Online research in philosophy

When working with a first-order theory, it is often convenient to use definitions. That is, if ϕ(x) is a first-order formula with the free variables shown, one can introduce a new relation symbol R to abbreviate ϕ, with defining axiom ∀x (R(x) ↔ ϕ(x)). Of course, this definition can later be eliminated from a proof, simply by replacing every instance of R by ϕ. But suppose the proof involves nested definitions, with a sequence of relation symbols R0, . . . , Rk abbreviating formulae ϕ0, . . . , ϕk, where each ϕi may have multiple occurrences of R0, . . . , Ri−1. In that case, the naive elimination procedure described above can yield an exponential increase in the length of the proof. In Section 2, I show that if the underlying theory proves that there are at least two elements in the universe, a more careful translation allows one to eliminate the new definitions with at most a polynomial increase in length. In fact, I will describe an explicit algorithm that can be implemented in polynomial time. The proof is not difficult, but it relies on the assumption that equality is included in the logic. A similar trick has been used by Solovay in simulating iterated definitions efficiently, as discussed in Section 3.2 of [Pudl´ak 1998]. Consequently, the result proved here may be folklore, but to my knowledge it has not appeared in the literature, and it is needed in Section 3. It is also sometimes convenient, in a first-order setting, to introduce Skolem func-.

Overall, Max Black's defense of the inductive support of inductive rules succeeds. Circularity is best explained in terms of epistemic conditions of inference. When an inference is circular, another inference token of the same type may, because of a difference of surrounding circumstances, not be circular. Black's inductive arguments in support of inductive rules fit this pattern: a token circular in some circumstances may be noncircular in other circumstances.

Gupta’s Rule of Revision theory of truth builds on insights to be found in Martin and Woodruff (1975) and Kripke (1975) (who in turn build on Tarski) in order to permanently deepen our understanding of truth, of paradox (and of the absence of it), and of how we work our language while our language is working us. His concept of a predicate deriving its meaning by way of a Rule of Revision ought to impact significantly on the philosophy of language. Still, fortunately, he has left me something to..

This essay defends the view that inductive reasoning involves following inductive rules against objections that inductive rules are undesirable because they ignore background knowledge and unnecessary because Bayesianism is not an inductive rule. I propose that inductive rules be understood as sets of functions from data to hypotheses that are intended as solutions to inductive problems. According to this proposal, background knowledge is important in the application of inductive rules and Bayesianism qualifies as an inductive rule. Finally, I consider a Bayesian formulation of inductive skepticism suggested by Lange. I argue that while there is no good Bayesian reason for judging this inductive skeptic irrational, the approach I advocate indicates a straightforward reason not to be an inductive skeptic.

In the four papers available on our web site (of which this is the first), we propose to develop an inductive logic. By “inductive logic” we mean a set of principles that distinguish between successful and unsuccessful strategies for scientific inquiry. Our logic will have a technical character, since it is built from the concepts and terminology of (elementary) model theory. The reader may therefore wish to know something about the kind of results on offer before investing time in definitions and notation. Providing such an informal overview is the purpose of the present essay. We begin with discussion of the central concept under investigation, namely, theory acceptance.

This original and enticing book provides a fresh, unifying perspective on many old and new logico-philosophical conundrums. Its basic thesis is that many concepts central in ordinary and philosophical discourse are inherently circular and thus cannot be fully understood as long as one remains within the confines of a standard theory of definitions. As an alternative, the authors develop a revision theory of definitions, which allows definitions to be circular without this giving rise to contradiction (but, at worst, to “vacuous” uses of definienda). The theory is applied with varying levels of detail to a circular analysis of concepts as diverse as truth, predication, necessity, physical object, etc. The focus is on truth, and hope is expressed that a deeper understanding of the Liar and related paradoxes has been provided: “We have tried to show that once the circularity of truth is recognized, a great deal of its behavior begins to make sense. In particular, from this viewpoint, the existence of the paradoxes seems as natural as the existence of the eclipses” (p. 142). We think that this hope is fully justified, although some problems remain that future research in this field should take into account. The following assumptions constitute the typical background in which the truth paradoxes arise: (i) classical first-order logic, (ii) a language allowing for self-reference, and (iii) the “semantic” Tarskian schema: (TS) T ‘A’ ↔ A (where ‘T’ is the truth predicate, and the single quotes are a nominalization device applicable to sentences; for simplicity, we only consider homophonic versions of TS). This background can be seen as somehow part of our ordinary linguistic and conceptual background and yet, to avoid inconsistency, one or more of these assumptions must be suitably weakened. The classical, Tarskian strategy is to forbid self-reference, whereas the fixed-point approaches stemming from the work of Saul Kripke (1975) and Robert Martin and Peter Woodruff (1975) weaken the logic..

Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.

In this rigorous investigation into the logic of truth Anil Gupta and Nuel Belnap explain how the concept of truth works in both ordinary and pathological ...

In this paper we argue that Revision Rules, introduced by Anil Gupta and Nuel Belnap as a tool for the analysis of the concept of truth, also provide a useful tool for defining computable functions. This also makes good on Gupta's and Belnap's claim that Revision Rules provide a general theory of definition, a claim for which they supply only the example of truth. In particular we show how Revision Rules arise naturally from relaxing and generalizing a classical construction due to Kleene, and indicate how they can be employed to reconstruct the class of the general recursive functions. We also point at how Revision Rules can be employed to access non-minimal fixed points of partially defined computing procedures.

In this paper we apply the idea of Revision Rules, originally developed within the framework of the theory of truth and later extended to a general mode of definition, to the analysis of the arithmetical hierarchy. This is also intended as an example of how ideas and tools from philosophical logic can provide a different perspective on mathematically more “respectable” entities. Revision Rules were first introduced by A. Gupta and N. Belnap as tools in the theory of truth, and they have been further developed to provide the foundations for a general theory of (possibly circular) definitions. Revision Rules are non-monotonic inductive operators that are iterated into the transfinite beginning with some given “bootstrapper” or “initial guess.” Since their iteration need not give rise to an increasing sequence, Revision Rules require a particular kind of operation of “passage to the limit,” which is a variation on the idea of the inferior limit of a sequence. We then define a sequence of sets of strictly increasing arithmetical complexity, and provide a representation of these sets by means of an operator G(x, φ) whose “revision” is carried out over ω2 beginning with any total function satisfying certain relatively simple conditions. Even this relatively simple constraint is later lifted, in a theorem whose proof is due to Anil Gupta.