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 objectualquantification 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)
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. (shrink)
For various reasons several authors have enriched classical first order syntax by adding a predicate abstraction operator. “Conservatives” have done so without disturbing the syntax of the formal quantifiers but “revisionists” have argued that predicate abstraction motivates the universal quantifier’s re-classification from an expression that combines with a variable to yield a sentence from a sentence, to an expression that combines with a one-place predicate to yield a sentence. My main aim is to advance the cause of predicate abstraction while (...) cautioning against revisionism. In so doing, however, I shall pursue a secondary aim by conveying mixed blessings to those who hold the view that in the logical sense of “existence” some existing object is such as to exist contingently. Advocates of this view must concede Williamson’s recent contention that the domain of unrestricted objectualquantification could not have been narrower than it is actually, but predicate abstraction affords them some hope of accommodating this concession. (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.
The question of the origin of polyadic expressivity is explored and the results are brought to bear on Bertrand Russell's 1903 theory of denoting concepts, which is the main object of criticism in his 1905 "On Denoting." It is shown that, appearances to the contrary notwithstanding, the background ontology of the earlier theory of denoting enables the full-blown expressive power of first-order polyadic quantification theory without any syntactic accommodation of scopal differences among denoting phrases such as 'all φ', 'every (...) φ', and 'any φ' on the one hand, and 'some φ' and 'a φ' on the other. The case provides an especially vivid illustration of the general point that structural (or ideological) austerity can be paid for in the coin of ontological extravagance. (shrink)
Ordinary English contains different forms of quantification over objects. In addition to the usual singular quantification, as in 'There is an apple on the table', there is plural quantification, as in 'There are some apples on the table'. Ever since Frege, formal logic has favored the two singular quantifiers ∀x and ∃x over their plural counterparts ∀xx and ∃xx (to be read as for any things xx and there are some things xx). But in recent decades it (...) has been argued that we have good reason to admit among our primitive logical notions also the plural quantifiers ∀xx and ∃xx. More controversially, it has been argued that the resulting formal system with plural as well as singular quantification qualifies as ‘pure logic’; in particular, that it is universally applicable, ontologically innocent, and perfectly well understood. In addition to being interesting in its own right, this thesis will, if correct, make plural quantification available as an innocent but extremely powerful tool in metaphysics, philosophy of mathematics, and philosophical logic. For instance, George Boolos has used plural quantification to interpret monadic second-order logic and has argued on this basis that monadic second-order logic qualifies as “pure logic.” Plural quantification has also been used in attempts to defend logicist ideas, to account for set theory, and to eliminate ontological commitments to mathematical objects and complex objects. (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)
In “Mathematics is megethology,” Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that (...) mereology and plural quantification are, in some ways, particularly relevant to a certain conception of the infinite. More precisely, though the principles of mereology and plural quantification do not guarantee the existence of an infinite number of objects, nevertheless, once the existence of any infinite object is admitted, they are able to assure the existence of an uncountable infinity of objects. So, if—as Lewis maintains—MPQ were parts of logic, the implausible consequence would follow that, given a countable infinity of individuals, logic would be able to guarantee an uncountable infinity of objects. (shrink)
Standard first-order logic plus quantifiers of all finite orders ("SFOLω") faces four well-known difficulties when used to characterize the behavior of certain English quantifier phrases. All four difficulties seem to stem from the typed structure of SFOLω models. The typed structure of SFOLω models is in turn a product of an asymmetry between the meaning of names and the meaning of predicates, the element-set asymmetry. In this paper we examine a class of models in which this asymmetry of meaning is (...) removed. The models of this class permit definitions of the quantifiers which allow a desirable flexibility in fixing the domain of quantification. Certain SFOLω type restrictions are thereby avoided. The resulting models of English validate all of the standard first-order logical truths and are free of the four deficiencies of SFOLω models. (shrink)
In my book, Worlds and Individuals, Possible and Otherwise , I use the novel idea of modal tense to respond to a number of arguments against modal realism. Peter van Inwagen’s million-carat-diamond objection is one of them. It targets the version of modal realism by David Lewis and exploits the fact that Lewis accepts absolutely unrestricted quantification. The crux of my response is to use modal tense to neutralize absolutely unrestricted quantification. Seahwa Kim says that even when equipped (...) with modal tense, I am unsuccessful, given my view of reality and the proper use of modal tense in speaking of reality. I counter her attempt at resurrecting van Inwagen’s objection and clarify how we should use modal tense and how we should talk about reality. (shrink)
We consider collective quantification in natural language. For many years the common strategy in formalizing collective quantification has been to define the meanings of collective determiners, quantifying over collections, using certain type-shifting operations. These type-shifting operations, i.e., lifts, define the collective interpretations of determiners systematically from the standard meanings of quantifiers. All the lifts considered in the literature turn out to be definable in second-order logic. We argue that second-order definable quantifiers are probably not expressive enough to formalize (...) all collective quantification in natural language. (shrink)
This paper examines the quest for the quantification of the predicate, as discussed by W.S. Jevons, and relates it to the discussion about universals and particulars between Plato and Aristotle. We conclude that the quest for the quantification of the predicate can only be achieved by stripping the syllogism from its metaphysical heritage.
. Three logical squares of predication or quantification, which one can even extend to logical hexagons, will be presented and analyzed. All three squares are based on ideas of the non-traditional theory of predication developed by Sinowjew and Wessel. The authors also designed a non-traditional theory of quantification. It will be shown that this theory is superfluous, since it is based on an obscure difference between two kinds of quantification and one pays a high price for differentiating (...) in this way: losing the definability between the existence- and all-quantifier. Therefore, a combination of non-traditional predication and classical quantification is preferred here. (shrink)
This paper deals with the logical form of quantified sentences. Its purpose is to elucidate one plausible sense in which quantified sentences can adequately be represented in the language of first-order logic. Section 1 introduces some basic notions drawn from general quantification theory. Section 2 outlines a crucial assumption, namely, that logical form is a matter of truth-conditions. Section 3 shows how the truth-conditions of quantified sentences can be represented in the language of first-order logic consistently with some established (...) undefinability results. Section 4 sketches an account of vague quantifier expressions along the lines suggested. Finally, section 5 addresses the vexed issue of logicality. (shrink)
Connectionist attention to variables has been too restricted in two ways. First, it has not exploited certain ways of doing without variables in the symbolic arena. One variable-avoidance method, that of logical combinators, is particularly well established there. Secondly, the attention has been largely restricted to variables in long-term rules embodied in connection weight patterns. However, short-lived bodies of information, such as sentence interpretations or inference products, may involve quantification. Therefore short-lived activation patterns may need to achieve the effect (...) of variables. The paper is mainly a theoretical analysis of some benefits and drawbacks of using logical combinators to avoid variables in short-lived connectionist encodings without loss of expressive power. The paper also includes a brief survey of some possible methods for avoiding variables other than by using combinators. (shrink)
It is common in metaphysical discourse to make claims like “Everything is self-identical” in which “everything” is intended to range over everything. This sort of “unrestricted” generality appears central to metaphysical discourse. But there is debate whether such generality, which appears to involve quantification over an all-inclusive domain, is even meaningful. To address this concern, Shaughan Lavine and Vann McGee supply competing accounts of the generality expressed by this use of “everything.” I argue that, from the perspective of the (...) metaphysician, neither of their proposals is entirely suitable. But their central insights, as well as their shortcomings, suggest an account which is suitable for metaphysical discourse, what I call “genuinely unrestricted quantification.” For while genuinely unrestricted quantification does not directly supply an account of the metaphysician's “everything,” it disentangles what are properly metaphysical issues from issues concerning how we ought to understand the quantifiers. It makes clear that whether the metaphysician's “everything” is meaningful cannot simply be resolved by supplying an appropriate account of our understanding of quantification. It depends on addressing certain substantive ontological questions. (shrink)
If a company’s share price rises when it sacks workers, or when it makes money from polluting the environment, it would seem that the accounting is not being done correctly. Real costs are not being paid. People’s ethical claims, which in a smaller-scale case would be legally enforceable, are not being measured in such circumstances. This results from a mismatch between the applied ethics tradition and the practice of the accounting profession. Applied ethics has mostly avoided quantification of rights, (...) while accounting practice has embraced quantification, but has been excessively conservative about what may be counted. The two traditions can be combined, by using some of the ideas economists have devised to quantify difficult-to-measure costs and benefits in environmental accounting. (shrink)
Proportional quantification and progressive aspect interact in English in revealing ways. This paper investigates these interactions and draws conclusions about the semantics of the progressive and telicity. In the scope of the progressive, the proportion named by a proportionality quantifier (e.g. most in The software was detecting most errors) must hold in every subevent of the event so described, indicating that a predicate in the scope of the progressive is interpreted as an internally homogeneous activity. Such an activity interpretation (...) is argued to be available for telic predicates (e.g. cross the street) because these receive a partitive interpretation except in combination with a completive operator, which asserts that the event so described has culminated. The obligatoriness of the completive operator in the preterit is shown to parametrically distinguish those languages that show completion entailments in the preterit, e.g. English, from those that do not, e.g. Malagasy, Hindi, and Japanese. (shrink)
We develop a variant of Least Fixed Point logic based on First Orderlogic with a relaxed version of guarded quantification. We develop aGame Theoretic Semantics of this logic, and find that under reasonableconditions, guarding quantification does not reduce the expressibilityof Least Fixed Point logic. But we also find that the guarded version ofa least fixed point algorithm may have a greater time complexity thanthe unguarded version, by a linear factor.
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 objectualquantification 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 objectualquantification, when properly cashed out, deflates. (shrink)
Higginbotham (1986) argues that conditionals embedded under quantifiers (as in ‘no student will succeed if they goof off’) constitute a counterexample to the thesis that natural language is semantically compositional. More recently, Higginbotham (2003) and von Fintel and Iatridou (2002) have suggested that compositionality can be upheld, but only if we assume the validity of the principle of Conditional Excluded Middle. I argue that these authors’ proposals (...) deliver unsatisfactory results for conditionals that, at least intuitively, do not appear to obey Conditional Excluded Middle. Further, there is no natural way to extend their accounts to conditionals containing ‘unless’. I propose instead an account that takes both ‘if’ and ‘unless’ statements to restrict the quantifiers in whose scope they occur, while also contributing a covert modal element to the semantics. In providing this account, I also offer a semantics for unquantified statements containing ‘unless’. (shrink)
The semantic rules governing natural language quantifiers (e.g. "all," "some," "most") neither coincide with nor resemble the semantic rules governing the analogues of those expressions that occur in the artificial languages used by semanticists. Some semanticists, e.g. Peter Strawson, have put forth data-consistent hypotheses as to the identities of the semantic rules governing some natural-language quantifiers. But, despite their obvious merits, those hypotheses have been universally rejected. In this paper, it is shown that those hypotheses are indeed correct. Moreover, data-consistent (...) hypotheses are put forth as to the identities of the semantic rules governing the words "most" and "many," the semantic rules governing which semanticists have thus far been unable to identify. The points made in this paper are anticipated in a paper, published in the same issue of the Journal of Pragmatics, by Andrzej Boguslawski. (shrink)
In this paper I revive two important formal approaches to the interpretation of natural language, that of Montague and that of Karttunen and Peters. Armed with insights from dynamic semantics (Heim, Krifka) the two turn out to stand up against age-old criticisms in an orthodox fashion. The plan is mainly methodological, as I only want to illustrate the technical feasibility of the revived proposals. Even so, there are illuminating and welcome empirical consequences on the subject of scope islands (as discussed (...) by Abusch and Kratzer, among many others), as well as unintended theoretical implications in the contextualist debate (Grice, Recanati, Simons, Stanley, and many others again). (shrink)
This paper describes quantificational structures in Greenlandic Eskimo (Kalaallisut), a language where familiar quantificational meanings are expressed in ways that are quite different from English. Evidence from this language thus poses some formidable challenges for cross-linguistic theories of compositional semantics.
Many of those who accept the universalist thesis that mereological composition is unrestricted also maintain that the folk typically restrict their quantifiers in such a way as to exclude strange fusions when they say things that appear to conflict with universalism. Despite its prima facie implausibility, there are powerful arguments for universalism. By contrast, there is remarkably little evidence for the thesis that strange fusions are excluded from the ordinary domain of quantification. Furthermore, this reconciliatory strategy seems hopeless when (...) applied to the more fundamental conflict between universalism and the intuitions that tell against it. (shrink)
There are four broad grounds upon which the intelligibility of quantification over absolutely everything has been questioned—one based upon the existence of semantic indeterminacy, another on the relativity of ontology to a conceptual scheme, a third upon the necessity of sortal restriction, and the last upon the possibility of indefinite extendibility. The argument from semantic indeterminacy derives from general philosophical considerations concerning our understanding of language. For the Skolem–Lowenheim Theorem appears to show that an understanding of quanti- fication over (...) absolutely everything (assuming a suitably infinite domain) is semantically indistinguishable from the understanding of quantification over something less than absolutely everything; the same first-order sentences are true and even the same first-order conditions will be satisfied by objects from the narrower domain. From this it is then argued that the two kinds of understanding are indistinguishable tout court and that nothing could count as having the one kind of understanding as opposed to the other. (shrink)
Jonathan Kvanvig has argued that “objectual” understanding, i.e. the understanding we have of a large body of information, cannot be reduced to explanatory concepts. In this paper, I show that Kvanvig fails to establish this point, and then propose a framework for reducing objectual understanding to explanatory understanding.
The currently standard philosophical conception of existence makes a connection between three things: certain ways of talking about existence and being in natural language; certain natural language idioms of quantification; and the formal representation of these in logical languages. Thus a claim like ‘Prime numbers exist’ is treated as equivalent to ‘There is at least one prime number’ and this is in turn equivalent to ‘Some thing is a prime number’. The verb ‘exist’, the verb phrase ‘there is’ and (...) the quantifier ‘some’ are treated as all playing similar roles, and these roles are made explicit in the standard common formalization of all three sentences by a single formula of first-order logic: ‘(∃ x )[P( x ) & N( x )]’, where ‘P( x )’ abbreviates ‘ x is prime’ and ‘N( x )’ abbreviates ‘ x is a number’. The logical quantifier ‘∃’ accordingly symbolizes in context the role played by the English words ‘exists’, ‘some’ and ‘there is’. (shrink)
Duncan Pritchard (2008, 2009, 2010, forthcoming) has argued for an elegant solution to what have been called the value problems for knowledge at the forefront of recent literature on epistemic value. As Pritchard sees it, these problems dissolve once it is recognized that that it is understanding-why, not knowledge, that bears the distinctive epistemic value often (mistakenly) attributed to knowledge. A key element of Pritchard’s revisionist argument is the claim that understanding-why always involves what he calls strong cognitive achievement—viz., cognitive (...) achievement that consists always in either (i) the overcoming of a significant obstacle or (ii) the exercise of a significant level of cognitive ability. After outlining Pritchard’s argument, we show (contra Pritchard) that understanding-why does not essentially involve strong cognitive achievement. Interestingly, in the cases in which understanding-why is distinctively valuable, it is (we argue) only because there is sufficiently rich objectual understanding in the background. If that’s right, then a plausible revisionist solution to the value problems must be sensitive to different kinds of understanding and what makes them valuable, respectively. (shrink)
This paper argues that ‘that’-clauses are not singular terms (without denying that their semantical values are propositions). In its first part, three arguments are presented to support the thesis, two of which are defended against recent criticism. The two good arguments are based on the observation that substitution of ‘the proposition that p’ for ‘that p’ may result in ungrammaticality. The second part of the paper is devoted to a refutation of the main argument for the claim that ‘that’-clauses are (...) singular terms, namely that this claim is needed in order to account for the possibility of quantification into ‘that’-clause position. It is shown that not all quantification in natural languages is quantification into the position of singular terms, but that there is also so-called ‘non-nominal quantification’. A formal analysis of non-nominal quantification is given, and it is argued that quantification into ‘that’-clause position can be treated as another kind non-nominal quantification. (shrink)
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. (shrink)
I first show that most authors who developed Plural Quantification Logic (PQL) argued it could capture various features of natural language better than can other logic systems. I then show that it fails to do so: it radically departs from natural language in two of its essential features; namely, in distinguishing plural from singular quantification and in its use of an relation. Next, I sketch a different approach that is more adequate than PQL for capturing plural aspects of (...) natural language semantics and logic. I conclude with a criticism of the claim that PQL should replace natural language for specific philosophical or scientific purposes. (shrink)
When viewed as the most comprehensive theory of collections, set theory leaves no room for classes. But the vocabulary of classes, it is argued, provides us with compact and, sometimes, irreplaceable formulations of largecardinal hypotheses that are prominent in much very important and very interesting work in set theory. Fortunately, George Boolos has persuasively argued that plural quantification over the universe of all sets need not commit us to classes. This paper suggests that we retain the vocabulary of classes, (...) but explain that what appears to be singular reference to classes is, in fact, covert plural reference to sets. (shrink)
Alternative readings of quantification are considered. The absence of an unequivocal translation into ordinary speech is noted. Some examples are cited which, in the opinion of the author, are a result of equivocal readings of quantification, or unnecessarily restrictive readings which obscure its primary function.
Formal semantics has so far focused on three categories of quantifiers, to wit, Q-determiners (e.g. 'every'), Q-adverbs (e.g. 'always'), and Q-auxiliaries (e.g. 'would'). All three can be analyzed in terms of tripartite logical forms (LF). This paper presents evidence from verbs with distributive affixes (Q-verbs), in Kalaallisut, Polish, and Bininj Gun-wok, which cannot be analyzed in terms of tripartite LFs. It is argued that a Q-verb involves discourse reference to a distributive verbal dependency, i.e. an episode-valued function that sends different (...) semantic objects in a contextually salient plural domain to different episodes. (shrink)
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. (shrink)
Next SectionWe discuss the thesis formulated by Hintikka (1973) that certain natural language sentences require non-linear quantification to express their meaning. We investigate sentences with combinations of quantifiers similar to Hintikka's examples and propose a novel alternative reading expressible by linear formulae. This interpretation is based on linguistic and logical observations. We report on our experiments showing that people tend to interpret sentences similar to Hintikka sentence in a way consistent with our interpretation.
In “Descriptions as Predicates” (Graff 2001) I argued that definite and indefinite descriptions should be given a uniform semantic treatment as predicates rather than as quantifier phrases. The aim of the current paper is to clarify and elaborate one of the arguments for the descriptions-as-predicates view, one that concerns the interaction of descriptions with adverbs of quantification.
Truth is a stable, epistemically unconstrained property of propositions, and the concept of truth admits of a non-reductive explanation: that, in a nutshell, is the view for which I argued in Conceptions of Truth. In this paper I try to explain that explanation in a more detailed and, hopefully, more perspicuous way than I did in Ch. 6.2 of the book and to defend its use of sentential quantification against some of the criticisms it has has come in for.