Natural recursion in syntax is recursion by linguistic value, which is not syntactic in nature but semantic. Syntax-specific recursion is not recursion by name as the term is understood in theoretical computer science. Recursion by name is probably not natural because of its infinite typeability. Natural recursion, or recursion by value, is not species-specific. Human recursion is not syntax-specific. The values on which it operates are most likely domain-specific, including those for syntax. (...) Syntax seems to require no more (and no less) than the resource management mechanisms of an embedded push-down automaton (EPDA). We can conceive EPDA as a common automata-theoretic substrate for syntax, collaborative planning, i-intentions, and we-intentions. They manifest the same kind of dependencies. Therefore, syntactic uniqueness arguments for human behavior can be better explained if we conceive automata-constrained recursion as the most unique human capacity for cognitive processes. (shrink)

The recursive aspect of process reliabilism has rarely been examined. The regress puzzle, which illustrates infinite regress arising from the combination of the recursive structure and the no-defeater condition incorporated into it, is a valuable exception. However, this puzzle can be dealt with in the framework of process reliabilism by reconsidering the relationship between the recursion and the no-defeater condition based on the distinction between prima facie and ultima facie justification. Thus, the regress puzzle is not a basis for (...) abandoning process reliabilism. A genuinely intractable problem for recursive reliabilism lies in the gap between the reliability of the entire path to a belief and that of its parts. Confronted with this puzzle, reliabilists can orient themselves toward ‘reliable-as-a-whole reliabilism’ instead of ‘reliable-in-every-part reliabilism’, including recursive reliabilism, which is found to be not well-motivated. (shrink)

Transfinite ordinal numbers enter mathematical practice mainly via the method of definition by transfinite recursion. Outside of axiomatic set theory, there is a significant mathematical tradition in works recasting proofs by transfinite recursion in other terms, mostly with the intention of eliminating the ordinals from the proofs. Leaving aside the different motivations which lead each specific case, we investigate the mathematics of this action of proof transforming and we address the problem of formalising the philosophical notion of elimination (...) which characterises this move. (shrink)

It is quite well-known from Kurt G¨odel’s (1931) ground-breaking Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are representable in sufficiently strong arithmetical theories. It is also known, though perhaps not as well-known as the former one, that some primitive recursive relations are not rudimentary. We present a simple and elementary proof of this fact in the first part of the paper. In the second part, we review some possible (...) notions of representability of functions studied in the literature, and give a new proof of the equivalence of the weak representability with the (strong) representability of functions in sufficiently strong arithmetical theories. (shrink)

At the phenomenal level, consciousness can be described as a singular, unified field of recursive self-awareness, consistently coherent in a particualr way; that of a subject located both spatially and temporally in an egocentrically-extended domain, such that conscious self-awareness is explicitly characterized by I-ness, now-ness and here-ness. The psychological mechanism underwriting this spatiotemporal self-locatedness and its recursive processing style involves an evolutionary elaboration of the basic orientative reference frame which consistently structures ongoing spatiotemporal self-location computations as i-here-now. Cognition computes action-output (...) in the midst of ongoing movement, and consequently requires a constant self-locating spatiotemporal reference frame as basis for these computations. Over time, constant evolutionary pressures for energy efficiency have encouraged both the proliferation of anticipative feedforward processing mechansims, and the elaboration, at the apex of the sensorimotor processing hierarchy, of self-activating, highly attenuated recursively-feedforward circuitry processing the basic orientational schema independent of external action output. As the primary reference frame of active waking cognition, this recursive i-here-now processing generates a zone of subjective self-awareness in terms of which it feels like something to be oneself here and now. This is consciousness. (shrink)

Recursion or self-reference is a key feature of contemporary research and writing in semiotics. The paper begins by focusing on the role of recursion in poststructuralism. It is suggested that much of what passes for recursion in this field is in fact not recursive all the way down. After the paradoxical meaning of radical recursion is adumbrated, topology is employed to provide some examples. The properties of the Moebius strip prove helpful in bringing out the dialectical (...) nature of radical recursion. The Moebius is employed to explore the recursive interplay of terms that are classically regarded as binary opposites: identity and difference, object and subject, continuity and discontinuity, etc. To realize radical recursion in an even more concrete manner, a higher-dimensional counterpart of the Moebius strip is utilized, namely, the Klein bottle. The presentation concludes by enlisting phenomenological philosopher Maurice Merleau-Ponty’s concept of depth to interpret the Klein bottle’s extra dimension. (shrink)

Recent advances in neuroscience lead to a wider realm for philosophy to include the science of the Darwinian-evolved computational brain, our inner world producing organ, a non-recursive super- Turing machine combining 100B synapsing-neuron DNA-computers based on the genetic code. The whole system is a logos machine offering a world map for global context, essential for our intentional grasp of opportunities. We start from the observable contrast between the chaotic universe vs. our orderly inner world, the noumenal cosmos. So far, philosophy (...) has been rehearsing our thoughts, our human-internal world, a grand painting of the outer world, how we comprehend subjectively our experience, worked up by the logos machine, but now we seek a wider horizon, how humans understand the world thanks to Darwinian evolution to adapt in response to the metaphysical gap, the chasm between the human animal and its environment, shaping the organism so it can deal with its variable world. This new horizon embraces global context coded in neural structures that support the noumenal cosmos, our inner mental world, for us as denizens of the outer environment. Kant’s inner and outer senses are fundamental ingredients of scientific philosophy. Several sections devoted to Heidegger, his lizard debunked, but his version of the metaphysical gap & his doctrine of the logos praised. Rorty and others of the behaviorist school discussed also. (shrink)

We discuss some basic issues underlying the natural numbers: induction and recursion. We examine recursive formulations and their use in establishing universal and particular properties.

A considerable literature has emerged around the idea of using ‘personal responsibility’ as an allocation criterion in healthcare distribution, where a person's being suitably responsible for their health needs may justify additional conditions on receiving healthcare, and perhaps even limiting access entirely, sometimes known as ‘responsibilisation’. This discussion focuses most prominently, but not exclusively, on ‘luck egalitarianism’, the view that deviations from equality are justified only by suitably free choices. A superficially separate issue in distributive justice concerns the two–way relationship (...) between health and other social goods: deficits in health typically undermine one's abilities to secure advantage in other areas, which in turn often have further negative effects on health. This paper outlines the degree to which this latter relationship between health and other social goods exacerbates an existing problem for proponents of responsibilisation (the ‘harshness objection’) in ways that standard responses to this objection cannot address. Placing significant conditions on healthcare access because of a person's prior responsibility risks trapping them in, or worsening, negative cycles where poor health and associated lack of opportunity reinforce one another, making further poor yet ultimately responsible choices more likely. It ends by considering three possible solutions to this problem. (shrink)

This article explores the metaphor of Science as provider of sharp images of our environment, using the epistemological framework of Objective Cognitive Constructivism. These sharp images are conveyed by precise scientific hypotheses that, in turn, are encoded by mathematical equations. Furthermore, this article describes how such knowledge is pro-duced by a cyclic and recursive development, perfection and reinforcement process, leading to the emergence of eigen-solutions characterized by the four essential properties of precision, stability, separability and composability. Finally, this article discusses (...) the role played by ontology and metaphysics in the scientific production process, and in which sense the resulting knowledge can be considered objective. (shrink)

Thomas Hurka, in his book Virtue, Vice, and Value, and elsewhere, develops a recursive analysis of higher-order pleasures and pains. The account leads Hurka to some potentially controversial conclusions. For instance, Hurka argues on its basis that some states are both good and evil and also that the view he calls the conditionality view is false. In this paper, I argue that Hurka’s formulation of the recursive account is unusual and inelegant, and that Hurka reaches his conclusions only because of (...) its peculiar aspects. I provide an alternative recursive account which is similar in spirit to Hurka’s but which is theoretically elegant and does not entail those conclusions. (shrink)

By formalizing some classical facts about provably total functions of intuitionistic primitive recursive arithmetic (iPRA), we prove that the set of decidable formulas of iPRA and of iΣ1+ (intuitionistic Σ1-induction in the language of PRA) coincides with the set of its provably ∆1-formulas and coincides with the set of its provably atomic formulas. By the same methods, we shall give another proof of a theorem of Marković and De Jongh: the decidable formulas of HA are its provably ∆1-formulas.

Focusing upon the five books of his early Hermes series, this article argues that Michel Serres furnishes an accomplished, unconventional philosophical account of communication and mediation-a structuralist epistemology designed to comprehend the sciences in their complexity and plurality-that, even decades after its first publication, has significant value for media theory. Two key themes within this pentalogy are highlighted: firstly, its emphasis upon motifs of communication, transport, and circulation, attempting to grasp the scientific field in topological terms, as a kind of (...) networked encyclopaedia; and secondly, its attempt to account for the intricate relationship between the formal and the empirical in all theorization. (shrink)

The component structures of two distinct neuropsychological systems are described. "System-Y" depends upon "system-X" which, on the other hand, can operate independently of system-Y. System-X provides a matrix upon which system-Y must operate, and, system-Y is transformed by the operations of system-X. In addition these neuropsychological structures reverberate in political history and in the cosmos. The most fundamental structure in the soul, in society, and in the cosmos, has the form of a conical spiral. It can be described mathematically as (...) an harmonic system and mythologically in terms of the birth, marriage and death of the divine-king. ;System-Y corresponds to the neocortex. System-X corresponds to the region below the cortex including the limbic brain. Many of the essential structures of system-Y are captured by Plato's image of the helmsman: orientation, locomotion, manual control and dexterity, visual guidance, verbal command, intention, and volition. System-X on the other hand has been systematically neglected by Western culture, beginning with Plato. ;The dissertation builds upon Yakovlev's distinction between "teleokinesis," "ectokinesis" and "endokinesis." "Teleokinesis" refers to goal-directed action in external space, and belongs to system-Y. Ectokinesis and endokinesis are components of system-X. "Endokinesis" refers to movements within the body. "Ectokinesis" refers to the emotions, which are expressions of endokinesis. The mechanics of ectokinesis is that of an harmonic or vibrational system, as is the mechanics of a musical instrument. ;Within a trance the structure of system-Y is temporarily altered, such that system-X enters the foreground of awareness. Because ectokinesis is analogous to music cultures inspired by trance experience understand the universe in terms of music. The structure of ectokinesis in a trance is that of a conical spiral. Plato inherited the mystical traditions of the ancient Near East but replaced the spiral of emotions with a spiral of ideas , by understanding the axis of the cone as the paradigmatic dimension, and the angular rotation of the spiral as the syntagmatic dimension, of language. Finally I explain how the mythology and political structure of theocratic society imitated the neuropsychological structures of trance experience. (shrink)

The abstract basis of modern computation is the formal description of a finite state machine, the Universal Turing Machine, based on manipulation of integers and logic symbols. In this contribution to the discourse on the computer-brain analogy, we discuss the extent to which analog computing, as performed by the mammalian brain, is like and unlike the digital computing of Universal Turing Machines. We begin with ordinary reality being a permanent dialog between continuous and discontinuous worlds. So it is with computing, (...) which can be analog or digital, and is often mixed. The theory behind computers is essentially digital, but efficient simulations of phenomena can be performed by analog devices; indeed, any physical calculation requires implementation in the physical world and is therefore analog to some extent, despite being based on abstract logic and arithmetic. The mammalian brain, comprised of neuronal networks, functions as an analog device and has given rise to artificial neural networks that are implemented as digital algorithms but function as analog models would. Analog constructs compute with the implementation of a variety of feedback and feedforward loops. In contrast, digital algorithms allow the implementation of recursive processes that enable them to generate unparalleled emergent properties. We briefly illustrate how the cortical organization of neurons can integrate signals and make predictions analogically. While we conclude that brains are not digital computers, we speculate on the recent implementation of human writing in the brain as a possible digital path that slowly evolves the brain into a genuine (slow) Turing machine. (shrink)

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: R.S. (...) Cohen et al. (Eds.), Potentiality, Entanglement, and Passion-at-a-Distance: Quantum Mechanical Studies fo Abner Shimony, Vol. 2, Kluwer Academic Publishers, Great Britain, 1997, pp. 177–190]. Unlike some others in the literature, these notions apply not only to certain nice sets, but to general sets in Rn and other appropriate spaces. We consider some motivations for these concepts and the logical relations between them. It has been argued that d.m.z. is especially appropriate for physical applications, and on Rn with the standard measure, it is strictly stronger than r.a. [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382]. Here we show that this is the only implication that holds among our three decidabilities in that setting. Under arbitrary measures, even this implication fails. Yet for intervals of non-zero length, and more generally, convex sets of non-zero measure, the three concepts are equivalent. (shrink)

The objective of this paper is to analyze the broader significance of Frege’s logicist project against the background of Wittgenstein’s philosophy from both Tractatus and Philosophical Investigations. The article draws on two basic observations, namely that Frege’s project aims at saying something that was only implicit in everyday arithmetical practice, as the so-called recursion theorem demonstrates, and that the explicitness involved in logicism does not concern the arithmetical operations themselves, but rather the way they are defined. It thus represents (...) the attempt to make explicit not the rules alone, but rather the rules governing their following, i.e. rules of second-order type. I elaborate on these remarks with short references to Brandom’s refinement of Frege’s expressivist and Wittgenstein’s pragmatist project. (shrink)

We describe a software system for the analysis of defined benefit actuarial plans. The system uses a recursive formulation of the actuarial stochastic processes to implement precise and efficient computations of individual and group cash flows.

Neuroscience has studied deductive reasoning over the last 20 years under the assumption that deductive inferences are not only de jure but also de facto distinct from other forms of inference. The objective of this research is to verify if logically valid deductions leave any cerebral electrical trait that is distinct from the trait left by non-valid deductions. 23 subjects with an average age of 20.35 years were registered with MEG and placed into a two conditions paradigm (100 trials for (...) each condition) which each presented the exact same relational complexity (same variables and content) but had distinct logical complexity. Both conditions show the same electromagnetic components (P3, N4) in the early temporal window (250–525 ms) and P6 in the late temporal window (500–775 ms). The significant activity in both valid and invalid conditions is found in sensors from medial prefrontal regions, probably corresponding to the ACC or to the medial prefrontal cortex. The amplitude and intensity of valid deductions is significantly lower in both temporal windows (p = 0.0003). The reaction time was 54.37% slower in the valid condition. Validity leaves a minimal but measurable hypoactive electrical trait in brain processing. The minor electrical demand is attributable to the recursive and automatable character of valid deductions, suggesting a physical indicator of computational deductive properties. It is hypothesized that all valid deductions are recursive and hypoactive. (shrink)

Abstract: This article presents a model of self-improving AI in which improvement could happen on several levels: hardware, learning, code and goals system, each of which has several sublevels. We demonstrate that despite diminishing returns at each level and some intrinsic difficulties of recursive self-improvement—like the intelligence-measuring problem, testing problem, parent-child problem and halting risks—even non-recursive self-improvement could produce a mild form of superintelligence by combining small optimizations on different levels and the power of learning. Based on this, we analyze (...) how self-improvement could happen on different stages of the development of AI, including the stages at which AI is boxed or hiding in the internet. (shrink)

It's often hypothesized that the structure of mental representation is map-like rather than language-like. The possibility arises as a counterexample to the argument from the best explanation of productivity and systematicity to the language of thought hypothesis—the hypothesis that mental structure is compositional and recursive. In this paper, I argue that the analogy with maps does not undermine the argument, because maps and language have the same kind of compositional and recursive structure.

(English then french abstract) -/- This article, which can be read by non-psychoanalysts, intends to browse in four stages through the issue offered to our thinking : two (odd-numbered) stages analyzing the argument that provides its context, and two (even-numbered) of propositions presenting our views on what could be the content of the analytic discourse in the coming years. After this introduction, a first reading will point by point but informally review the argument of J.-P. Journet by showing that each (...) of its clauses may generate a “bifurcated” comment able to serve or go against the analytic discourse. Hence the interest of the “differential diagnosis” mentioned in our title, which makes glimpse the traps that the homonymy may set to this speech. Then, to prepare a second scan which sticks neither to the analytical doxa, nor to the even authoritative opinions of our tenors and seniors, it will be proposed two attempts at redefinitions (“apophatic” and “recursive”) of what psychoanalysis is, as well as methodological tools operating downstream from these redefinitions to thwart the obstacles of “external” and “internal” homonymy (starting from a syllogism which can make consensus).The third stage will precisely be made of our second scan of the issue, whose elements will be reviewed and analyzed more methodically :“external differential diagnosis” between the analytic discourse and psychology, philosophy, sociology, and modern science, and “internal differential diagnosis” on the entanglement between theoretical advances of psychoanalysts and the repeated survival of fantasmatic elements...Finally, a fourth part will present propositions and perspectives resulting from these analysis (principle of economy as to the source of psychoanalytical theorizations; dialogue with the other fields, but without compromising; specific relations with the discourse of science), all this leading to an invitation, beyond disputes, to renew the content of the analytic discourse on some points... -/- --------------------------------------------------------------------- -/- Cet article, qui se veut lisible aux non-analystes, se propose de parcourir en quatre temps la problématique offerte à notre réflexion : deux temps (de rang impair) d'analyse de l'argument qui en fournit le contexte, et deux (de rang pair) de propositions présentant nos vues sur ce que pourrait être la teneur du discours analytique dans les prochaines années. Après cette introduction, un premier parcours réexaminera point par point mais informellement l'argument de J.-P. Journet en montrant que chacune de ses propositions peut donner lieu à un commentaire “bifide” à même de servir ou de desservir le discours analytique. D'où l'intérêt du “diagnostic différentiel” évoqué dans notre titre, qui fait entrevoir les pièges que l'homonymie peut tendre à ce discours.Puis, pour préparer un second balayage qui ne s'en tienne ni à la doxa analytique, ni aux opinions même autorisées de nos ténors et seniors, il sera proposé deux tentatives de redéfinitions (“apophatique” et “récursive”) de ce qu'est l'analyse, ainsi que des outils méthodologiques fonctionnant en aval de ces redéfinitions pour déjouer les embûches de l'homonymie “externe” et “interne” (à partir d'un syllogisme pouvant faire consensus). Le troisième temps sera fait justement de ce second balayage de la problématique, dont les éléments seront reconsidérés et analysés plus méthodiquement : “diagnostic différentiel externe” entre le discours analytique et les discours psychologique, philosophique, sociologique et celui de la science moderne ; et “diagnostic différentiel interne” portant sur l'intrication entre avancées théoriques des analystes et survivance à répétition d'éléments fantasmatiques...Enfin une quatrième partie exposera propositions et perspectives résultant de ces analyses (principe d'économie quant à la source des théorisations analytiques ; dialogue avec les autres champs, mais sans compromissions ; relations spécifiques avec le discours de la science), l'ensemble débouchant sur une invitation, au delà des différends, à renouveler sur certains points la teneur du discours analytique... (shrink)

This paper sums up the fundamental features of intelligence through the common features stated by various definitions of "intelligence": Intelligence is the ability of achieving systematic goals (functions) of brain and nerve system through selecting, and artificial intelligence or machine intelligence is an imitation of life intelligence or a replication of features and functions. Based on the definition mentioned above, this paper discusses and summarizes the development routes of ideas on computable intelligence, including Godel's "universal recursive function", the computation activities (...) of "selection", recursive function and Turing machine, mathematical expression of computable intelligence, core nature of computable intelligence, and computability and strong artificial intelligence. At the end of this paper is the conclusion drawn by the authors. (shrink)

The paper intends to zoom in and find a uniqueness in human language by narrowing down the range of cognitive domains to human computational mind having a property of recursion which is exclusively unique to human and not in any other species in animalia kingdom.This notion of recursion is the centrality of the paper. There has been an opposition to the notion of recursion being only unique to human and the paper makes an attempt to reply to (...) such arguments using experimental findings from modern neuroscience. The existing controversies over the proposed minimalist language and its future remains open to the future of modern neuroscience and modern physics. (shrink)

This article addresses three questions about well-being. First, is well-being future-sensitive? I.e., can present well-being depend on future events? Second, is well-being recursively dependent? I.e., can present well-being depend on itself? Third, can present and future well-being be interdependent? The third question combines the first two, in the sense that a yes to it is equivalent to yeses to both the first and second. To do justice to the diverse ways we contemplate well-being, I consider our thought and discourse about (...) well-being in three domains: everyday conversation, social science, and philosophy. This article’s main conclusion is that we must answer the third question with no. Present and future well-being cannot be interdependent. The reason, in short, is that a theory of well-being that countenances both future-sensitivity and recursive dependence would have us understand a person’s well-being at a time as so intricately tied to her well-being at other times that it would not make sense to consider her well-being an aspect of her state at particular times. It follows that we must reject either future-sensitivity or recursive dependence. I ultimately suggest, especially in light of arguments based on assumptions of empirical research on well-being, that the balance of reasons favors rejecting future-sensitivity. (shrink)

Corcoran’s 27 entries in the 1999 second edition of Robert Audi’s Cambridge Dictionary of Philosophy [Cambridge: Cambridge UP]. -/- ancestral, axiomatic method, borderline case, categoricity, Church (Alonzo), conditional, convention T, converse (outer and inner), corresponding conditional, degenerate case, domain, De Morgan, ellipsis, laws of thought, limiting case, logical form, logical subject, material adequacy, mathematical analysis, omega, proof by recursion, recursive function theory, scheme, scope, Tarski (Alfred), tautology, universe of discourse. -/- The entire work is available online free at more (...) than one website. Paste the whole URL. http://archive.org/stream/RobertiAudi_The.Cambridge.Dictionary.of.Philosophy/Robert.Audi_The.Cambrid ge.Dictionary.of.Philosophy -/- The 2015 third edition will be available soon. Before you think of buying it read some reviews on Amazon and read reviews of its competition: For example, my review of the 2008 Oxford Companion to Philosophy, History and Philosophy of Logic,29:3,291-292. URL: http://dx.doi.org/10.1080/01445340701300429 -/- Some of the entries have already been found to be flawed. For example, Tarski’s expression ‘materially adequate’ was misinterpreted in at least one article and it was misused in another where ‘materially correct’ should have been used. The discussion provides an opportunity to bring more flaws to light. -/- Acknowledgements: Each of these entries was presented at meetings of The Buffalo Logic Dictionary Project sponsored by The Buffalo Logic Colloquium. The members of the colloquium read drafts before the meetings and were generous with corrections, objections, and suggestions. Usually one 90-minute meeting was devoted to one entry although in some cases, for example, “axiomatic method”, took more than one meeting. Moreover, about half of the entries are rewrites of similarly named entries in the 1995 first edition. Besides the help received from people in Buffalo, help from elsewhere was received by email. We gratefully acknowledge the following: José Miguel Sagüillo, John Zeis, Stewart Shapiro, Davis Plache, Joseph Ernst, Richard Hull, Concha Martinez, Laura Arcila, James Gasser, Barry Smith, Randall Dipert, Stanley Ziewacz, Gerald Rising, Leonard Jacuzzo, George Boger, William Demopolous, David Hitchcock, John Dawson, Daniel Halpern, William Lawvere, John Kearns, Ky Herreid, Nicolas Goodman, William Parry, Charles Lambros, Harvey Friedman, George Weaver, Hughes Leblanc, James Munz, Herbert Bohnert, Robert Tragesser, David Levin, Sriram Nambiar, and others. -/- . (shrink)

Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solution in non-negative integers is not recursive. Craig Smoryński's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. Let R be a subring of Q with or without 1. By H_{10}(R), we denote the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether (...) or not the equation has a solution in R. We prove that a positive solution to H_{10}(R) implies that the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. We show the converse implication for every infinite set R \subseteq Q such that there exist computable functions \tau_1,\tau_2:N \to Z which satisfy (\forall n \in N \tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). This implication for R=N guarantees that Smoryński's theorem follows from Matiyasevich's theorem. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have a rational solution is not recursive. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have only finitely many rational solutions is not recursively enumerable. These conjectures are equivalent by our results for R=Q. (shrink)

Frames, i.e., recursive attribute-value structures, are a general format for the decomposition of lexical concepts. Attributes assign unique values to objects and thus describe functional relations. Concepts can be classified into four groups: sortal, individual, relational and functional concepts. The classification is reflected by different grammatical roles of the corresponding nouns. The paper aims at a cognitively adequate decomposition, particularly, of sortal concepts by means of frames. Using typed feature structures, an explicit formalism for the characterization of cognitive frames is (...) developed. The frame model can be extended to account for typicality effects. Applying the paradigm of object-related neural synchronization, furthermore, a biologically motivated model for the cortical implementation of frames is developed. Cortically distributed synchronization patterns may be regarded as the fingerprints of concepts. (shrink)

Causation is a macroscopic phenomenon. The temporal asymmetry displayed by causation must somehow emerge along with other asymmetric macroscopic phenomena like entropy increase and the arrow of radiation. I shall approach this issue by considering ‘causal inference’ techniques that allow causal relations to be inferred from sets of observed correlations. I shall show that these techniques are best explained by a reduction of causation to structures of equations with probabilistically independent exogenous terms. This exogenous probabilistic independence imposes a recursive order (...) on these equations and a consequent distinction between dependent and independent variables that lines up with the temporal asymmetry of causation. (shrink)

Perceptual and recursion-based faculties have long been recognized to be vital constituents of human (and, in general, animal) cognition. However, certain faculties such as the visual and the linguistic faculty have come to receive far more academic and experimental attention, in recent decades, than other recognized categories of faculties. This paper seeks to highlight the imbalance in these studies and bring into sharper focus the need for further in-depth philosophical treatments of faculties such as especially hearing, touch, and proprioception, (...) besides other faculties such as the olfactory and gustatory ones. It also seeks to bring to bear the debate on the role of qualia in perception and overall in cognition in its thesis of the significance of these other modular faculties for genuine insights into cognition as a (now) technologically expanded domain. (shrink)

This paper articulates and defends a conception of logical form as semantic form revealed by a compositional meaning theory. On this conception, the logical form of a sentence is determined by the semantic types of its primitive terms and their mode of combination as it relates to determining under what conditions it is true. We develop this idea in the framework of truth-theoretic semantics. We argue that the semantic form of a declarative sentence in a language L is revealed by (...) a (canonical) proof of its T- sentence in an interpretive truth theory for L. We give a precise characterization of sameness of logical form between any two declarative sentences in any two languages in terms of the notion of corresponding proofs in interpretive truth theories for the languages. We illustrate the utility of this approach with a number of examples. We then extend the characterization to non-declaratives in a generalization of truth-theoretic semantics that appeals to fulfillment conditions, of which truth conditions are one variety. On this approach, logical forms are not reified, and the notion of sameness of logical form is treated as conceptually basic. We discuss the relation of this conception of logical form to the project of identifying logical constants, reviewing two approaches, one of which takes topic neutrality as central, the other recursion. We argue that the project of identifying logical constants for the purposes of classifying together valid arguments is largely independent of that identifying logical form of sentences, and urge an ecumenical approach to extending talk of logical constants beyond where it is currently well-grounded. (shrink)

A recursive definition of knowledge is proposed, to deal with Gettier style difficulties, and provide a framework for a variety of recursionist theories of knowledge.

The field of the philosophy of biology is flourishing in its aim to evaluate and rethink the view inherited from the previous century ---the Modern Synthesis. Different research areas and theories have come to the fore in the last decades in order to account for different biological phenomena that, in the first instance, fall beyond the explanatory scope of the Modern Synthesis. This thesis is anchored and motivated by this revolt in the philosophy of biology. -/- The central target in (...) this context is the possibility of naturalizing teleology, a classical nightmare for the history of biology itself. This requires, principally, understanding the causes of teleological explanations without assuming an unfashioned backward causation of sorts. As the riddles of teleological explanations are about their temporal dimension, I analyze different temporal scales of biological processes: evolutionary, developmental, and physiological. -/- The first one is the one defended in the context of the Modern Synthesis. As expected, one of the aims of this thesis is to evaluate the adequacy of an evolutionary account of teleological explanation. The scrutiny is negative. Evolutionary explanations in the context of the Modern Synthesis lack the necessary causal roots to naturalize teleology. Concerning the physiological scale, a long tradition pushed up by Kant and the organicist movement in the 20th century allows us to better understand how teleological explanations can be naturalized in physiological process. The key notions in this temporal scale are self-organization and the recursive, looped character of physiological process. While the physiological scale may be suitably accounted by contemporaries views, such as Autonomous Systems Theory, different central teleological phenomena remain unexplained from a purely physiological perspective. In particular, different issues concerning the (adaptive) construction of organism ---such as plasticity, robustness, variation, novelty, inheritance--- deserve an ontogenetic analysis. -/- The principal aim of this thesis is to provide a theory of teleological development that falls beyond the Modern Synthesis' framework and is prompted by different insights from the history of biology. I call it Agential Teleosemantics. It rests on two central pillars. First, that developmental processes, beyond any gene-centered stance, can be understood in informational terms; i.e. developmental processes are about the interaction of developmental resources conveying biological information. The second ingredient is agentivity, namely the idea that development is regulated by an agentive system according to the adaptiveness of the phenotypic outcomes produced. The role of agency in Agential Teleosemantic is equivalent to the role of genes in the Modern Synthesis: it is responsible for explaining the order and the adaptive complexity in the living realm. -/- The second target of this thesis regards the possibility of naturalizing intentional explanations in cognitive science. The central project involved in such an aim is known as teleosemantics. Classical teleosemantics however is etiological: it explains the teleofunctions of representational systems in terms of evolutionary processes. The different disputes in the contemporary philosophy of biology provide two insights to analyze teleosemantics in cognitive science. First, the challenges against the Modern Synthesis must be extended to the evolutionary approach of etiological teleosemantics. Second, as Agential Teleosemantics suggests an alternative source for teleofunctions ---ontogeny, I offer an attempt to integrate Agential Teleosemantics into cognitive science in order to provide an alternative teleosemantic project to understand intentional explanations in cognitive science. (shrink)

I here investigate the sense in which diagonalization allows one to construct sentences that are self-referential. Truly self-referential sentences cannot be constructed in the standard language of arithmetic: There is a simple theory of truth that is intuitively inconsistent but is consistent with Peano arithmetic, as standardly formulated. True self-reference is possible only if we expand the language to include function-symbols for all primitive recursive functions. This language is therefore the natural setting for investigations of self-reference.

In Recursivity and Contingency, Yuk Hui prompts a rigorous historical and philosophical analysis of today’s algorithmic culture. As evidenced by highspeed AI trading, predictive processing algorithms, elastic graph-bunching biometrics, Hebbian machine learning and thermographic drone warfare, we are privy to an epochal technological transition. As these technologies, stilted on inductive learning, demonstrate, we no longer occupy the moment of the ‘storage-and-retrieval’ static database but are increasingly engaged with technologies that are involved in the ‘manipulable arrangement’ (p204) of the indeterminable. It (...) is, in fact, extricating the indeterminable or the Inhuman and its cosmic anti-capitalist imperative that concerns the core of Hui’s project of technodiversity. (shrink)

Until the late 19th century scientists almost always assumed that the world could be described as a rule-based and hence deterministic system or as a set of such systems. The assumption is maintained in many 20th century theories although it has also been doubted because of the breakthrough of statistical theories in thermodynamics (Boltzmann and Gibbs) and other fields, unsolved questions in quantum mechanics as well as several theories forwarded within the social sciences. Until recently it has furthermore been assumed (...) that a rule-based and deterministic system was also predictable if only the rules were known, but this assumption has now been undermined by modern chaos-theory describing rule-based and deterministic, but unpredictable systems, while catastrophe-theory delivers a set of types describing various kinds of instability and conditions for the stability of a given system. Hence the main trait in the theoretical development in the 20th-century science can be described as a basic modification and limitation of some of the fundamental and strong assumptions forwarded in the previous epochs of modern science. Ironically, the very same process has been a process in which the human capacity to intervene in nature has expanded dramatically and mainly with the help of the very same theories, and not least because they allow nature to be described and made manipulable on a lower level and a more fine-grained scale. While the overall theoretical consistency between the various theories has gone, the reach of human intervention in nature has increased based on quite new dimensions whether in the area of physics (e.g.: energy technologies, chemical technologies, nanotechnologies etc.) or biology (ge¬netic manipulation) or in the area of psychology, sociology and culture (artificial simulations of mental processes, new means of communication, implying changes in the social infrastructure and cultural behaviour etc.). While some of these changes and new conditions can be reflected from within the conceptual framework of rule-based systems, albeit more complex than formerly recognized, others seem to give rise to the question of whether there are »systems« and relations between different systems in the world which are not rule-based? For instance, it seems to be obvious that the notion of instability represents a major conceptual break with former theories of rule-based systems, as the stability of the latter is an axiomatically given property implied in the very notion of rule-based systems, while instability can only be the result of external influence which should be explained as the result of another rule-based sy¬stem. While there are no difficulties implied concerning the stability of rule-based systems, the notion of unstable states of a system raises the question of how there can be a system at all if there are no invariant stabilising principles? This is the first question which I will address. And I shall do so by taking two examples of such systems as my point of departure. The first example will be the computer and the second will be ordinary language. In both cases I will argue that the stability of these systems (which are both defined by the existence/presence of human intentions) are provided by the help of - differently organised - redundancy functions which both allow the maintenance of systems in unstable macro-states, suspension of previous rules, underdetermination and overdetermination and generation/emergence/creation of new rules more or less independent of previous rules by the help of optional recursions to the permanently accessible underlying levels as for instance the level of binary representation in computers. Since the notion of redundancy is both controversial as such and often avoided, the concept is discussed (as defined in Claude Shannon's mathematical theory of information and in the semiotic framework of J. J. Greimas) and leading to a more general definition in which the redundancy functions serve to overcome noisy conditions, but at the cost of rule-based stability, determination, and predictability. A second question will be how the notion of rule-generating systems relates to the notion of anticipatory systems and it will be argued that rule-generating systems share some features with an¬ticipatory systems and that the former from a certain viewpoint can be seen as a subclass of the latter, although anticipative features are not necessarily a part of the definition of rule-generating systems. On the other hand, it will be discussed whether anticipatory systems which are not rule-generating systems can exist and it will be argued that the capacity to anticipate is strongly limited if it is not part of a rule-generating system. Therefore, it is concluded that the most powerful anticipatory systems need to be rule-generating systems. (shrink)

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That (...) series rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. (shrink)

Simple taste predications come with an acquaintance requirement: they require the speaker to have had a certain kind of first-hand experience with the object of predication. For example, if I tell you that the creme caramel is delicious, you would ordinarily assume that I have actually tasted the creme caramel and am not simply relying on the testimony of others. The present essay argues in favor of a 'lightweight' expressivist account of the acquaintance requirement. This account consists of a recursive (...) semantics and an account of assertion; it is compatible with a number of different accounts of truth and content, including contextualism, relativism, and purer forms of expressivism. The principal argument in favor of this account is that it correctly predicts a wide range of data concerning how the acquaintance requirement interacts with Boolean connectives, generalized quantifiers, epistemic modals, and attitude verbs. (shrink)

This chapter discusses different perspectives and trends in social decision making, especially the actual processes used by humans when they make decisions in their everyday lives or in business situations. The chapter uses cognitive psychological techniques to break down these processes and set them in their social context. Most of our decisions are made in a social context and are therefore influenced by other people. If you are at an auction and bidding on a popular item, you will try to (...) guess how long the others will keep in the bidding race. Likewise, when you and another person have to coordinate your decision to win a card game you will also try to think about what the other person know and what s/he will do, so you can adjust your choice. In other words, you are trying to come up with a model of the person and his decision-making process. This involves the capability to think about the thoughts and intentions of others, to put yourself in another person's shoes (perspective taking), and to think recursively about your and another person's over several steps (like in a game of chess). Especially, recursive social reasoning is one of the hallmarks of social decision-making. Whereas observational, competitive and cooperative decision-making can all be seen as trying to maximize the own reward or profit, altruistic decision-making blatantly violates this axiomatic principle of behavioral economics. Yet, altruistic behavior is quite common throughout human society. -/- . (shrink)

One theoretical charge (of Optimality Theory in its early conception) must have been to retain that sense of qualitative particularity as affecting as constraining theory relevant to a proscribed field when clearly a motivation was to divine in circumscriptions operational consequences conceived on a deferred abstractive level. An attraction of the theory's embodying results of constraint interactions as responsive to theory-internal qualitative implementation, as being in fact supplementarily transparent to co-ordinations of variously language specific implementations, qualitative identifications, was apparent naturalistic (...) coordination of defined features taken here—in OT—at one fell-swoop. Differently put, the uses of analyses of the relevant types of spaces (‘primitive from the viewpoint of linguistic theories’) will (latterly) involve a period of time-stretch or compression over a previously normal (I take it) process of embeddings in a new theorised and recursively enacted invocation/re-invocation of base as digestively present in results of corrected theories. Or as Prince and Smolensky have it -/- 'What we have done, in essence, is to replace the iterative procedure (act/evaluate, act/evaluate, . . . ) with a recursive scheme: collect the results of all possible actions, then sort recursively.' OT, 16. (shrink)

Take a strip of paper with 'once upon a time there'‚ written on one side and 'was a story that began'‚ on the other. Twisting the paper and joining the ends produces John Barth’s story Frame-Tale, which prefixes 'once upon a time there was a story that began'‚ to itself. I argue that the ability to understand this sentence cannot be explained by tacit knowledge of a recursive theory of truth in English.

This essay aims to vindicate the thesis that cognitive computational properties are abstract objects implemented in physical systems. I avail of Voevodsky's Univalence Axiom and function type equivalence in Homotopy Type Theory, in order to specify an abstraction principle for epistemic (hyper-)intensions. The homotopic abstraction principle for epistemic (hyper-)intensions provides an epistemic conduit for our knowledge of (hyper-)intensions as abstract objects. Higher observational type theory might be one way to make first-order abstraction principles defined via inference rules, although not higher-order (...) abstraction principles, computable. The truth of my first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable i.e. Turing computable and the Turing machine being physically implementable. Epistemic modality and hyperintensionality can thus be shown to be both a compelling and a materially adequate candidate for the fundamental structure of mental representational states, comprising a fragment of the language of thought. (shrink)

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural (...) numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals. (shrink)

In this paper, Nāgārjuna’s philosophical interpretation of the terms kāraṇa and kārya is analysed after having methodologically confined the specific field of interest to the MMK. From the study of all the occurrences of kāraṇa and kārya in the MMK (listed in paragraph 2), it emerges that Nāgārjuna makes use of these two terms to refer to skandhas as causes (kāraṇa) of further skandhas as effects (kārya), hence conveying with this words the idea of, so to speak, subjectivity and (re)birth. (...) From the principal commentaries on the MMK (particularly Bhāviveka and Candrakīrti), we know also that, even when the reference of kāraṇa and kārya to skandhas is not explicitly pointed out by Nāgārjuna, it nonetheless can be inferred from the peculiar context in which these terms are employed. This conclusion seems to be confirmed also by the crosscheck analysis of the philosophical usage of the parallel terms hetu and phala, which refer either to, so to speak, objective (i.e., concerning general bhāvas) causes and effects, or – in the case of phala as consequence of karman/kriyā – to moral results (these two meanings are confirmed also by some passages from the Vigrahavyāvartanī), but in no cases Nāgārjuna makes recurse to hetu and phala to refer to what could be called a subjective (i.e., concerning skandhas) level of causality. To the paper an addendum is added, in which the commentaries on MMK 8.4 are taken into consideration. From this excursus it appears that at least Buddhapālita, Bhāviveka and Candrakīrti, by ‘overinterpreting’ the meaning of hetu in the kārika, are consequently forced to readapt the philosophical significance of kāraṇa and kārya in a way that does not fit so much with Nāgārjuna’s original message. (shrink)

The evolution from pre-human primates to modern Homo sapiens is a complex one involving many domains, ranging from the material to the social to the cognitive, both at the individual and the community levels. This article focuses on a critical qualitative transition that took place during this evolution involving both the social and the cognitive domains. For the social domain, the transition is from the face-to-face forms of social interaction and organization that characterize the non-human primates that reached, with Pan, (...) a hiatus due to the centripetal effects that highly individualized behavior has on a social system. The transition is to the relation-based forms of social organization that evolved in the hominins ancestral to Homo sapiens and are universal in human societies today. For the cognitive domain, this transition involves going from behavior responding mainly to phenomenal level sensory inputs to behavior formed in accordance with the concept of a relation, initially abstracted from behavior patterns, then extending the concept of a relation beyond abstraction from behavior patterns to the concept of a relation generated recursively through constructing the relation of a relation. This extension made possible the construction of systems of relations; initially genealogical systems of relations constructed culturally using the logic of recursion, and subsequently, the symbolic, computational systems of kin term relations referred to by anthropologists as kinship terminologies. The latter are “constructed realities” in the sense this term is used by cultural anthropologists. It follows that the evolution of relation-based systems of social interaction is not adequately accounted for through population model evolutionary accounts such as the Dual Inheritance Theory of human evolution since “constructed realities” constitute collectively and publicly shared cultural knowledge rather than the individually and privately possessed knowledge that is assumed in the population model framework for human evolution. (shrink)

SNePS, the Semantic Network Processing System 45, 54], has been designed to be a system for representing the beliefs of a natural-language-using intelligent system (a \cognitive agent"). It has always been the intention that a SNePS-based \knowledge base" would ultimatelybe built, not by a programmeror knowledge engineer entering representations of knowledge in some formallanguage or data entry system, but by a human informing it using a natural language (NL) (generally supposed to be English), or by the system reading books or (...) articles that had been prepared for human readers. Because of this motivation, the criteria for the development of SNePS have included: it should be able to represent anything and everything expressible in NL; it should be able to represent generic, as well as speci c information; it should be able to use the generic and the speci c information to reason and infer information implied by what it has been told; it cannot count on any particular order among the pieces of information it is given; it must continue to act reasonably even if the information it is given includes circular de nitions, recursive rules, and inconsistent information. (shrink)

What is history about? This Element shows that answers centred on the keyword 'past events' are incomplete, even if they are not simply wrong. Interweaving theoretical and historical perspectives, it provides an abstract overview of the thematic plurality that characterizes contemporary academic historiography. The reflection on different sorts of pasts that can be at focus in historical research and writing encompasses events as well as non-events, especially recursive social structures and cultural webs. Some consequences of such plurality for discussions concerning (...) historical methodology, explanation, exemplification, and representation are also outlined. The basic message, reinforced throughout, is that the great relevance of non-event-centred approaches should prompt us to talk more about “histories” in the plural and less about “history” in the singular. (shrink)