.This basic idea is not new; it has been discussed by Gödel and by Kreisel. Kreisel's investigations [Kr] were based on Herbrand's theorem; in unpublished memoranda he has also used Gödel's methods on some examples. These methods of Gödel (the double-negation and Dialectica interpretations) lie at the root of our work here. Previous work, however, has been limited to traditional formal systems of number theory and analysis. It is only recently that formal systems adequate to formalize constructive mathematics as a whole have been developed. Thus, for the first time we are in a position to formulate logical theorems which are easily applicable to mathematical practice. It is this program which we here carry out.Now that the work has been placed in some historical context, let us return to the main question: which φ have the property (*)? Upon first considering the problem, one might guess that any φ which makes no existential assertions (including disjunctions) should have the property (*).

Existence and feasibility in arithmetic

“From two integers k, l one passes immediately to kl; this process leads in a few steps to numbers which are far larger than any occurring in experience, e.g., 67(257729).Intuitionism, like ordinary mathematics, claims that this number can be represented by an arabic numeral. Could not one press further the criticism which intuitionism makes of existential assertions and raise the question: What does it mean to claim the existence of an arabic numeral for the foregoing number, since in practice we are not in a position to obtain it?