Online research in philosophy

The paper purports to show, against Quine, that one can construct a language , which results from the extension of the theory of truth functions by introducing sentence letter quantification. Next a semantics is provided for this language. It is argued that the quantification is neither substitutional nor requires one to consider the sentence letters as taking entities as values.

Turing’s famous 1936 paper “On computable numbers, with an application to the Entscheidungsproblem” defines a computable real number and uses Cantor’s diagonal argument to exhibit an uncomputable real. Roughly speaking, a computable real is one that one can calculate digit by digit, that there is an algorithm for approximating as closely as one may wish. All the reals one normally encounters in analysis are computable, like π, √2 and e. But they are much scarcer than the uncomputable reals because, as Turing points out, the computable reals are countable, whilst the uncomputable reals have the power of the continuum. Furthermore, any countable set of reals has measure zero, so the computable reals have measure zero. In other words, if one picks a real at random in the unit interval with uniform probability distribution, the probability of obtaining an uncomputable real is unity. One may obtain a computable real, but that is in- finitely improbable. But how about individual examples of uncomputable reals? We will show two: H and the halting probability Ω, both contained in the unit interval. Their construction was anticipated in..

Quineans have taken the basic expression of ontological commitment to be an assertion of the form '' x '', assimilated to theEnglish ''there is something that is a ''. Here I take the existential quantifier to be introduced, not as an abbreviation for an expression of English, but via Tarskian semantics. I argue, contrary to the standard view, that Tarskian semantics in fact suggests a quite different picture: one in which quantification is of a substitutional type apparently first proposed by Geach. The ontological burden is borne by constant symbols, and truth is defined separately from reference.

In the Begriffschrift Frege drew no distinction—or anyway signalled no importance to the distinction—between quantifying into positions occupied by what he called eigennamen—singular terms—in a sentence and quantification into predicate position or, more generally, quantification into open sentences—into what remains of a sentence when one or more occurrences of singular terms are removed. He seems to have conceived of both alike as perfectly legitimate forms of generalisation, each properly belonging to logic. More accurately: he seems to have conceived of quantification as such as an operation of pure logic, and in effect to have drawn no distinction between first-order, second-order and higherorder quantification in general.

We show that for any real number, the class of real numbers less random than it, in the sense of rK-reducibility, forms a countable real closed subfield of the real ordered field. This generalizes the well-known fact that the computable reals form a real closed field. With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also proved nearly simultaneously and independently by Ng (Keng Meng Ng. Master's Thesis. National University of Singapore, in preparation). Lastly, we show that the class of d.c.e. reals is properly contained in the class or reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rK-reducibility).

A case against Prior’s theory of propositions goes thus: (1) everyday propositional generalizations are not substitutional; (2) Priorean quantifications are not objectual; (3) quantifications are substitutional if not objectual; (4) thus, Priorean quantifications are substitutional; (5) thus that Priorean quantifications are not ontologically committed to propositions provides no basis for a similar claim about our everyday propositional generalizations. Prior agrees with (1) and (2). He rejects (3), but fails to support that rejection with an account of quantification on which there could be quantifications that are neither substitutional nor objectual. The paper draws from the work of Lorenzen an alternative conception of quantification in terms of which that needed account can be given.

Whereas arithmetical quantification is substitutional in the sense that a some-quantification is true only if some instance of it is true, it does not follow (and, in fact, is not true) that an account of the truth-conditions of the sentences of the language of arithmetic can be given by a substitutional semantics. A substitutional semantics fails in a most fundamental fashion: it fails to articulate the truth-conditions of the quantifications with which it is concerned. This is what is defended in the paper. In particular, it is defended against remarks to the contrary in a well known paper on the subject.

Fundamental to Quine’s philosophy of logic is the thesis that substitutional quantification does not express existence. This paper considers the content of this claim and the reasons for thinking it is true.

We here establish two theorems which refute a pair of what we believe to be plausible assumptions about differences between objectual and substitutional quantification. The assumptions (roughly stated) are as follows: (1) there is at least one set d and denumerable first order language L such that d is the domain set of no interpretation of L in which objectual and substitutional quantification coincide. (2) There exist interpreted, denumerable, first order languages K with indenumerable domains such that substitutional quantification deviates from objectual quantification in K and this deviance remains for all name extensions I of K. We show these assumptions have actually been made, and then prove the refuting theorems.

There are plausible objections to substitutional construals of generalization. But these objections do not apply to a substitutional construal of generalization proposed by Peter Geach several years ago. This paper examines Geach’s conception.