Axiomatic Theories of Partial Ground I

This is part one 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 us to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we develop an axiomatization of the relation of partial ground over the truths of arithmetic and show that the theory is a proof-theoretically conservative extension of the theory PT of positive truth. We construct models for the theory and draw some conclusions for the semantics of conceptualist ground.     

Remarks on Compositionality and Weak Axiomatic Theories of Truth

The paper draws attention to an important, but apparently neglected distinction relating to axiomatic theories of truth, viz. the distinction between weakly and strongly truth-compositional theories of truth. The paper argues that the distinction might be helpful in classifying weak axiomatic theories of truth and examines some of them with respect to it.     

Theories of Truth Which Have No Standard Models

This papers deals with the class of axiomatic theories of truth for semantically closed languages, where the theories do not allow for standard models; i.e., those theories cannot be interpreted as referring to the natural number codes of sentences only (for an overview of axiomatic theories of truth in general, see Halbach[6]). We are going to give new proofs for two well-known results in this area, and we also prove a new theorem on the nonstandardness of a certain theory of truth. The results indicate that the proof strategies for all the theorems on the nonstandardness of such theories are "essentially" of the same kind of structure.     

Truth and Reduction

The proof-theoretic results on axiomatic theories oftruth obtained by different authors in recent years are surveyed.In particular, the theories of truth are related to subsystems ofsecond-order analysis. On the basis of these results, thesuitability of axiomatic theories of truth for ontologicalreduction is evaluated.     

Supervaluation-Style Truth Without Supervaluations

Kripke’s theory of truth is arguably the most influential approach to self-referential truth and the semantic paradoxes. The use of a partial evaluation scheme is crucial to the theory and the most prominent schemes that are adopted are the strong Kleene and the supervaluation scheme. The strong Kleene scheme is attractive because it ensures the compositionality of the notion of truth. But under the strong Kleene scheme classical tautologies do not, in general, turn out to be true and, as a consequence, classical reasoning is no longer admissible once the notion of truth is involved. The supervaluation scheme adheres to classical reasoning but violates compositionality. Moreover, it turns Kripke’s theory into a rather complicated affair: to check whether a sentence is true we have to look at all admissible precisification of the interpretation of the truth predicate we are presented with. One consequence of this complicated evaluation condition is that under the supervaluation scheme a more proof-theoretic characterization of Kripke’s theory becomes inherently difficult, if not impossible. In this paper we explore the middle ground between the strong Kleene and the supervaluation scheme and provide an evaluation scheme that adheres to classical reasoning but retains many of the attractive features of the strong Kleene scheme. We supplement our semantic investigation with a novel axiomatic theory of truth that matches the semantic theory we have put forth.     

Continuity in Semantic Theories of Programming

Continuity is perhaps the most familiar characterization of the finitary character of the operations performed in computation. We sketch the historical and conceptual development of this notion by interpreting it as a unifying theme across three main varieties of semantical theories of programming: denotational, axiomatic and event-based. Our exploration spans the development of this notion from its origins in recursion theory to the forms it takes in the context of the more recent event-based analyses of sequential and concurrent computations, touching upon the relations of continuity with non-determinism.     

Reliabilism and antirealist theories of truth

In order to shed light on the question of whether reliabilism entails or excludes certain kinds of truth theories, I examine two arguments that purport to establish that reliabilism cannot be combined with antirealist and epistemic theories of truth. I take antirealism about truth to be the denial of the recognition-transcendence of truth, and epistemic theories to be those that identify truth with some kind of positive epistemic status. According to one argument, reliabilism and antirealism are incompatible because the former takes epistemic justification to be recognition-transcendent in a certain sense that conflicts with the latter’s denial of the recognition-transcendence of truth. I show that, because the recognition-transcendence of reliabilist justification is significantly weaker than the recognition-transcendence required by a realist conception of truth, antirealist theories of truth that deny the strong transcendence of truth do not threaten the externalist character of reliabilism. According to the second argument, reliabilism cannot be combined with an epistemic truth theory because reliabilists analyze positive epistemic status in terms of truth but epistemic theorists analyze truth in terms of positive epistemic status. However, I argue that reliabilists who wish to adopt an epistemic theory of truth can avoid circularity by appealing to a multiplicity of positive epistemic statuses.     

Completeness for systems including real numbers

The usual completeness theorem for first-order logic is extended in order to allow for a natural incorporation of real analysis. Essentially, this is achieved by building in the set of real numbers into the structures for the language, and by adjusting other semantical notions accordingly. We use many-sorted languages so that the resulting formal systems are general enough for axiomatic treatments of empirical theories without recourse to elements of set theory which are difficult to interprete empirically. Thus we provide a way of applying model theory to empirical theories without tricky detours. Our frame is applied to axiomatizations of three empirical theories: classical mechanics, phenomenological thermodynamics, and exchange economics.     

Conservative Theories of Classical Truth

Some axiomatic theories of truth and related subsystems of second-order arithmetic are surveyed and shown to be conservative over their respective base theory. In particular, it is shown by purely finitistically means that the theory PA ÷ "there is a satisfaction class" and the theory FS of [2] are conservative over PA.     

Theories of Truth without Standard Models and Yablo’s Sequences

The aim of this paper is to show that it’s not a good idea to have a theory of truth that is consistent but ω-inconsistent. In order to bring out this point, it is useful to consider a particular case: Yablo’s Paradox. In theories of truth without standard models, the introduction of the truth-predicate to a first order theory does not maintain the standard ontology. Firstly, I exhibit some conceptual problems that follow from so introducing it. Secondly, I show that in second order theories with standard semantics the same procedure yields a theory that doesn’t have models. So, while having an ω- inconsistent theory is a bad thing, having an unsatisfiable theory of truth is actually worse. This casts doubts on whether the predicate in question is, after all, a truthpredicate for that language. Finally, I present some alternatives to prove an inconsistency adding plausible principles to certain theories of truth.     

Weak and Strong Theories of Truth

A subtheory of the theory of self-referential truth known as FS is shown to be weak as a theory of truth but equivalent to full FS in its proof-theoretic strength.     

Theorizing about truth outside of one’s own language

A theory of truth is language-transcendent if it ascribes truth conditions to truth-bearers that are not expressible in (or translatable to) our natural language; a theory is language-immanent if it is not language-transcendent. In this paper, I argue for the following theses. (1) Whether the correct theory of truth is language-transcendent or language-immanent will have significant consequences for general philosophy. Prima facie, a language-transcendent theory is preferable. However, (2) language-transcendent theories tend to require substantive metaphysical commitments concerning truth. Deflationist theories are particularly interesting in this context. I argue that, if deflationism is to remain consistent with its motivations, then it must opt for a language-immanent conception of truth.

     

Deflationism beyond arithmetic

The conservativeness argument poses a dilemma to deflationism about truth, according to which a deflationist theory of truth must be conservative but no adequate theory of truth is conservative. The debate on the conservativeness argument has so far been framed in a specific formal setting, where theories of truth are formulated over arithmetical base theories. I will argue that the appropriate formal setting for evaluating the conservativeness argument is provided not by theories of truth over arithmetic but by those over subject matters ‘richer’ than arithmetic, such as set theory. The move to this new formal setting provides deflationists with better defence and brings a broader perspective to the debate.

     

A note on three-valued logic and Tarski theorem on truth definitions

We introduce a notion of semantical closure for theories by formalizing Nepeivoda notion of truth. [10]. Tarski theorem on truth definitions is discussed in the light of Kleene's three valued logic (here treated with a formal reinterpretation of logical constants). Connections with Definability Theory are also established.A first version was prepared for sec. 5-Gnsaga-CNR and presented at the Logic Seminar, Institute of Philosophy, University of Firenze, in the academic year 1978–79. The author would like to thank the referee for his criticism and the Alexander von Humboldt-Stiftung for the support during the revision of the paper.     

Frege's Elucidatory Holism

I argue against the two most influential readings of Frege's methodology in the philosophy of logic. Dummett's “semanticist” reading sees Frege as taking notions associated with semantical content—and in particular, the semantical notion of truth—as primitive and as intelligible independently of their connection to the activity of judgment, inference, and assertion. Against this, the “pragmaticist” reading proposed by Brandom and Ricketts sees Frege as beginning instead from the independent and intuitive grasp that we allegedly have on the latter activity and only then moving on to explain semantical notions in terms of the nature of such acts. Against both readings, I argue, first, that Frege gives clear indication that he takes semantical and pragmatical notions to be equally primitive, such that he would reject the idea that either sort of notion could function as the base for a non-circular explanation of the other. I argue, secondly, that Frege's own method for conveying the significance of these primitive notions—an activity that Frege calls “elucidation”—is, in fact, explicitly circular in nature. Because of this, I conclude that Frege should be read instead as conceiving of our grasp of the semantical and pragmatical dimensions of logic as far more of a holistic enterprise than either reading suggests.     

带良好条件句的真理不动点理论

各种真理不动点理论的一个主要问题在于：它们通常缺乏像样的条件句。本文旨在提出两种新的想法去处理这个问题。在介绍克里普克的真理不动点理论之后,本文的作者探讨了对真理不动点理论的种种反对意见,并回答了其中的一部分。然后,作者说明了菲尔德的真理理论,并说明该理论如何在克里普克的真理不动点理论至上增加了条件句。不过,本文的作者争论说,菲尔德的条件句仍有许多不尽理想之处。为了弥补这些缺点,作者提出了两个想法：一个是在不动点理论至上增加亚当斯条件句,另一个则是在其上增加史多内克条件句。     

Success and truth in the realism/anti-realism debate

I aim to clarify the relationship between the success of a theory and the truth of that theory. This has been a central issue in the debates between realists and anti-realists. Realists assume that success is a reliable indicator of truth, but the details about the respects in which success is a reliable indicator or test of truth have been largely left to our intuitions. Lewis (Synthese 129:371–380, 2001) provides a clear proposal of how success and truth might be connected, comparing a test of success of our theories to medical tests with low rates of false positives and false negatives. But, contrary to what Lewis claims, I argue that it is not enough for the realist to undercut the claim that success is not a reliable indicator of truth. Rather, the realist must show that our current best theories are likely true. Further, I argue that tests in science are unlike medical tests in a number of important ways.     

DEFLATIONISM,CONSERVATIVENESS AND MAXIMALITY

We discuss two desirable properties of deflationary truth theories: conservativeness and maximality. Joining them together, we obtain a notion of a maximal conservative truth theory – a theory which is conservative over its base, but can’t be enlarged any further without losing its conservative character. There are indeed such theories; we show however that none of them is axiomatizable, and moreover, that there will be in fact continuum many theories of this sort. It turns out in effect that the deflationist still needs some additional principles, which would permit him to construct his preferred theory of truth.     

Confirming Mathematical Theories: an Ontologically Agnostic Stance

The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject of significant criticism. In this paper I explore an alternative to the Quine/Putnam indispensability approach. I begin with a van Fraassen-like distinction between accepting the adequacy of a mathematical theory and believing in the truth of a mathematical theory. Finally, I consider the problem of moving from the adequacy of a mathematical theory to its truth. I argue that the prospects for justifying this move are qualitatively worse in mathematics than they are in science. This revised version was published online in June 2006 with corrections to the Cover Date.