According to the traditional bundle theory, particulars are bundles of compresent universals. I think we should reject the bundle theory for a variety of reasons. But I will argue for the thesis at the core of the bundle theory: that all the facts about particulars are grounded in facts about universals. I begin by showing how to meet the main objection to this thesis (which is also the main objection to the bundle theory): that it is inconsistent with the possibility (...) of distinct qualitative indiscernibles. Here, the key idea appeals to a non-standard theory of haecceities as non-well-founded properties of a certain sort. I will then defend this theory from a number of objections, and finally argue that we should accept it on the basis of considerations of parsimony about the fundamental. (shrink)

Public announcement logic is an extension of multiagent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: after which , does it hold that Kφ? We give various semantic results and show completeness for a Hilbert-style axiomatization of this logic. There is a natural generalization to a logic for arbitrary events.

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, I argue (...) that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

Information is a notion of wide use and great intuitive appeal, and hence, not surprisingly, different formal paradigms claim part of it, from Shannon channel theory to Kolmogorov complexity. Information is also a widely used term in logic, but a similar diversity repeats itself: there are several competing logical accounts of this notion, ranging from semantic to syntactic. In this chapter, we will discuss three major logical accounts of information.

Are there ‘self-referential’ propositions? That is, propositions that say of themselves that they have a certain property, such as that of being false. There can seem reason to doubt that there are. At the same time, there are a number of reasons why it matters. For suppose that there are indeed no such propositions. One might then hope that while paradoxes such as the Liar show that many plausible principles about sentences must be given up, no such fate will befall (...) principles about propositions. But the existence of self-referential propositions would dash such hopes. Further, the existence of such propositions would also seem to challenge the widespread claim that Liar sentences fail to express propositions. The aim of this paper is thus to settle the question–at least given an assumption. In particular, I argue that if propositions are structured, then self-referential propositions exist. (shrink)

My aim in this paper is to propose what seems to me a distinctive approach to set theoretic methodology. By ‘methodology’ I mean the study of the actual methods used by practitioners, the study of how these methods might be justified or reformed or extended. So, for example, when the intuitionist's philosophical analysis recommends a wholesale revision of the methods of proof used in classical mathematics, this is a piece of reformist methodology. In contrast with the intuitionist, I will focus (...) more narrowly on the methods of contemporary set theory, and, more importantly, I will certainly recommend no sweeping reforms. Rather, I begin from the assumption that the methodologist's job is to account for set theory as it is practiced, not as some philosophy would have it be. This credo lies at the very heart of the so-called ‘naturalism’ to be described here.A philosopher looking at set theoretic practice from the outside, so to speak, might notice any number of interesting methodological questions, beginning with the intuitionist's ‘why use classical logic?’, but this sort of question is not a live issue for most practicing set theorists. One central question on which the philosopher's and the practitioner's interests converge is this: what is the status of independent statements like the continuum hypothesis (CH)? A number of large questions arise in its wake: what criteria should guide the search for new axioms? For that matter, what reasons support our adoption of the old axioms? (shrink)

According to traditional theism, God alone exists a se, independent of all other things, and all other things exist ab alio, i.e., God both creates them and sustains them in existence. On the face of it, divine "aseity" is inconsistent with classical Platonism, i.e., the view that there are objectively existing, abstract objects. For according to the classical Platonist, at least some abstract entities are wholly uncreated, necessary beings and, hence, as such, they also exist a se. The thesis of (...) theistic activism purports to reconcile divine aseity with a robust Platonism. Specifically, the activist holds that God creates the abstract objects no less than the contingent concrete objects of the physical universe and hence that, like all created things, they exist ab alio after all, their necessity notwithstanding. But many philosophers believe a severe roadblock for activism remains — a problem known as the bootstrapping objection. Despite widespread faith in the deliverances of this argument, in this paper I show that the bootstrapping objection is open to significant objections on several fronts. (shrink)

David Lewis (1980) proposed the Principal Principle (PP) and a “reformulation” which later on he called ‘OP’ (Old Principle). Reacting to his belief that these principles run into trouble, Lewis (1994) concluded that they should be replaced with the New Principle (NP). This conclusion left Lewis uneasy, because he thought that an inverse form of NP is “quite messy”, whereas an inverse form of OP, namely the simple and intuitive PP, is “the key to our concept of chance”. I argue (...) that, even if OP should be discarded, PP need not be. Moreover, far from being messy, an inverse form of NP is a simple and intuitive Conditional Principle (CP). Finally, both PP and CP are special cases of a General Principle (GP); it follows that so are PP and NP, which are thus compatible rather than competing. (shrink)

In this article we study the systems KF and VF of truth over set theory as well as related systems and compare them with the corresponding systems over arithmetic.

We can introduce singular terms for ordered pairs by means of an abstraction principle. Doing so proves useful for a number of projects in the philosophy of mathematics. However there is a question whether we can appeal to the abstraction principle in good faith, since a version of the Caesar Problem can be generated, posing the worry that abstraction fails to introduce expressions which refer determinately to the requisite sort of object. In this note I will pose the difficulty, and (...) then propose a solution. Since my solution appeals to a plausible constraint on the introduction of new expressions to a language, it is of interest independently of the particular case of terms for pairs. Since these provide the occasion for discussion we should nonetheless review the use of abstraction for pairs before the argumentative business of the paper commences. (shrink)

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N (...) or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1+ 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most OrdM; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HFM, ∈M〉 of M. Indeed, 〈 HFM, ∈M〉 is universal for all countable acyclic binary relations. (shrink)

Ambiguity is a property of syntactic expressions which is ubiquitous in all informal languages–natural, scientific and mathematical; the efficient use of language depends to an exceptional extent on this feature. Disambiguation is the process of separating out the possible meanings of ambiguous expressions. Ambiguity is typical if the process of disambiguation can be carried out in some systematic way. Russell made use of typical ambiguity in the theory of types in order to combine the assurance of its (apparent) consistency (“having (...) the cake”) with the freedom of the informal untyped theory of classes and relations (“eating it too”). The paper begins with a brief tour of Russell’s uses of typical ambiguity, including his treatment of the statement Cls ∈ Cls. This is generalized to a treatment in simple type theory of statements of the form A ∈ B where A and B are class expressions for which A is prima facie of the same or higher type than B. In order to treat mathematically more interesting statements of self membership we then formulate a version of typical ambiguity for such statements in an extension of Zermelo-Fraenkel set theory. Specific attention is given to how the“naive” theory of categories can thereby be accounted for. (shrink)

Here we investigate the classes RCA $^\uparrow_\alpha$ of representable directed cylindric algebras of dimension α introduced by Nemeti[12]. RCA $^\uparrow_\alpha$ can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, "purely cylindric algebraic" proof for the following theorems of Nemeti: (i) RCA $^\uparrow_\alpha$ is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one can obtain (...) a strong representation theorem for RCA $^\uparrow_\alpha$ if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories. (shrink)

Pavel Materna proposed valuable explications of concept and conceptual system. After their introduction, we contrast conceptual systems with (a novel notion of) derivation systems. Derivation systems differ from conceptual systems especially in including derivation rules. This enables us to show close connections among the realms of objects, their concepts, and reasoning with concepts.

Ontology is the study of what there is, what kinds of things make up reality. Ontology seems to be a very difficult, rather speculative discipline. However, it is trivial to conclude that there are properties, propositions and numbers, starting from only necessarily true or analytic premises. This gives rise to a puzzle about how hard ontological questions are, and relates to a puzzle about how important they are. And it produces the ontologyobjectivity dilemma: either (certain) ontological questions can be trivially (...) answered using only uncontroversial premises, or the uncertainties of ontology are really a threat to the truth of basically everything we say or believe. The main aim of this dissertation is to resolve these puzzles and to shed some light on the discipline of ontology. I defend a view inspired by Carnap’s internal-external distinction about what there is, but one according to which both internal and external questions are fully factual and meaningful. In particular, I argue that the trivial arguments are valid, but they do not answer any ontological questions. Furthermore, I propose an account of the function of our talk about properties, propositions and natural numbers. According to this account our talk about them has no ontological presuppositions for its literal and objective truth. This avoids the ontology-objectivity dilemma, and solves the puzzles about ontology. To do this I look at quantification and noun phrases in general, and at their relation to ontology. I argue that quantifiers are semantically underspecified in a certain respect, and play two different roles in communication. I discuss the relation between syntactic form and information structure, the function of certain non-referential, non-quantificational noun phrases, the uses of bare number determiners, and how arithmetic truths are learned and taught. The more metaphysical issues discussed include: inexpressible properties, logicism about arithmetic, nominalism, Carnap’s view about ontology, the problem of universals, the relationship between ontology and objectivity, different projects within ontology, non-existent.... (shrink)

Imre Lakatos’ Proofs and Refutations is a book well known to those who work in the philosophy of mathematics, though it is perhaps not widely referred to. Its general thrust is out of tenor with the foundationalist perspective that has dominated work in the philosophy of mathematics since the early years of the 20th century. It seems to us, though, that the book contains striking insights into the nature of proof, and the purpose of this paper is to explore and (...) apply some of these. (shrink)

The problem is addressed of establishing the satisfiability of prenex formulas involving a single universal quantifier, in diversified axiomatic set theories. A rather general decision method for solving this problem is illustrated through the treatment of membership theories of increasing strength, ending with a subtheory of Zermelo-Fraenkel which is already complete with respect to the ∀*∀ class of sentences. NP-hardness and NP-completeness results concerning the problems under study are achieved and a technique for restricting the universal quantifier is presented.

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)

A new logic is presented without predicates—except equality. Yet its expressive power is the same as that of predicate logic, and relations can faithfully be represented in it. In this logic we also develop an alternative for set theory. There is a need for such a new approach, since we do not live in a world of sets and predicates, but rather in a world of things with relations between them.

Many bioethical arguments rely implicitly on the assumption that the concept of “human part” is one on which everyone must agree, because it is unambiguous. But various parties interpret this “unambiguous” term in incompatible ways, leading to contention. This article is an informal presentation of a topomereological system on whose preferred interpretation several distinct but related meanings of “human part” can be isolated: part of a human body, part of the completion of a human body, and part of a human (...) being. A case is analyzed (the first total artificial heart (TAH) implantation), demonstrating in the process much of the apparatus of the system. By means of a casuistic methodology, the analysis is translated into recommendations for the ethical conduct of future TAH research. The more general conclusion, however, is that formal methods may provide useful tools for clarifying thought processes and organizing arguments in debates over bioethical issues. (shrink)

In this paper we give an account of the rise and development of coalgebraic thinking in mathematics and computer science as an illustration of the way mathematical frameworks may be transformed. Originating in a foundational dispute as to the correct way to characterise sets, logicians and computer scientists came to see maximizing and minimizing extremal axiomatisations as a dual pair, each necessary to represent entities of interest. In particular, many important infinitely large entities can be characterised in terms of such (...) axiomatisations. We consider reasons for the delay in arriving at the coalgebraic framework, despite many unrecognised manifestations occurring years earlier, and discuss an apparent asymmetry in the relationship between algebra and coalgebra. (shrink)

Transitive extensional well founded relations provide an intuitionistic notion of ordinals which admits transfinite induction. However these ordinals are not directed and their successor operation is poorly behaved, leading to problems of functoriality. We show how to make the successor monotone by introducing plumpness, which strengthens transitivity. This clarifies the traditional development of successors and unions, making it intuitionistic; even the (classical) proof of trichotomy is made simpler. The definition is, however, recursive, and, as their name suggests, the plump ordinals (...) grow very rapidly. Directedness must be defined hereditarily. It is orthogonal to the other four conditions, and the lower powerdomain construction is shown to be the universal way of imposing it. We treat ordinals as order-types, and develop a corresponding set theory similar to Osius' transitive set objects. This presents Mostowski's theorem as a reflection of categories, and set-theoretic union is a corollary of the adjoint functor theorem. Mostowski's theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of replacement, but this seems to be unavoidable for the plump rank. The comparison between sets and toposes is developed as far as the identification of replacement with completeness, and there are some suggestions for further work in this area. Each notion of set or ordinal defines a free algebra for one of the theories discussed by Joyal and Moerdijk, namely joins of a family of arities together with an operation s satisfying conditions such as x ≤ sx, monotonicity or s(x ∨ y) ≤ sx ∨ sy. Finally we discuss the fixed point theorem for a monotone endofunction s of a poset with least element and directed joins. This may be proved under each of a variety of additional hypotheses. We explain why it is unlikely that any notion of ordinal obeying the induction scheme for arbitrary predicates will prove the pure result. (shrink)

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding `forward looking' and `backward looking' modal operators. We show that this `bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper concludes with some (...) discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects. (shrink)

[1] You have a crystal ball. Unfortunately, it’s defective. Rather than predicting the future, it gives you the chances of future events. Is it then of any use? It certainly seems so. You may not know for sure whether the stock market will crash next week; but if you know for sure that it has an 80% chance of crashing, then you should be 80% confident that it will—and you should plan accordingly. More generally, given that the chance of a (...) proposition A is x%, your conditional credence in A should be x%. This is a chance-credence principle: a principle relating chance (objective probability) with credence (subjective probability, degree of belief). Let’s call it the Minimal Principle (MP). (shrink)

A definition of nonstandard universe which gets over the limitation to the finite levels of the cumulative hierarchy is proposed. Though necessarily nonwellfounded, nonstandard universes are arranged in strata in the likeness of superstructures and allow a rank function taking linearly ordered values. Nonstandard universes are also constructed which model the whole ZFC theory without regularity and satisfy the κ-saturation property.

In this paper, we explore the idea that sets depend on, or are grounded in, their members. It is said that a set depends on each of its members, and not vice versa. Members do not depend on the sets that they belong to. We show that the intuitive modal truth conditions for dependence, given in terms of possible worlds, do not accurately capture asymmetric dependence relations between sets and their members. We extend the modal truth conditions to include impossible (...) worlds and give a more satisfactory account of the dependence of a set on its members. Focusing on the case of singletons, we articulate a logical framework in which to evaluate set-theoretic dependence claims, using a normal first-order modal logic. We show that on this framework the dependence of a singleton on its single members follows from logic alone. However, the converse does not hold. (shrink)

We outline a novel solution to Moore’s paradox within the framework of update semantics, which explains Moorean absurdity in terms of non-cohesiveness. It is argued that, unlike the outlined solution, Gillies’ treatment of the paradox within this framework is not satisfactory.

A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of a greatest fixpoint formula of the modal μ-calculus. This generalizes the standard result in the literature where a finitary modal characterization is provided only for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and non-well-founded sets.

This paper is concerned with the way different axiom systems for set theory can be justified by appeal to such intuitions as limitation of size, predicativity, stratification, etc. While none of the different conceptions historically resulting from the impetus to provide a solution to the paradoxes turns out to rest on an intuition providing an unshakeable foundation,'each supplies a picture of the set-theoretic universe that is both useful and internally well motivated. The same is true of more recently proposed axiom (...) systems for non-well-founded universes, and an attempt is made to motivate such axiom systems on the basis of an old and respected ‘algebraic’ intuition. (shrink)

In this paper we introduce the notion of a set algebra S satisfying a system E of equations. After defining a notion of freeness for such algebras, we show that, for any system E of equations, set algebras that are free in the class of structures satisfying E exist and are unique up to a bisimulation. Along the way, analogues of classical set-theoretic and algebraic properties are investigated.

I characterize the deductivist ideal of justification and, following to a great extent Toulmin’s work The Uses of Argument, I try to explain why this ideal is erroneous. Then I offer an alternative model of justification capable of making our claims to knowledge about substantial matters sound and reasonable. This model of justification will be based on a conception of justification as the result of good argumentation, and on a model of argumentation which is a pragmatic linguistic reconstruction of Toulmin’s (...) model of argument. (shrink)

are considered with a view toward analyzing operational semantics from the perspective of predicate logic. The notion of a bisimulation is employed in two distinct ways: (i) as an extensional notion of equivalence on programs (or processes) generalizing input/output equivalence (at a cost exceeding II' ,over certain transition predicates computable in log space). and (ii) as a tool for analyzing the dependence of transitions on data (which can be shown to be elementary or nonelementary. depending on the formulation of the (...) transitions). (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)

Approaches to truth and the Liar paradox seem invariably to face a dilemma: either appeal to some sort of hierarchy, or declare apparently perfectly coherent concepts incoherent. But since both options lead to severe expressive restrictions, neither seems satisfactory. The aim of this paper is a new approach, which avoids the dilemma and the resulting expressive restrictions. Previous approaches tend to appeal to some new sort of semantic value for the truth predicate to take. I argue that such approaches inevitably (...) lead to the dilemma in question. In contrast, the present proposal sticks with classical semantic values, but allows the compositional rules associated with these to admit of exceptions. I show how such an approach can be developed systematically. (shrink)

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

We explore a non-classical, universal set theory, based on a purely 'structural' conception of sets. A set is a transfinite process of unfolding of an arbitrary binary structure, with identity of sets given by the observational equivalence between such processes. We formalize these notions using infinitary modal logic, which provides partial descriptions for set structures up to observational equivalence. We describe the comprehension and topological properties of the resulting set-theory, and we use it to give non-classical solutions to classical paradoxes, (...) to prove fixed-point theorems that relate recursion and corecursion, to formalize 'super-large', reflexive categories and 'super-large' circular modes, and to provide 'natural' solutions for domain equations. (shrink)

We examine the question of whether scientific theories can ever be complete. For two closely related reasons, we will argue that they cannot. The first reason is the inability to determine what are “valid empirical observations”, a result that is based on a self-reference Gödel/Tarski-like proof. The second reason is the existence of “meta-empirical” evidence of the inherent incompleteness of observations. These reasons, along with theoretical incompleteness, are intimately connected to the notion of belief and to theses within the philosophy (...) of science: the Quine-Duhem (and underdetermination) thesis and the observational/theoretical distinction failure. Some puzzling aspects of the philosophical theses will become clearer in light of these connections. Other results that follow are: no absolute measure of the informational content of empirical data, no absolute measure of the entropy of physical systems, and no complete computer simulation of the natural world are possible. The connections with the mathematical theorems of Gödel and Tarski reveal the existence of other connections between scientific and mathematical incompleteness: computational irreducibility, complexity, infinity, arbitrariness and self-reference. Finally, suggestions will be offered of where a more rigorous (or formal) “proof” of scientific incompleteness can be found. (shrink)

Purpose: The paper discusses the concept of a reflexive domain, an arena where the apparent objects as entities of the domain are actually processes and transformations of the domain as a whole. Human actions in the world partake of the patterns of reflexivity, and the productions of human beings, including science and mathematics, can be seen in this light. Methodology: Simple mathematical models are used to make conceptual points. Context: The paper begins with a review of the author's previous work (...) on eigenforms - objects as tokens for eigenbehaviors, the study of recursions and fixed points of recursions. The paper also studies eigenforms in the Boolean reflexive models of Vladimir Lefebvre. Findings: The paper gives a mathematical definition of a reflexive domain and proves that every transformation of such a domain has a fixed point. Thus eigenforms exist in reflexive domains. We discuss a related concept called a "magma." A magma is composed entirely of its own structure-preserving transformations. Thus a magma can be regarded as a model of reflexivity and we call a magma "reflexive" if it encompasses all of its structure-preserving transformations. We prove a fixed point theorem for reflexive magmas. We then show how magmas are related to knot theory and to an extension of set theory using knot diagrammatic topology. This work brings formalisms for self- reference into a wider arena of process algebra, combinatorics, non-standard set theory and topology. The paper then discusses how these findings are related to lambda calculus, set theory and models for self- reference. The last section of the paper is an account of a computer experiment with a variant of the Life cellular automaton of John H. Conway. In this variant, 7-Life, the recursions lead to self-sustaining processes with very long evolutionary patterns. We show how examples of novel phenomena arise in these patterns over the course of large time scales. Value: The paper provides a wider context and mathematical conceptual tools for the cybernetic study of reflexivity and circularity in systems. (shrink)