Substitutional Quantification

It is argued that kripke has not shown that an explanatory truth theory for quantifiers which employs a substitutional approach does not require the hypothesis and that everything in the domain has a name, As wallace had claimed. It is further argued that kripke's substitutional quantifiers are best regarded as an extension of a device for abbreviating conjunctions and disjunctions.

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. (shrink)

Peter van Inwagen has long claimed that he doesn’t understand substitutional quantification and that the notion is, in fact, meaningless. Van Inwagen identifies the source of his bewilderment as an inability to understand the proposition expressed by a simple sentence like “,” where “$\Sigma$” is the existential quantifier understood substitutionally. I should think that the proposition expressed by this sentence is the same as that expressed by “.” So what’s the problem? The problem, I suggest, is that van Inwagen takes (...) traditional existential quantification to be ontologically committing and substitutional quantification to be ontologically noncommitting, which requires that the two quantifiers have different meanings—but no different meaning for the substitutional quantifier is forthcoming. What van Inwagen fails to appreciate is that substitutional quantification is directed at a criterion of ontological commitment, namely, W. V. O. Quine’s, which is quite different from van Inwagen’s criterion. Substitutional quantification successfully avoids the commitments Quine’s criterion would engender but has the same commitments as existential quantification given van Inwagen’s criterion. The question, then, is whether the existential quantifier is ontologically committing, as van Inwagen believes. The answer to that question will depend on whether the ordinary language “there is/are,” which is codified by the existential quantifier, is ontologically committing. There are good reasons to doubt that it is. (shrink)

Contribution to book symposium on C. Hill's: Thought and World. Focus is primarily on the intelligibility of Hill's substitutional quantification into propositions.

Truth and Meaning is a classic collection of original essays on fundamental questions in the philosophy of language.

The standard truth-conditional semantics for substitutional quantification, due to Saul Kripke, does not specify what proposition is expressed by sentences containing the particular substitutional quantifier. In this paper, I propose an alternative semantics for substitutional quantification that does. The key to this semantics is identifying an appropriate propositional function to serve as the content of a bound occurrence of a formula containing a free substitutional variable. I apply this semantics to traditional philosophical reasons for interest in substitutional quantification, namely, theories (...) of truth and ontological commitment. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

This chapter argues that special quantifiers such as 'something' when occurring in argument position are not ordinary or substitutional quantifiers; rather they have a reifying force introducing a domain of tropes or kinds of tropes to quantify over.

This paper argues that certain complements in philosophically significant constructions, especially predicative and clausal complements and intensional NPs, should not be analysed as providing an argument for a relation expressed by the verb, but rather as forming a complex predicate together with the verb. Apparent evidence for the traditional relational analyses, namely the possibility of replacing the complement by quantifiers such as 'something', will be shown to be misguided. Quantifiers like 'something' rather act as nominalizing expressions introducing ‘new’, derived objects (...) into the semantic structure of a sentence. The resulting analyses will have several philosophical ramifications. (shrink)

Quantified expressions in natural language generally are taken to act like quantifiers in logic, which either range over entities that need to satisfy or not satisfy the predicate in order for the sentence to be true or otherwise are substitutional quantifiers. I will argue that there is a philosophically rather important class of quantified expressions in English that act quite differently, a class that includes something, nothing, and several things. In addition to expressing quantification, such expressions act like nominalizations, introducing (...) a new domain of objects that would not have been present in the semantic structure of the sentence otherwise. The entities those expressions introduce are of just the same sort as those that certain ordinary nominalizations refer to (such as John's wisdom or John's belief that S), namely they are tropes or entities related to tropes. Analysing certain quantifiers as nominalizing quantifiers will shed a new light on philosophical issues such as the status of properties and the nature of propositional attitudes. (shrink)

Kripke’s substitutional interpretation of quantifiers is usually said to be unsatisfactory for independence-friendly (IF) languages. The purpose of this paper is to question this claim. Two accounts of substitutional semantics for IF sentences will be written down, and the objection of the so-called ‘dummy variables’ will be ruled out. Moreover, it will be argued, against the traditional view, that Game-Theoretical Semantics (GTS) should be conceived of as substitutional. The paper ends with some remarks concerning the reasons why substitution is especially (...) suitable for dynamic semantics. (shrink)

In order to know what a belief is, we need to know when it is appropriate to say that two subjects (or the same subject at two different times) believe(s) the same or entertain the same thought. This is not entirely straightforward. Consider for instance1. Tom thinks that he himself is the smartest and Tim believes the same2. In 2001, Bill believed that some action had to be taken to save the rain forest and today he believes the same.What does (...) Tim think? That he, Tim, is the smartest, or that Tom is? And what does Bill believe today? That action had to be taken in 2001 or that it has to be taken now? Both answers are intuitively acceptable. This has to be accounted for somehow.Building on Mark Richard's work on tense, Scott Soames 1 claims that the substitutional interpretation of the quantifiers is unable to account for the intended meaning of such statements as (2) and the validity of some inferences involving them. I will show that his argument is not convincing. Not only does the substitutional interpretation fare no worse than the objectual one, but it seems to be able to avoid a problem which could be seriously damaging for any account of the sameness of thoughts based on the notion of structured proposition. In the first section, I state the problem allegedly raised by tensed belief ascriptions to the substitutional interpretation of the quantifiers. In the second, Soames's argument is shown to be flawed. I also show that the content of the that-clause in (2) is not faithfully represented by any kind of structured proposition. Finally, I show how the substitutional interpretation can handle all such statements as (1) and (2) and the inferences involving them. (shrink)

Brentano's innovations in logical theory are considered in the context of his descriptive psychology, with its distinction between differences in quality and in object of mental phenomena. Objections are raised to interpretations that depend on a parallel between Urteil and assertion of a proposition. A more appropriate parallel is drawn between the assertion as subject to description in a metalanguage and the Urteil as secondary object in inner perception. This parallel is then applied so as to suggest a reinterpretation of (...) substitutional quantification, rendering the substitutional interpretation immune to problems that often arise as to the relation between substitutional range and referential range. (shrink)