In this paper, I present what I call the symmetry conception of God within 1st order, extensional, non-well-founded set theory. The symmetry conception comes in two versions. According to the first, God is that unique being that is universally symmetrical with respect to set membership. According to the second, God is the universally symmetrical set of all sets that are universally symmetrical with respect to set membership. I present a number of theorems, most importantly that any universally symmetrical set is (...) identical to its essence, that show that the two symmetry conceptions intersect with some dominant theological conceptions of God. The theorems also show that both of the symmetry conceptions of God entail that God has a fractal like structure. (shrink)

Aristotle's categorial scheme had an unparalleled effect not only on his own philosophical system, but also on the systems of many of the greatest philosophers in the Western tradition. The set of doctrines in the Categories, known as categorialism, play, for instance, a central role in Aristotle's discussion of change in the Physics, in the science of being qua being in the Metaphysics, and in the rejection of Platonic ethics in the Nicomachean Ethics. Plainly, the enterprise of categorialism inaugurated by (...) Aristotle runs deep in the philosophical psyche. Even so, despite its wide-reaching influence—and, indeed owing to that influence—any attempt to describe categorialism faces a significant difficulty: experts disagree on many of its most important and fundamental aspects. This article argues that Aristotle's categorial scheme, as is the case with many works in the history of philosophy, is best illuminated by opposing beams of interpretive light. It examines how Aristotle arrived at his list of categories and considers the connection between Aristotle's categories and his hylomorphism. (shrink)

According to Augustine, abstract objects are ideas in the mind of God. Because numbers are a type of abstract object, it would follow that numbers are ideas in the mind of God. Call such a view the “Augustinian View of Numbers” (AVN). In this paper, I present a formal theory for AVN. The theory stems from the symmetry conception of God as it appears in Studtmann (2021). I show that the theory in Studtmann’s paper can interpret the axioms of Peano (...) Arithmetic minus the induction schema. This fact allows for the development of arithmetic in a natural way. The development eventuates in a theory that can interpret second-order arithmetic. The conception of God that emerges by the end of the discussion is a conception of an infinite, ineffable, self-cause that contains objects that not only serve as numbers but also encode information about each other. (shrink)

The traditional problem of free will has reached an impasse; we are unlikely to see progress without rethinking the terms in which the problem had been cast. Our approach offers an alternative to the standard terms of the debate, by developing an authorially parameterized approach articulated within a two-dimensional semantics for temporal predicates.

Empiricism and the Problem of Metaphysics develops and defends an empiricist solution to the problem of metaphysics, then examines the implications of such a solution for skeptical arguments and the is-ought gap. At the heart of the solution is an empirically verifiable empiricist view of the a priori.

In this note, I discuss a very simple time-dependent formula and its relation to the cosmological constant. I first note that the the formula yields an exact match of the cosmological constant on the assumption that the age of the universe is 8.08×10^60 in Planck units. I then suggest a physical interpretation of the formula according to which the cosmological constant is the result of a magnetic field.

According to Augustine, abstract objects are ideas in the Mind of God. Because numbers are a type of abstract object, it would follow that numbers are ideas in the Mind of God. Let us call such a view the Augustinian View of Numbers (AVN). In this paper, I present a formal theory for AVN. The theory stems from the symmetry conception of God as it appears in Studtmann (2021). I show that Robinson’s Arithmetic is a conservative extension of the axioms (...) in Studtmann’s original paper. The extension is made possible by identifying the set of natural numbers with God, 0 with Being, and the successor function with the essence function. The resulting theory can then be augmented to include Peano Arithmetic by adding a set-theoretic version of induction and a comprehension schema restricted to arithmetically definable properties. In addition to these formal matters, the paper provides a characterization of the mind of God. According to the characterization, the Being essences that constitute God’s mind act as both numbers and representations – each (except for Being itself) has all the properties of some number and encodes all the properties of that number’s predecessor. The conception of God that emerges by the end of the discussion is a conception of an infinite, ineffable, axiologically and metaphysically ultimate entity that contains objects that not only serve as numbers but also encode information about each other. (shrink)

In this paper, we provide a game-theoretic examination of indirect utilitarianism by comparing the expected payoffs of attempts to apply a deontological principle and a utilitarian principle within the context of the Prisoner’s Dilemma (PD). Although many of the best-known utilitarians and consequentialists have accepted some indirect form of their respective views, the results in this paper suggest that they have been overly quick to dismiss altogether the benefits of directly enacting utilitarian principles. We show that for infallible moral agents, (...) what we call ‘non-autonomous agents’, direct utilitarianism dominates indirect utilitarianism via deontology in terms of achieving the maximized utilitarian outcome, but only in underlying games where the maximized utilitarian outcome involves unequal payoffs. In other situations, indirect utilitarianism implemented through Kantian deontology either ties or dominates direct utilitarianism in terms of achieving the maximized utilitarian outcome. We also examine the two different moralities on the assumption that fallibility, which is a form of autonomy, is an aspect of moral agency by introducing Endogenized Morality Models (EMM’s). We believe that just as indirect utilitarians worry about the cost of applying moral principles, so too they should worry about the fact that humans have both pro-social and materialistically selfish motivations and hence are fallible moral agents. We show that there are conditions under which fallible autonomous utilitarians achieve higher expected material and psychic payoffs than fallible autonomous deontologists and conditions under which they do not. (shrink)

The use of evolutionary game theory to explain the evolution of human norms and the behavior of humans who act according to those norms is widespread. Both the aims and motivation for its use are clearly articulated by Harms and Skyrms (2008) in the following passage: "A good theory of evolution of norms might start by explaining the evolution of altruism in Prisoner’s Dilemma, of Stag Hunting, and of the equal split in the symmetric bargaining game. These are not well-explained (...) by classical game theory based on rational choice. From a technical point of view, they present different theoretical challenges. In the bargaining game, there are an infinite number of equilibria with no principled (rational choice) way to select the cooperative one. In Stag Hunt there are only two, but the non-cooperative one is selected by risk-dominance. In Prisoner’s Dilemma the state of mutual cooperation is not a Nash equilibrium at all, and cooperation flies in the face of the rational-choice principle that one does not choose less rather than more. In contrast to rational choice theory, the most common tool of evolutionary game theory is the replicator dynamics, in which the propagation rate of each strategy is determined by its current payoffs. These dynamics have a rationale in both biological and cultural evolutionary modeling, and sometimes tell us things that rational choice theory does not." We agree with the first sentence in this quotation: a good theory about the behavior under norms ought to explain altruism in the Prisoner’s Dilemma (PD), playing Stag in Stag Hunt (SH), and offering equal splits in the symmetric Nash bargaining game (NB). We also agree with Harms and Skyrms about the difference in technical challenges each of these games poses. Finding a single mechanism, even one as broadly understood as evolution, that could solve these challenges en masse is no doubt a tall order. Nonetheless, in this paper, we present a single, simple, modification to SH, NB, and a general n-player PD that does just that: we introduce deontological autonomy into the models. (shrink)

The idea that a being can be identical to its own essence has a long and venerable history in theological speculation. As with many ideas in theology, however, such an idea has never been given an adequate mathematical formulation. The key to such a formulation, I argue, is introducing an essence axiom into non-well-founded set theory. According to such an axiom, for every set, x, there is a set that contains all and only those sets that contain x. With such (...) an axiom, it becomes possible to introduce an essence function that takes each set to its essence. An object that is identical to its essence, then, is simply a fixed point of the essence function. In this paper, I first discuss a theorem according to which any fixed point of the essence function is universally symmetrical with respect to the set membership relation. After proving the theorem, I discuss its theological implications. I go on to argue that the process of counting the members of a set that is identical to its essence is a Sisyphusian process. Such a fact I suggest shows that the absurdity of human existence is not alleviated by the existence of God but rather is entailed by it. (shrink)

The Nash counterfactual considers the question: what would happen were I to change my behaviour assuming no one else does. By contrast, the Kantian counterfactual considers the question: what would happen were everyone to deviate from some behaviour. We present a model that endogenizes the decision to engage in this type of Kantian reasoning. Autonomous agents using this moral framework receive psychic payoffs equivalent to the cooperate-cooperate payoff in Prisoner’s Dilemma regardless of the other player’s action. Moreover, if both interacting (...) agents play Prisoner’s Dilemma using this moral framework, their material outcomes are a Pareto improvement over the Nash equilibrium. (shrink)

In The Divine Fractal, Studtmann (2021) introduced a novel conception of God, what he calls the symmetry conception, and showed that such a conception not only can be formalized within extensional non-well-founded set theory but also entails the Thomistic view that God is identical to her essence. In this paper, I show that Studtmann’s symmetry conception of God can be integrated into a recent approach to quantum gravity, namely causal set theory. The theory that results has two significant consequences. First, (...) God is the necessarily existing set of spacetime events. Second, the square root of the probability that a spacetime event randomly chosen from N spacetime events is uncaused is given by the following formula: ((Harmonic [N])/N)^(1/2). I conclude the paper by briefly discussing a connection between this formula and the cosmological constant. (shrink)

There is a very deep tension that exists at the heart of Aristotle’s metaphysical system in virtue of the fact that his works seem to contain two distinct and not obviously commensurate ontologies: an early ontology that Aristotle outlines in the Categories and a later ontology that he develops in his physical–metaphysical treatises. In this paper I briefly describe the two ontologies, discuss the sources of conflict and outline different scholarly speculations about the relationship between the two.

In 'Subjunctive Conditionals: Two Parameters vs. Three' Pavel Tichy articulates and defends a three-parameter account of counterfactuals. In the paper, he responds to a well known objection against the validity of various forms of inference, in particular strengthening of the antecedent, contraposition, and hypothetical syllogism. In this paper, I argue that his response to the objection is inadequate. I then propose an alternative form of the three-parameter account of counterfactuals that avoids the objection in question.

In this paper, I address both the interpretive and philosophical issues concerning prime matter. My aim is to show that a philosophically interesting account of prime matter can be articulated that strongly coheres with, even if it is not necessitated by, Aristotle’s texts. In articulating the interpretation, I first examine a view defended by both Richard Sorabji and Robert Sokolowski according to which prime matter is extension. Such a view, I argue, is problematic for a number of reasons. Nonetheless, it (...) provides a convenient starting point for the view I defend according to which prime matter is intimately linked to, though not identical with, extension. (shrink)

In this paper, I address both the interpretive and philosophical issues concerning prime matter. My aim is to show that a philosophically interesting account of prime matter can be articulated that strongly coheres with, even if it is not necessitated by, Aristotle’s texts. In articulating the interpretation, I first examine a view defended by both Richard Sorabji and Robert Sokolowski according to which prime matter is extension. Such a view, I argue, is problematic for a number of reasons. Nonetheless, it (...) provides a convenient starting point for the view I defend according to which prime matter is intimately linked to, though not identical with, extension. (shrink)

Whence the categories? -- The body problem in Aristotle -- Form -- Prime matter -- Quality -- Quantity -- Substance.