A Logically Formalized Axiomatic Epistemology System Σ + C and Philosophical Grounding Mathematics as a Self-Sufficing System

The subject matter of this research is Kant’s apriorism underlying Hilbert’s formalism in the philosophical grounding of mathematics as a self-sufficing system. The research aim is the invention of such a logically formalized axiomatic epistemology system, in which it is possible to construct formal deductive inferences of formulae—modeling the formalism ideal of Hilbert—from the assumption of Kant’s apriorism in relation to mathematical knowledge. The research method is hypothetical–deductive (axiomatic). The research results and their scientific novelty are based on a logically formalized axiomatic system of epistemology called Σ + C, constructed here for the first time. In comparison with the already published formal epistemology systems X and Σ, some of the axiom schemes here are generalized in Σ + C, and a new symbol is included in the object-language alphabet of Σ + C, namely, the symbol representing the perfection modality, C: “it is consistent that…”. The meaning of this modality is defined by the system of axiom schemes of Σ + C. A deductive proof of the consistency of Σ + C is submitted. For the first time, by means of Σ + C, it is deductively demonstrated that, from the conjunction of Σ + C and either the first or second version of Gödel’s theorem of incompleteness of a formal arithmetic system, the formal arithmetic investigated by Gödel is a representation of an empirical knowledge system. Thus, Kant’s view of mathematics as a self-sufficient, pure, a priori knowledge system is falsified.

Transcendental Knowability and A Priori Luminosity

This paper draws out and connects two neglected issues in Kant’s conception of a priori knowledge. Both concern topics that have been central to contemporary epistemology and to formal epistemology in particular: knowability and luminosity. Does Kant commit to some form of knowability principle according to which certain necessary truths are in principle knowable to beings like us? Does Kant commit to some form of luminosity principle according to which, if a subject knows a priori, then they can know that they know a priori? I defend affirmative answers to both of these questions, and by considering the special kind of modality involved in Kant’s conceptions of possible experience and the essential completability of metaphysics, I argue that his combination of knowability and luminosity principles leads Kant into difficulty.

Justification awareness

We offer a new semantic approach to formal epistemology that incorporates two principal ideas: (i) justifications are prime objects of the model: knowledge and belief are defined evidence-based concepts; (ii) awareness restrictions are applied to justifications rather than to propositions, which allows for the maintaining of desirable closure properties. The resulting structures, Justification Awareness Models, JAMs, naturally include major justification models, Kripke models and, in addition, represent situations with multiple possibly fallible justifications which, in full generality, were previously off the scope of rigorous epistemic modeling.

Explication, Description and Enlightenment

In the first chapter of his book Logical Foundations of Probability, Rudolf Carnap introduced and endorsed a philosophical methodology which he called the method of ‘explication’. P.F. Strawson took issue with this methodology, but it is currently undergoing a revival. In a series of articles, Patrick Maher has recently argued that explication is an appropriate method for ‘formal epistemology’, has defended it against Strawson’s objection, and has himself put it to work in the philosophy of science in further clarification of the very concepts on which Carnap originally used it (degree of confirmation, and probability), as well as some concepts to which Carnap did not apply it (such as justified degree of belief). We shall outline Carnap’s original idea, plus Maher’s recent application of such a methodology, and then seek to show that the problem Strawson raised for it has not been dealt with. The method is indeed, we argue, problematic and therefore not obviously superior to the ‘descriptive’ method associated with Strawson. Our targets will not only be Carnapians, though, for what we shall say also bears negatively on a project that Paul Horwich has pursued under the name ‘therapeutic’, or ‘Wittgensteinian’ Bayesianism. Finally, explication, as we shall suggest and as Carnap recognised, is not the only route to philosophical enlightenment.

Probabilistic Representations of Belief

Formal epistemology is epistemology that uses mathematical tools. Foremost among them is probability theory. We can represent the strength of a belief by assigning it a number between zero and one, with one representing belief with maximal strength, and zero with minimal strength. Using these, and other, formal tools, we can investigate a range of epistemological questions, such as: What justifies beliefs? How should evidence inform belief? How should we update our beliefs over time?