Context: The thoroughly second-order cybernetic underpinnings of naturalist theatre have gone almost entirely unremarked in the literature of both theatre studies and cybernetics itself. As a result, rich opportunities for the two fields to draw mutual benefit and break new ground through both theoretical and empirical investigations of these underpinnings have, thus far, gone untapped. Problem: The field of cybernetics continues to remain academically marginalized for, among other things, its alleged lack of experimental rigor. At the same time, (...) the field of theatre studies finds itself at an impasse between post-structuralist semioticians and embodied cognitivists regarding key onto-epistemological issues. A program of research framing and utilizing naturalist theatre as a second-order cybernetic/cybersemiotic laboratory holds much promise in addressing both matters and lending credence to Ross Ashby’s assertion that “the discovery that two fields are related leads to each branch helping in the development of the other.” Method: After establishing the nature of the onto-epistemological deadlock within theatre studies, this article examines the application of cybernetic heuristics within naturalistic theatre, leading to a second-order cybernetic analysis of its processes of production and reception and the outline of an experimental program for exploring these processes further. Results: Foundations for a model of naturalist theatre as a cybersemiotic laboratory grounded in a novel operationalization of Gordon Pask’s conversation theory. Implications: The proposed laboratory could result in the generation of quantitative and qualitative research pertaining to several dimensions of second-order cybernetics; particularly cybersemiotics, which, as a result, may end up better positioned to help dissolve onto-epistemological deadlocks between constructivists and realists of all stripes across the academy and beyond. Constructivist content: I argue that an analysis of naturalistic theatre’s processes of meaning-making filtered through the constructivist ontological agnosticism of second-order cybernetics offers a productive middle way forward for those on both sides of the social constructivist/embodied cognitive realist divide, within and beyond theatre studies. The article draws upon the works of Gregory Bateson, Søren Brier, Ranulph Glanville, Heinz von Foerster, and Niklas Luhmann. (shrink)

This paper begins with a problem stemming from Hume regarding credences about credences. Suppose one has a credence of .95 in p, and suppose one assesses the credence to be such. But suppose one’s second-order credence in this assessment is less than 1. Then, by a standard conditionalization rule, one’s credence in p becomes less than .95. Moreover, such “erosion” can iterate by considering one’s, third-, fourth-, fifth-order credences, etc. (In light of this, some (...) have rejected higher-order credences; however, it is argued that Lewis’ “Principal Principle” forbids this.) The paper then offers a partial solution by describing circumstances under which a person has reason to assign credence 1 to a first-order credence. The solution takes the form of a “transparency” view of self-knowledge from Parent (2017, chapter 8). Briefly, it is argued that when one “reflex-like” utters a sentence “The probability of p is n”, then as a matter of psychological law, the utterance is an expression of one’s own judgment. The Principal Principle then necessitates that one’s credence in p is n, assuming one is being rational. Thus, upon perceiving such an utterance, one can thereby have a reason to assign a second-order credence of 1. (shrink)

This paper begins with a problem stemming from Hume regarding credences about credences. Suppose one has a credence of .95 in p, and suppose one assesses the credence to be such. But suppose one’s second-order credence in this assessment is less than 1. Then, by a standard conditionalization rule, one’s credence in p becomes less than .95. Moreover, such “erosion” can iterate by considering one’s, third-, fourth-, fifth-order credences, etc. (In light of this, some (...) have rejected higher-order credences; however, it is argued that Lewis’ “Principal Principle” forbids this.) The paper then offers a partial solution by describing circumstances under which a person has reason to assign credence 1 to a first-order credence. The solution takes the form of a “transparency” view of self-knowledge from Parent (2017, chapter 8). Briefly, it is argued that when one “reflex-like” utters a sentence “The probability of p is n”, then as a matter of psychological law, the utterance is an expression of one’s own judgment. The Principal Principle then necessitates that one’s credence in p is n, assuming one is being rational. Roughly, upon perceiving such an utterance, one can thereby have a reason to assign a second-order credence of 1. (shrink)

David Lewis's influential work on the epistemology and metaphysics of objective chance has convinced many philosophers of the central importance of the following two claims: First, it is a serious cost of reductionist positions about chance (such as that occupied by Lewis) that they are, apparently, forced to modify the Principal Principle--the central principle relating objective chance to rational subjective probability--in order to avoid contradiction. Second, it is a perhaps more serious cost of the rival non-reductionist position that, (...) unlike reductionism, it can give no coherent explanation for why the Principal Principle should hold. I argue that both of these claims are fundamentally mistaken. (shrink)

Some have argued that chance and determinism are compatible in order to account for the objectivity of probabilities in theories that are compatible with determinism, like Classical Statistical Mechanics (CSM) and Evolutionary Theory (ET). Contrarily, some have argued that chance and determinism are incompatible, and so such probabilities are subjective. In this paper, I argue that both of these positions are unsatisfactory. I argue that the probabilities of theories like CSM and ET are not chances, but also that they (...) are not subjective probabilities either. Rather, they are a third type of probability, which I call counterfactual probability. The main distinguishing feature of counterfactual-probability is the role it plays in conveying important counterfactual information in explanations. This distinguishes counterfactual probability from chance as a second concept of objective probability. (shrink)

According to the traditional Bayesian view of credence, its structure is that of precise probability, its objects are descriptive propositions about the empirical world, and its dynamics are given by conditionalization. Each of the three essays that make up this thesis deals with a different variation on this traditional picture. The first variation replaces precise probability with sets of probabilities. The resulting imprecise Bayesianism is sometimes motivated on the grounds that our beliefs should not be more precise than the evidence (...) calls for. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In the first essay I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid without compromising the initial evidentialist motivation. The second variation replaces descriptive claims with moral claims as the objects of credence. I consider three standard arguments for probabilism with respect to descriptive uncertainty—representation theorem arguments, Dutch book arguments, and accuracy arguments—in order to examine whether such arguments can also be used to establish probabilism with respect to moral uncertainty. In the second essay, I argue that by and large they can, with some caveats. First, I don’t examine whether these arguments can be given sound non-cognitivist readings, and any conclusions therefore only hold conditional on cognitivism. Second, decision-theoretic representation theorems are found to be less convincing in the moral case, because there they implausibly commit us to thinking that intertheoretic comparisons of value are always possible. Third and finally, certain considerations may lead one to think that imprecise probabilism provides a more plausible model of moral epistemology. The third variation considers whether, in addition to conditionalization, agents may also change their minds by becoming aware of propositions they had not previously entertained, and therefore not previously assigned any probability. More specifically, I argue that if we wish to make room for reflective equilibrium in a probabilistic moral epistemology, we must allow for awareness growth. In the third essay, I sketch the outline of such a Bayesian account of reflective equilibrium. Given that this account gives a central place to awareness growth, and that the rationality constraints on belief change by awareness growth are much weaker than those on belief change by conditionalization, it follows that the rationality constraints on the credences of agents who are seeking reflective equilibrium are correspondingly weaker. (shrink)

“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. Pp. 61–76. -/- Abstract. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is (...) actually a familiar part of our traditional intuitive logical framework and that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as radical: an incipient paradigm shift involving radical repudiation of a part of our scientific tradition, a tradition that is defended by classical logicians. But it is also viewed as reactionary: as being analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may someday be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor. People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic. And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject. As Aristotle first said, we do not understand a discipline until we have seen its development. It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been. (shrink)

In the language of second-order logic, first- and second-order variables are distinguished syntactically and cannot be grammatically substituted. According to a prominent argument for the deployment of these languages, these substitution failures are necessary to block the derivation of paradoxes that result from attempts to generalize over predicate interpretations. I first examine previous approaches which interpret second-order sentences using expressions of natural language and argue that these approaches undermine these syntactic restrictions. I then examine (...) Williamson’s primitivist approach according to which second-order sentences are not offered readings in a previously understood language. I argue that the syntactic restrictions alone do not block the derivation of the paradox, unless they are backed by a principled reason that the language cannot be expanded to allow the grammatical substitution of first- and second- order variables. I argue that there is neither a syntactic nor a semantic principle that prohibits such an expansion. (shrink)

The characterization of second-order type isomorphisms is a purely syntactical problem that we propose to study under the enlightenment of game semantics. We study this question in the case of second-order λμ-calculus, which can be seen as an extension of system F to classical logic, and for which we define a categorical framework: control hyperdoctrines.Our game model of λμ-calculus is based on polymorphic arenas which evolve during the play. We show that type isomorphisms coincide with the (...) “equality” on arenas associated with types. Finally we deduce the equational characterization of type isomorphisms from this equality. We also recover from the same model Roberto Di Cosmo’s characterization of type isomorphisms for system F.This approach leads to a geometrical comprehension on the question of second-order type isomorphisms, which can be easily extended to some other polymorphic calculi including additional programming features. (shrink)

A rich literature has discussed what it is to respond to a reason, e.g., to believe or act on the basis of some consideration or another. In comparison, what it would be to respond to a second-order reason has been underexplored. Yet formulating an account of this is vital for maintaining the existence of second-order reasons in both the practical and epistemic domains. And indeed, there are reasons to doubt this is possible. For example, responding to (...) second-order reasons is meant to be different from simply coming to believe or act for a more richly described first-order reason but it's not clear what this would look like. This paper introduces an account of the second-order basing relation which meets these concerns. It therefore both develops our picture of second-order reasons and helps to defend their existence. (shrink)

Quine’s most important charge against second-, and more generally, higher-order logic is that it carries massive existential commitments. The force of this charge does not depend upon Quine’s questionable assimilation of second-order logic to set theory. Even if we take second-order variables to range over properties, rather than sets, the charge remains in force, as long as properties are individuated purely extensionally. I argue that if we interpret them as ranging over properties more reasonably (...) construed, in accordance with an abundant or deflationary conception, Quine’s charge can be resisted. This interpretation need not be seen as precluding the use of model-theoretic semantics for second-order languages; but it will preclude the use of the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach I recommend has revisionary implications which some may find unpalatable; it is, however, compatible with the quite different special case in which the second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, some important metalogical results obtainable under the standard semantics may still be obtained. In my final section, I discuss the relations between second-order logic, interpreted as I recommend, and a strong version of schematic ancestral logic promoted in recent work by Richard Heck. I argue that while there is an interpretation on which Heck’s logic can be contrasted with second-order logic as standardly interpreted, when it is so interpreted, its differences from the more modest form of second-order logic I advocate are much less substantial, and may be largely presentational. (shrink)

We examine second order intuitionistic propositional logic, IPC². Let $F_\exists $ be the set of formulas with no universal quantification. We prove Glivenko's theorem for formulas in $F_\exists $ that is, for φ € $F_\exists $ φ is a classical tautology if and only if ¬¬φ is a tautology of IPC². We show that for each sentence φ € $F_\exists $ (without free variables), φ is a classical tautology if and only if φ is an intuitionistic tautology. As (...) a corollary we obtain a semantic argument that the quantifier V is not definable in IPC² from ⊥, V, ^, →. (shrink)

A normative reason for a person to? is a consideration which favours?ing. A motivating reason is a reason for which or on the basis of which a person?s. This paper explores a connection between normative and motivating reasons. More specifically, it explores the idea that there are second-order normative reasons to? for or on the basis of certain first-order normative reasons. In this paper, I challenge the view that there are second-order reasons so understood. I (...) then show that prominent views in contemporary epistemology are committed to the existence of second-order reasons, specifically, views about the epistemic norms governing practical reasoning and about the role of higher-order evidence. If there are no second-order reasons, those views are mistaken. (shrink)

Intuitionistic dependence logic was introduced by Abramsky and Väänänen [1] as a variant of dependence logic under a general construction of Hodges’ (trump) team semantics. It was proven that there is a translation from intuitionistic dependence logic sentences into second order logic sentences. In this paper, we prove that the other direction is also true, therefore intuitionistic dependence logic is equivalent to second order logic on the level of sentences.

Sellars [1960, ‘Grammar and existence: A preface to ontology’] argues that the truth of a second-order sentence does not incur commitment to there being any sort of abstract entity. This paper begins by exploring the arguments that Sellars offers for the above claim. It then develops those arguments by pointing out places where Sellars has been unclear or ought to have said more. In particular, Sellars's arguments rely on there being a means by which language users could come (...) to understand sentences of a second-order language wherein the truth of second-order existential sentences do not require there to be abstract entities. In addition to this, as Sellars [1979, Naturalism and Ontology] notes, a formal account of quantification is required that does not make use of the apparatus of sequences. Both such a translation of and a formal account of quantification are provided in this article. (shrink)

This dissertation develops an inferentialist theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. From this starting point the dissertation develops a theory of quantification as marking (...) coherent ways a language can be expanded and modality as the means by which we can reflect on the norms governing the assertion and denial conditions of our language. If the view of quantification that is argued for is correct, then there is no tension between second-order quantification and nominalism. In particular, the ontological commitments one can incur through the use of a quantifier depend wholly on the ontological commitments one can incur through the use of atomic sentences. The dissertation concludes by applying the developed theory of meaning to the metaphysical issue of necessitism and contingentism. Two objections to a logic of contingentism are raised and addressed. The resulting logic is shown to meet all the requirement that the dissertation lays out for a theory of meaning for quantifiers and modal operators. (shrink)

The logic of paradox, LP, is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order LP is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, concluding (...) that it will be extremely difficult to appeal to second-order LP for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made. (shrink)

We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory (...) and second-order logic are not radically different: the latter is a major fragment of the former. (shrink)

A second-order preference is a preference over preferences. This paper addresses the role that second-order preferences play in a theory of instrumental rationality. I argue that second-order preferences have no role to play in the prescription or evaluation of actions aimed at ordinary ends. Instead, second-order preferences are relevant to prescribing or evaluating actions only insofar as those actions have a role in changing or maintaining first-order preferences. I establish these claims (...) by examining and rejecting the view that second-order preferences trump first-order preferences. I also examine and reject the view that second-order preferences give additional normative force to an agent’s preferred first-order preferences. I conclude by arguing that second-order preferences should be integrated into an agent’s object-level preference ordering, and by explaining how best to make sense of this integration. (shrink)

It seems natural to think of an unwilling addict as having a pattern of preferences that she does not endorse—preferences that, in some sense, she does not ‘identify’ with. Following Frankfurt (1971), Jeffrey (1974) proposed a way of modeling those features of an agent’s preferences by appealing to preferences among preferences.Th„e addict’s preferences are preferences she does not prefer to have. I argue that this modeling suggestion will not do, for it follows from plausible assumptions that a minimally rational agent (...) must prefer those first-order preferences she actually has. I close by considering two different but related ways to think about the initial phenomenon. (shrink)

We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be (...) in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenon we call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory. (shrink)

This paper investigates whether there is an acceptable version of Functionalism that avoids commitment to second-order properties. I argue that the answer is "no". I consider two reductionist versions of Functionalism, and argue that both are compatible with multiple realization as such. There is a more specific type of multiple realization that poses difficulties for these views, however. The only apparent Functionalist solution is to accept second-order properties.

Pure second-order logic is second-order logic without functional or first-order variables. In "Pure Second-Order Logic," Denyer shows that pure second-order logic is compact and that its notion of logical truth is decidable. However, his argument does not extend to pure second-order logic with second-order identity. We give a more general argument, based on elimination of quantifiers, which shows that any formula of pure second-order logic with (...) second-order identity is equivalent to a member of a circumscribed class of formulas. As a corollary, pure second-order logic with second-order identity is compact, its notion of logical truth is decidable, and it satisfies a pure second-order analogue of model completeness. We end by mentioning an extension to n th-order pure logics. (shrink)

Clarke and Beck argue that the ANS doesn't represent non-numerical magnitudes because of its second-order character. A sensory integration mechanism can explain this character as well, provided the dumbbell studies involve interference from systems that segment by objects such as the Object Tracking System. Although currently equal hypotheses, I point to several ways the two can be distinguished.

There is a widespread view that in order to be rational we must mostly know what we believe. In the probabilistic tradition this is defended by arguments that a person who failed to have this knowledge would be vulnerable to sure loss, or probabilistically incoherent. I argue that even gross failure to know one's own beliefs need not expose one to sure loss, and does not if we follow a generalization of the standard bridge principle between first-order and (...) second-order beliefs. This makes it possible for a subject to use probabilistic decision theory to manage in a rational way cases of potential failure of this self-knowledge, as we find in implicit bias. Through such cases I argue that it is possible for uncertainty about what our beliefs are to be not only rationally permissible but advantageous. (shrink)

Context: The term “second-order cybernetics” was introduced by von Foerster in 1974 as the “cybernetics of observing systems,” both the act of observing systems and systems that observe. Since then, the term has been used by many authors in articles and books and has been the subject of many conference panels and symposia. Problem: The term is still not widely known outside the fields of cybernetics and systems science and the importance and implications of the work associated with (...) second-order cybernetics is not yet widely discussed. I claim that the transition from cybernetics to second-order cybernetics is a fundamental scientific revolution that is not restricted to cybernetics or systems science. Second-order cybernetics can be regarded as a scientific revolution for the general methodology of science and for many disciplines as well. Method: I first review the history of cybernetics and second-order cybernetics. Then I analyze the major contents of von Foerster’s fundamental revolution in science and present it as a general model for an alternative methodology of science. Subsequently, I present an example of practicing second-order socio-cybernetics from within. I describe some consequences of doing science from within, and I suggest some new horizons for second-order cybernetics. Results: Second-order cybernetics leads to a new foundation for conducting science and offers important contributions for a new way of organizing science. It expands the conception of science so that it can more adequately deal with living systems. Implications: Second-order cybernetics extends the traditional scientific approach by bringing scientists within the domain of what is described and analyzed. It provides models of research processes for when the scientist is within the system being studied. In this way it offers a new foundation for research in the social sciences, in management science, and in other fields such as the environmental sciences or the life sciences. Keywords: Epistemology, general scientific methodology, cybernetics, social sciences, action research, Heinz von Foerster. (shrink)

We introduce a second-order system V1-Horn of bounded arithmetic formalizing polynomial-time reasoning, based on Grädel's 35) second-order Horn characterization of P. Our system has comprehension over P predicates , and only finitely many function symbols. Other systems of polynomial-time reasoning either allow induction on NP predicates , and hence are more powerful than our system , or use Cobham's theorem to introduce function symbols for all polynomial-time functions . We prove that our system is equivalent to (...) QPV and Zambella's P-def. Using our techniques, we also show that V1-Horn is finitely axiomatizable, and, as a corollary, that the class of Σ1b consequences of S21 is finitely axiomatizable as well, thus answering an open question. (shrink)

Context: The problems that are most in need of interdisciplinary collaboration are “wicked problems,” such as food crises, climate change mitigation, and sustainable development, with many relevant aspects, disagreement on what the problem is, and contradicting solutions. Such complex problems both require and challenge interdisciplinarity. Problem: The conventional methods of interdisciplinary research fall short in the case of wicked problems because they remain first-order science. Our aim is to present workable methods and research designs for doing second-order (...) science in domains where there are many different scientific knowledges on any complex problem. Method: We synthesize and elaborate a framework for second-order science in interdisciplinary research based on a number of earlier publications, experiences from large interdisciplinary research projects, and a perspectivist theory of science. Results: The second-order polyocular framework for interdisciplinary research is characterized by five principles. Second-order science of interdisciplinary research must: 1. draw on the observations of first-order perspectives, 2. address a shared dynamical object, 3. establish a shared problem, 4. rely on first-order perspectives to see themselves as perspectives, and 5. be based on other rules than first-order research. Implications: The perspectivist insights of second-order science provide a new way of understanding interdisciplinary research that leads to new polyocular methods and research designs. It also points to more reflexive ways of dealing with scientific expertise in democratic processes. The main challenge is that this is a paradigmatic shift, which demands that the involved disciplines, at least to some degree, subscribe to a perspectivist view. Constructivist content: Our perspectivist approach to science is based on the second-order cybernetics and systems theories of von Foerster, Maruyama, Maturana & Varela, and Luhmann, coupled with embodied theories of cognition and semiotics as a general theory of meaning from von Uexküll and Peirce. (shrink)

The traditional sciences have always had trouble with ambiguity. To overcome this barrier, ‘science’ has imposed “enabling constraints”—hidden assumptions which are given the status of ceteris paribus. Such assumptions allow ambiguity to be bracketed away at the expense of transparency. These enabling constraints take the form of uncritically examined presuppositions, which we refer to throughout the article as “uceps.” The meanings of the various uceps are shown via their applicability to the science of climate change. Second order science (...) examines variations in values assumed for these uceps and looks at the resulting impacts on related scientific claims. Second order science reveals hidden issues, problems and assumptions which all too often escape the attention of the practicing scientist. This article lays out initial foundations for second order science, its ontology, methodology, and implications. (shrink)

Context: Philosophy of science is the branch of philosophy that deals with methods, foundations, and implications of science. It is a theory of how to create scientific knowledge. Presently, there is widespread agreement on how to do science, namely conjectures, ideally in the form of a mathematical model, and refutations, testing the model using empirical evidence. Problem: Many social scientists are using a conception of science created for the physical sciences. Expanding philosophy of science so that it more successfully encompasses (...) social systems would create new avenues of inquiry. Two dimensions could be added to philosophy of science: the amount of attention paid to the observer and the amount of impact of a theory on the system described. Method: My approach is to illuminate underlying assumptions. I claim that there are at least three epistemologies and that they can be combined to form a more robust conception of knowledge and of how to do research. There are at least four models and four basic elements being used by scientists. Results: The article identifies the logical propositions underlying second-order science. It suggests strategies for developing second-order science. And it describes several methods that can be used to practice second-order science, including how past theories have not only described but also changed the phenomenon being studied. Implications: The task for members of the scientific community, particularly social scientists, is to practice second-order science and to develop further its theories and methods. A practical implication is to accept methods for acting as well as theories as a contribution to science, since methods explicitly define the role of an observer/ participant. Constructivist content: The paper is an extension of the work of Heinz von Foerster and other second-order cyberneticians. (shrink)

By the standards presented in the Introduction, CMFC2 is deficient on at least one ontological ground: ‘∀’ is a syncategorematic expression and so CMFC2 is not an ideal language. To some there may be an additional difficulty: any two wffs provably equivalent in the classical sense are provably identical. We hope in sequel to present systems free of these difficulties, free either of one or the other, or perhaps both.This work was done with the aid of Canada Council Grant S74-0551-S1.

Context: Many recent research areas such as human cognition and quantum physics call the observer-independence of traditional science into question. Also, there is a growing need for self-reflexivity in science, i.e., a science that reflects on its own outcomes and products. Problem: We introduce the concept of second-order science that is based on the operation of re-entry. Our goal is to provide an overview of this largely unexplored science domain and of potential approaches in second-order fields. (...) Method: We provide the necessary conceptual groundwork for explorations in second-order science, in which we discuss the differences between first- and second-order science and where we present a roadmap for second-order science. The article operates mainly with conceptual differentiations such as the separation between three seemingly identical concepts such as Science II, Science 2.0 and second-order science. Results: Compared with first-order science, the potential of second-order science lies in 1. higher levels of novelty and innovations, 2. higher levels of robustness and 3. wider integration as well as higher generality. As first-order science advances, second-order science, with re-entry as its basic operation, provides three vital functions for first-order science, namely a rich source of novelty and innovation, the necessary quality control and greater integration and generality. Implications: Second-order science should be viewed as a major expansion of traditional scientific fields and as a scientific breakthrough towards a new wave of innovative research. Constructivist content: Second-order science has strong ties with radical constructivism, which can be qualified as the most important root/origin of second-order science. Moreover, it will be argued that a new form of cybernetics is needed to cope with the new problems and challenges of second-order science. (shrink)

We study elementary second order extensions of the theoryID 1 of non-iterated inductive definitions and the theoryPA Ω of Peano arithmetic with ordinals. We determine the exact proof-theoretic strength of those extensions and their natural subsystems, and we relate them to subsystems of analysis with arithmetic comprehension plusΠ 1 1 comprehension and bar induction without set parameters.

Taking the example of the Basel carnival Fasnacht, the paper shows in what way ritual can maintain the impression of being traditional and unchanging, and yet be open to changes and innovations. As the basic conceptual framework we use the notion of liminality, which Victor Turner identified as the creative moment of ritual. In Fasnacht, this liminal dimension appears in two degrees. The carnival as such represents a reflexive liminal counterpart to the standard social structure, yet it is itself also (...) highly structured and bound by conservative traditional rules. As a reaction to this, there arises within Fasnacht a second-order liminality in the form of the so-called wild Fasnacht, which turns the official antistructural Fasnacht rules upside down once again, testing their validity. While the carnival structure offers a regulated opportunity for questioning everyday social behaviour, wild Fasnacht questions even this regulated manner of questioning as such, opening space for its reflection and modification. We use detailed examples to demonstrate how the subtle dialectics of structure and antistructure keeps the tradition of Basel’s Fasnacht alive and allows it to evolve without destroying its aura of traditionality. (shrink)

We prove that every monadic second-order property of the unfolding of a transition system is a monadic second-order property of the system itself. An unfolding is an instance of the general notion of graph covering. We consider two more instances of this notion. A similar result is possible for one of them but not for the other.

The algebraic structure of the 8-spinor formalism is discussed, and the general form of the 8-component wave equation, equivalent to the second-order 4-component one, is presented. This allows a canonical formulation that will be the first stage of the future Clebsch parametrization, i.e., a relativistic generalization of the Bohm-Schiller-Tiomno pioneering work on the Pauli equation.

Second or higher order probabilities have commonly been viewed with scepticism by those working within the realm of probability and decision theory. The aim of the present note is to show how the notion of second order probabilities can add to our understanding of judgmental and decision processes and how the traditional framework of Bayesian decision theory can be extended in a fruitful way by taking such entities into account. Section one consists of a brief account (...) of arguments put forth against higher order probabilities as well as of counterarguments. In order to provide an example of the applicability of second order probabilities a decision theory encompassing such probabilities will be presented in section two. In section three I will try to emphasize the value of second order probabilities for a deeper and more complete understanding of the notion of risk. (shrink)

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second (...) order number theory is forcible from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

Second-order quantifier elimination in the context of classical logic emerged as a powerful technique in many applications, including the correspondence theory, relational databases, deductive and knowledge databases, knowledge representation, commonsense reasoning and approximate reasoning. In the current paper we first generalize the result of Nonnengart and Szałas [17] by allowing second-order variables to appear within higher-order contexts. Then we focus on a semantical analysis of conditionals, using the introduced technique and Gabbay’s semantics provided in [10] (...) and substantially using a third-order accessibility relation. The analysis is done via finding correspondences between axioms involving conditionals and properties of the underlying third-order relation. (shrink)

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\'e to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to (...) Weak K\"onig's Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others. (shrink)