Axiomatic Theories of Partial Ground I. The Base Theory (original) (raw)

Axiomatic Theories of Partial Ground II. Partial Ground and Typed Truth

Journal of Philosophical Logic, 2017

This is part two of a two-part paper, in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. This allows me to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we extend the base theory from the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to 0 and thus is consistent. We argue that this theory provides a natural solution to Fine's " puzzle of ground " about the interaction of truth and ground. Finally, we show that if we drop the typing of our truth-predicate, we run into similar paradoxes as in the case of truth: we get ground-theoretical paradoxes of self-reference.

Axioms for Grounded Truth [Review of Symbolic Logic]

2013

We axiomatize Leitgeb's (2005) theory of truth and show that this theory proves all arithmetical sentences of the system of ramified analysis up to \epsilon_0. We also give alternative axiomatizations of Kripke's (1975) theory of truth (Strong Kleene and supervaluational version) and show that they are at least as strong as the Kripke-Feferman system KF and Cantini's VF, respectively.

Ground First: Against the Proof-Theoretic Definition of Ground

This paper evaluates the proof-theoretic definition of ground developed by Poggiolesi in a range of recent publications and argues that her proposed definition fails. The paper then outlines an alternative approach where logical consequence relations and the logical operations are defined in terms of ground.

Better Semantics for the Pure Logic of Ground

Philosophers have spilled a lot of ink over the past few years exploring the nature and significance of grounding. Kit Fine has made several seminal contributions to this discussion, including an exact treatment of the formal features of grounding [Fine, 2012a]. He has specified a language in which grounding claims may be expressed, proposed a system of axioms which capture the relevant formal features, and offered a semantics which interprets the language. Unfortunately, the semantics Fine offers faces a number of problems. In this paper, I review the problems and offer an alternative that avoids them. I offer a semantics for the pure logic of ground that is motivated by ideas already present in the grounding literature, and for which a natural axiomatization capturing central formal features of grounding is sound and complete. I also show how the semantics I offer avoids the problems faced by Fine’s semantics.

the pure logic of ground

I set up a system of structural rules for reasoning about ground and prove soundness and completeness for an appropriate truthmaker semantics.

Notes on Formal Theories of Truth

Mathematical Logic Quarterly, 1989

8 0. Introduction I n this paper we investigate formal systems, which are related to KRIPKE's theory of truth and to its subsequent extensions via four-valued logic (see KRIPKE [22], VISSER [33], WOODRUFF [34]). There are several motivations for an axiomatic study: let us mention a few of them.

A theory of formal truth arithmetically equivalent to ID1

Journal of Symbolic Logic, 1990

We present a theory VF of partial truth over Peano arithmetic and we prove that VF and ID1, have the same arithmetical content. The semantics of VF is inspired by van Fraassen's notion of supervaluation.

The Logic of Ground

Journal of Philosophical Logic, 2019

I explore the logic of ground. I first develop a logic of weak ground. This logic strengthens the logic of weak ground presented by Fine in his ‘Guide to Ground.’ This logic, I argue, generates many plausible principles which Fine’s system leaves out. I then derive from this a logic of strict ground. I argue that there is a strong abductive case for adopting this logic. It’s elegant, parsimonious and explanatorily powerful. Yet, so I suggest, adopting it has important consequences. First, it means we should think of ground as a type of identity. Second, it means we should reject much of Fine’s logic of strict ground. I also show how the logic I develop connects to other systems in the literature. It is definitionally equivalent both to Angell’s logic of analytic containment and to Correia’s system G.

Does set theory really ground arithmetic truth

arXiv: Logic, 2019

We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth (even in a relative sense). Our method is to emphasize the incomplete picture of both theories and treat models as their syntactical counterparts. Insisting on the incomplete picture will allow us to argue in favor of the revisability of the standard model interpretation. We then show that it is hopeless to expect that the relative grounding provided by a standard interpretation can resist being revisable. We start briefly characterizing the expansion of arithmetic truth provided by the interpretation in a set theory. Further, we show that, for every well-founded interpretation of recursive extensions of PA in extensions of ZF, the interpreted version of arithmetic has more theorems than the original. This theorem expansion is not complete however. We continue by defining the coordination problem. T...

Predicative foundations of arithmetic

Predicative mathematics in the sense originating with Poincaré and Weyl begins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or set-theoretic standpoint, this appears problematic, for, as the story is usually told, impredicative principles seem to play an essential role in the foundations of arithmetic itself. 1 It is the main purpose of this paper to show that this appearance is illusory: as will emerge, a predicatively acceptable axiomatization of the natural number system can be formulated, and both the existence of structures of the relevant type and the categoricity of the relevant axioms can be proved in a predicatively acceptable way.