Online research in philosophy

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.

This paper is a reaction to G. Küng's and J. T. Canty's Substitutional Quantification and Leniewskian quantifiers'Theoria 36 (1970), 165–182. I reject their arguments that quantifiers in Ontology cannot be referentially interpreted but I grant that there is what can be called objectual — referential interpretation of quantifiers and that because of the unrestricted quantification in Ontology the quantifiers in Ontology should not be given a so-called objectual-referential interpretation. I explain why I am in agreement with Küng and Canty's recommendation that Ontology's quantifiers not be substitutionally interpreted even if Leniewski intended them to be so interpreted. A notion of an interpretation which is referential but yet which does not interpret as an assertor of existence of objects in a domain is developed. It is then shown that a first order version of Ontology is satisfied by those special kind of referential interpretations which read as Something as epposed to Something existing.

Call a quantifier unrestricted if it ranges over absolutely all things: not just over all physical things or all things relevant to some particular utterance or discourse but over absolutely everything there is. Prima facie, unrestricted quantification seems to be perfectly coherent. For such quantification appears to be involved in a variety of claims that all normal human beings are capable of understanding. For instance, some basic logical and mathematical truths appear to involve unrestricted quantification, such as the truth that absolutely everything is self-identical and the truth that the empty set has absolutely no members. Various metaphysical views too appear to involve unrestricted quantification, such as the physicalist view that absolutely everything is physical.

I investigate substitutional interpretations of quantifiers that count existential sentences true just in case they have true instances in a parametric extension of the language. I devise a semantics meeting four criteria: (1) it accounts adequately for natural language quantification; (2) it provides an account of justification in abstract sciences; (3) it constitutes a continuous semantics for natural and formal languages; and (4) it is purely substitutional, containing no appeal to referential interpretations. The prospects for a purely substitutional theory of quantification are thus no worse than for a referential account.

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 two doctrines for which Quine is particularly well known: the doctrine of ontological commitment and the inscrutability thesis—the thesis that reference and quantification are inscrutable. At first glance, the two doctrines are squarely at odds. If there is no fact of the matter as to what our expressions refer to, then it would appear that no determinate commitments can be read off of our best theories. We argue here that the appearance of a clash between the two doctrines is illusory. The reason that there is no real conflict is not simply that in determining our theories’ ontological commitments we naturally rely on our home language but also (and more importantly) that ontological commitment is not intimately tied to objectual quantification and a reference-first approach to language. Or so we will argue. We conclude with a new inscrutability argument which rests on the observation that the notion of objectual quantification, when properly cashed out, deflates.

There are two different ways of understanding the notion of ‘ontological commitment’. A question about ‘what is said to be’ by a theory or ‘what a theory says there is’ deals with ‘explicit’ commitment; a question about the ontological costs or preconditions of the truth of a theory concerns ‘implicit’ commitment. I defend a conception of ontological commitment as implicit commitment, and argue that existentially quantified idioms in natural language are implicitly, but not explicitly, committing. I use the distinction between the two kinds of ontological commitment to diagnose a flaw in a widely used argument to the effect that existential quantification is not ontologically committing.

Peter Geach proposed a substitutional construal of quantification over thirty years ago. It is not standardly substitutional since it is not tied to those substitution instances currently available to us; rather, it is pegged to possible substitution instances. We argue that (i) quantification over the real numbers can be construed substitutionally following Geach's idea; (ii) a price to be paid, if it is that, is intuitionism; (iii) quantification, thus conceived, does not in itself relieve us of ontological commitment to real numbers.

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.