Skolem’s “paradox” as logic of ground: The mutual foundation of both proper and improper interpretations

A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality.Its investigation needs philosophical means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction.A comparison to Mach’s doctrine is used to be revealed the fundamental and philosophical reductionism of Husserl’s phenomenology leading to a kind of Pythagoreanism in the final analysis.Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom.An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being (rather than reality for reality is external only to Gödel mathematics) is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem.

Grounding grounds necessity

Drawing from extensions of existing ideas in the logic of ground, a novel account of the grounds of necessity is presented, the core of which states that necessary truths are necessary because they stand in specific grounding connections.

Reality, fundamentality, grounding (and cognate notions)

The notion of grounding is one of the central notions in the debates concerning ontological dependence in contemporary metaphysics and metametaphysics. In this paper we have carried out a comparative analysis of grounding, supervenience, reality, fundamentality, and cognate notions, and we have demonstrated what their role should be in the context of neo-Aristotelian hierarchical ontologies and the project of metaphysical foundationalism. We have also sketched out some basic outlines of what Kit Fine calls ?the pure logic of ground? by establishing certain formal desiderata which grounding ought to meet in order to successfully carry out its specific ontologico- explanatory role. It is finally shown that grounding suffers from similar problems and shortcomings as supervenience, and that a satisfactory solution of those problems cannot be found by looking to metaphysical primitivism according to which grounding is a sui generis, primitive and unanalysable notion which is nonetheless essential for metaphysics. Even though grounding might turn out to be an ?essentially contested concept?, in the end we suggest how the aforementioned problems might be met by means of holistic considerations of grounding within the broader context of the entire (meta)metaphysical theory.

PURE LOGIC OF ITERATED FULL GROUND

This article develops the Pure Logic of Iterated Full Ground (plifg), a logic of ground that can deal with claims of the form “ϕ grounds that (ψ grounds θ)”—what we call iterated grounding claims. The core idea is that some truths Γ ground a truth ϕ when there is an explanatory argument (of a certain sort) from premisses Γ to conclusion ϕ. By developing a deductive system that distinguishes between explanatory and nonexplanatory arguments we can give introduction rules for operators for factive and nonfactive full ground, as well as for a propositional “identity” connective. Elimination rules are then found by using a proof-theoretic inversion principle.

ON WEAK GROUND

Though the study of grounding is still in the early stages, Kit Fine, in ”The Pure Logic of Ground”, has made a seminal attempt at formalization. Formalization of this sort is supposed to bring clarity and precision to our theorizing, as it has to the study of other metaphysically important phenomena, like modality and vagueness. Unfortunately, as I will argue, Fine ties the formal treatment of grounding to the obscure notion of a weak ground. The obscurity of weak ground, together with its centrality in Fine’s system, threatens to undermine the extent to which this formalization offers clarity and precision. In this paper, I show how to overcome this problem. I describe a system, the logic of strict ground (LSG) and demonstrate its adequacy; I specify a translation scheme for interpreting Fine’s weak grounding claims; I show that the interpretation verifies all of the principles of Fine’s system; and I show that derivability in Fine’s system can be exactly characterized in terms of derivability in LSG. I conclude that Fine’s system is reducible to LSG.

THE PURE LOGIC OF GROUND

I lay down a system of structural rules for various notions of ground and establish soundness and completeness.