A novel theory of reflective consciousness, will and self is presented, based on modeling each of these entities using self-referential mathematical structures called hypersets. Pattern theory is used to argue that these exotic mathematical structures may meaningfully be considered as parts of the minds of physical systems, even finite computational systems. The hyperset models presented are hypothesized to occur as patterns within the "moving bubble of attention" of the human brain and any roughly human-mind-like AI system. These ideas appear (...) to be compatible with both panpsychist and materialist views of consciousness, and probably other views as well. Their relationship with the CogPrime AI design and its implementation in the OpenCog software framework is elucidated in detail. (shrink)

The ecological approach to perception-action is unlike the standard approach in several respects. It takes the animal-in-its-environment as the proper scale for the theory and analysis of perception-action, it eschews symbol based accounts of perception-action, it promotes self-organization as the theory-constitutive metaphor for perception-action, and it employs self-referring, non-predicative definitions in explaining perception-action. The present article details the complexity issues confronted by the ecological approach in terms suggested by Rosen and introduces non-well-founded set theory as a potentially useful tool for (...) expressing them. The issues and the tool are brought to focus in the concept of affordance that is the basis for explanation of prospective control of action in the ecological approach. (shrink)

Knowledge modal formulas are interpreted by a universal modal world in the hypersets universe [A]. This remedies the limitation of the interpretation of knowledge formulas by a tower of modal worlds in the well founded universe [F], where each world can interpret only a portion of knowledge modal formulas.

Using hypersets as an analytic tool, we compare traditionally Gibsonian (Chemero 2003; Turvey 1992) and representationalist (Sahin et al. this issue) understandings of the notion ‘affordance’. We show that representationalist understandings are incompatible with direct perception and erect barriers between animal and environment. They are, therefore, scarcely recognizable as understandings of ‘affordance’. In contrast, Gibsonian understandings are shown to treat animal-environment systems as unified complex systems and to be compatible with direct perception. We discuss the fruitful connections between Gibsonian (...) affordances and dynamical systems explanation in the behavioral sciences and point to prior fruitful application of Gibsonian affordances in robotics. We conclude that it is unnecessary to re-imagine affordances as representations in order to make them useful for researchers in robotics. (shrink)

Aczel's theory of hypersets provides an interesting alternative to the standard view of sets as inductively constructed, well-founded objects, thus providing a convienent formalism in which to consider non-well-founded versions of classically well-founded constructions, such as the "circular logic" of [3]. This theory and ZFC are mutually interpretable; in particular, any model of ZFC has a canonical "extension" to a non-well-founded universe. The construction of this model does not immediately generalize to weaker set theories such as the theory of (...) admissible sets. In this paper, we formulate a version of Aczel's antifoundation axiom suitable for the theory of admissible sets. We investigate the properties of models of the axiom system KPU - , that is, KPU with foundation replaced by an appropriate strengthening of the extensionality axiom. Finally, we forge connections between "non-wellfounded sets over the admissible set A" and the fragment L A of the modal language L ∞. (shrink)

Bringing together powerful new tools from set theory and the philosophy of language, this book proposes a solution to one of the few unresolved paradoxes from antiquity, the Paradox of the Liar. Treating truth as a property of propositions, not sentences, the authors model two distinct conceptions of propositions: one based on the standard notion used by Bertrand Russell, among others, and the other based on J.L. Austin's work on truth. Comparing these two accounts, the authors show that while the (...) Russellian conception of the relation between sentences, propositions, and truth is crucially flawed in limiting cases, the Austinian perspective has fruitful applications to the analysis of semantic paradox. In the course of their study of a language admitting circular reference and containing its own truth predicate, Barwise and Etchemendy also develop a wide range of model-theoretic techniques--based on a new set-theoretic tool, Peter Aczel's theory of hypersets--that open up new avenues in logical and formal semantics. (shrink)

The three classic regress problems (the Extensive Regress of states, the Intensive Regress of contents, and the Fichte-Henrich-Shoemaker Regress of de se beliefs) related to the Self-Awareness Thesis (that one’s conscious states are the ones that one is aware of being in) can all be elegantly resolved by a self-acquaintance postulate. This resolution, however, entails that consciousness has an irreducibly circular structure and that self-acquaintance should not be conceived of in terms of an independent entity bearing an external or mediated (...) relation to itself but rather in terms of a realized relation-instance relating to itself as well as to something other than itself. Consciousness, on this account, has a categorially curious status. It is like a relation-particular hybrid. This can be formalized in terms of the theory of hypersets, which in turn can be used to elucidate the problem of individuality, one source of the conceptual difficulty with adequately characterizing de se content. (shrink)

This paper investigates an alternative set theory (due to Peter Aczel) called Hyperset Theory. Aczel uses a graphical representation for sets and thereby allows the representation of non-well-founded sets. A program, called HYPERSOLVER, which can solve systems of equations defined in terms of sets in the universe of this new theory is presented. This may be a useful tool for commonsense reasoning.

We approach the virtual reality phenomenon by studying its relationship to set theory. This approach offers a characterization of virtual reality in set theoretic terms, and we investigate the case where this is done using the wellfoundedness property. Our hypothesis is that non-wellfounded sets (so-called hypersets) give rise to a different quality of virtual reality than do familiar wellfounded sets. To elaborate this hypothesis, we describe virtual reality through Sommerhoff’s categories of first- and second-order self-awareness; introduced as necessary conditions (...) for consciousness in terms of higher cognitive functions. We then propose a representation of first- and second-order self-awareness through sets, and assume that these sets, which we call events, originally form a collection of wellfounded sets. Strong virtual reality characterizes virtual reality environments which have the limited capacity to create only events associated with wellfounded sets. In contrast, the logically weaker and more general concept of weak virtual reality characterizes collections of virtual reality mediated events altogether forming an entirety larger than any collection of wellfounded sets. By giving reference to Aczel’s hyperset theory we indicate that this definition is not empty because hypersets encompass wellfounded sets already. Moreover, we argue that weak virtual reality could be realized in human history through continued progress in computer technology. Finally, within a more general framework, we use Baltag’s structural theory of sets (STS) to show that within this hyperset theory Sommerhoff’s first- and second-order self-awareness as well as both concepts of virtual reality admit a consistent mathematical representation. To illustrate our ideas, several examples and heuristic arguments are discussed. (shrink)

This paper has two main purposes. First, it will provide an introductory discussion of hyperset theory, and show that it is useful for modeling complex systems. Second, it will use hyperset theory to analyze Robert Rosen’s metabolismrepair systems and his claim that living things are closed to efficient cause. It will also briefly compare closure to efficient cause to two other understandings of autonomy, operational closure and catalytic closure.

This paper has two primary aims. The first is to provide an introductory discussion of hyperset theory and its usefulness for modeling complex systems. The second aim is to provide a hyperset analysis of Robert Rosen’s metabolism-repair systems and his claim that living things are closed to efficient cause. Consequences of the hyperset models for Rosen’s claims concerning computability and life are discussed.