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)
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)
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.
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)
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)
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)
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)
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)
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)
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.
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)
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)
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)
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)
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.
We 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.
Argues for a minimal level of quantification for the "proof beyond reasonable doubt" standard of criminal law: if a jury asks "Is 60% enough?", the answer should be "No.".
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.
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)
Standard first-order logic plus quantifiers of all finite orders (SFOL) 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 SFOL models. The typed structure of SFOL 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 SFOL 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 SFOL models. (shrink)
The paper reviews a number of approaches for handling restricted quantification in relevant logic, and proposes a novel one. This proceeds by introducing a novel kind of enthymematic conditional.
In “Descriptions as Predicates” (Fara 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-aspredicates view, one that concerns the interaction of descriptions with adverbs of quantification.
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)
Anderson-like ontological proofs, studied in this paper, employ contingent identity, free principles of quantification of the 1st order variables and classical principles of quantification of the 2nd order variables. All these theories are strongly complete wrt. classes of modal structures containing families of world-varying objectual domains of the 1st order and constant conceptual domains of the 2nd order. In such structures, terms of the 1st order receive only rigid extensions, which are elements of the union of all (...) 1st order domains. Terms of the 2nd order receive extensions and intensions. Given a family of preselected world-varying objectual domains of the 2nd order, non-rigid extensions of the 2nd order terms belong always to a preselected domain connected with a given world. Rigid intensions of the 2nd order terms are chosen from among members of a conceptual domain of the 2nd order, which is the set of all functions from the set of worlds to the union of all 2nd order preselected domains such that values of these functions at a given world belong to a preselected domain connected with this world. (shrink)
In the opening sections of this paper, we defined ambiguity in terms of distinct sentences (for a single sentence-string) with, in particular, distinct sets of truth conditions for the corresponding negative sentence-string. Lexical vagueness was defined as equivalent to disjunction, for under conditions of the negation of a sentence-string containing such an expression, all the relevant more specific interpretations of the string had also to be negated. Yet in the case of mixed quantification sentences, the strengthened, more specific, interpretations (...) of some such positive string are not all of them necessarily implied to be false if the corresponding negative sentence-string is asserted. On the contrary, as we saw in section 6, a negative sentence-string can be used to deny one of the more specific interpretations of the corresponding positive string without also denying other weaker interpretations of that same string. One might therefore argue that the only empirical evidence availble for assessing quantified sentences suggests clearly that these sentence-strings are ambiguous. Indeed logicians, many of whom restrict their attention to propositions, MUST recognise logical ambiguity at this point. For the contextualisation of the negative sentences in section 6 showed that it was possible to assert the falsity of some proposition P expressed by the sentence S while asserting a further proposition which was compatible with the truth of S. However the corresponding conclusion that such sentence-strings are sententially ambiguous is not a necessary conclusion for the linguist: for the alternative account of postulating a single semantic representation plus a set of semantic procedures is also compatible with the negation evidence. Moreover we have seen independent reasons for thinking that if sentential ambiguity is assumed to be in one-to-one correspondence with what we should now call logical ambiguity, a considerable body of generalisations is lost. For the maximal ambiguity account, it should be recalled, is committed to assigning at least thirteen distinct propositions and hence thirteen distinct sentence outputs for every sentence-string containing no more than two quantifiers, for three out of the four interpretations originally outlined in this paper can be understood with each numeral taken either in an ’exactly’ sense or in an ’at least’ sense. Moreover there is no explanation of why just these interpretations are available-they are merely an arbitrary list, no more connected than are the two interpretations of John saw her duck, with no reason to predict that the ambiguity would carry over from language to language. If then it is granted that an ambiguity account fails to capture appropriate generalisations, only two alternative accounts of mixed quantification sentence-strings remain viable-an analysis proposing an initial co-ordinate logical form like the logical form III, which is the strongest form compatible with each of the propositional interpretations of sentence-string Two examiners marked six scripts, and the radically weak form in which only existential quantification (both over sets and over members of those sets) is invoked. Since there are strong arguments to suggest that the procedures which both analyses require are semantic, there seems no reason not to adopt the radical vagueness account, with its considerably greater simplicity. (shrink)
The present paper is an attempt at the investigation of the nature of polarity contrast in natural languages. Truth conditions for natural language sentences are incomplete unless they include a proper definition of the conditions under which they are false. It is argued that the tertium non datur principle of classical bivalent logical systems is empirically invalid for natural languages: falsity cannot be equated with non-truth. Lacking a direct intuition about the conditions under which a sentence is false, we need (...) an independent foundation of the concept of falsity. The solution I offer is a definition of falsity in terms of the truth of a syntactic negation of the sentence. A definition of syntactic negation is proposed for English (Section 1). The considerations are applied to the analysis of definites in non-generic sentences and the analysis of generic indefinites. These two domains are investigated in breadth and some depth and the analyses compared and connected. During the discussion of non-generic predications with definite arguments and their respective negations (Section 2), a theory of predication is developed, basic to which is the distinction between integrative and summative predication. Summative predication, e.g., distributive plural, leads to contrary, all-or-no-thing, polarity contrasts due to the fundamental Presupposition of Indivisibility. Further-more, levels of predication are distinguished that are built up by various processes of constructing macropredications from lexical predicates. Given this analysis, particular (i.e., non-generic) quantification (Section 3) can be reanalyzed as an integrative, first-order form of predication that fills the truth-value gaps created by summative predication. The account comprises both nominal and adverbial quantification and relates quantification to the simpler types of predication discussed in Section 2. (shrink)
A broad-scale quantification of the measure of quality for scholarship is under way. This trend has fundamental implications for the future of academic publishing and employment. In this essay we want to raise questions about these burgeoning practices, particularly how they affect philosophy of education and similar sub-disciplines. First, details are given of how an ‘impact factor’ is calculated. The various meanings that can be attached to it are scrutinised. Second, we examine how impact factors are used to make (...) various ‘high stakes’ academic decisions, such as hiring and promotion, funding of research projects and how much money is to be awarded to a particular area. By focusing on a particular practice, problems with the application of the metric generally are outlined. Finally, we offer some general observations about the unintended consequences and other problems arising from the widespread use of this metric, including attempts to ‘game the system’. We argue that the use of impact factors increasingly shapes the kind of topics and issues scholars write on, their choices of methodology, and their choice of publication venues for their work. Technical measures and mechanisms tend to ‘colonise’ the qualitative and professional judgments that must also be part of the process of evaluation, and for which bibliometrics alone cannot offer a substitute. (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)
In biscuit conditionals (BCs) such as If you’re hungry, there’s pizza in the fridge, the if clause appears to apply to the illocutionary act performed in uttering the main clause, rather than to its propositional content. Accordingly, previous analyses of BCs have focused on illocutionary acts, and, this, I argue, leads them to yield incorrect paraphrases. I propose, instead, that BCs involve existential quantification over potential literal acts such as assertions, questions, commands, and exclamations, the semantic objects associated with (...) declarative, interrogative, imperative, and exclamative sentences, respectively. Such an existential interpretation of BCs requires only that we add potential literal acts to our inventory of individuals, and it produces reasonable paraphrases in which if has its normal meaning: If you’re hungry,[there’s a (relevant/salient) assertion that] there’s pizza in the fridge. These potential literal act variables are introduced into semantic interpretations and then undergo Existential Closure. Hence, we would expect to see similar interpretations in contexts other than BCs, that is, with other if constructions, with connectives other than if, with potential literal acts other than assertion, and in root sentences. This prediction is borne out, along with the parallel prediction that we cannot quantify over purely illocutionary acts like offers, but only over potential literal acts, those conventionally associated with a particular morphosyntactic shape. (shrink)
In Ockhamist branching-time logic [Prior 67], formulas are meant to be evaluated on a specified branch, or history, passing through the moment at hand. The linguistic counterpart of the manifoldness of future is a possibility operator which is read as `at some branch, or history (passing through the moment at hand)'. Both the bundled-trees semantics [Burgess 79] and the $\langle moment, history\rangle$ semantics [Thomason 84] for the possibility operator involve a quantification over sets of moments. The Ockhamist frames are (...) (3-modal) Kripke structures in which this second-order quantification is represented by a first-order quantification. The aim of the present paper is to investigate the notions of modal definability, validity, and axiomatizability concerning 3-modal frames which can be viewed as generalizations of Ockhamist frames. (shrink)
So far, we have focused on discourse reference to atomic individuals and specific times, events, and states. The basic point of the argument was that all types of discourse reference involve attention-guided anaphora (in the sense of Bittner 2012: Ch. 2). We now turn to discourses involving anaphora to and by quantificational expressions. Today, we focus on quantification over individuals but the analysis we develop will directly generalize to other semantic types. The basic idea is that quantification is (...) one more species of top-level anaphora--to wit, anaphora to top-ranked sets. (shrink)
The aim of this paper is to argue that update semantics is a natural framework for contextually restricted quantification, and to illustrate its use in the analysis of anaphoric definite descriptions and certain other anaphoric terms.
Those who want to interpret the quantifier ? (3 x) (. . .x. . .)'as having no existence commitment often fail to distinguish between this objective and that of merely changing the values of the variables. The confusion vitiates solutions of the singular existence anomalies which purport to be based on a non?existential interpretation of the quantifier. An example of one who makes the distinction but still interprets the particular quantifier non?existentially is offered by Czeslaw Lejewski. Objection to the classical (...) interpretation of the quantifiers often runs hand in hand with aversion to extensional logic. However, it is at least arguable that such an aversion is the result of underestimating the resources of extensional logic. These points arc discussed in the wake of Professor Marcus's recent paper in this journal ?Interpreting Quantification? (shrink)
In “Descriptions as Predicates” (Fara 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-aspredicates view, one that concerns the interaction of descriptions with adverbs of quantification.
Temporal logic is one of the many areas in which a possible world semantics is adopted. Prior's Ockhamist and Peircean semantics for branching-time, though, depart from the genuine Kripke semantics in that they involve a quantification over histories, which is a second-order quantification over sets of possible worlds. In the paper, variants of the original Prior's semantics will be considered and it will be shown that all of them can be viewed as first-order counterparts of the original semantics.
Within Linguistics the semantic analysis of natural languages (English, Swahili, for example) has drawn extensively on semantical concepts first formulated and studied within classical logic, principally first order logic. Nowhere has this contribution been more substantive than in the domain of quantification and variable binding. As studies of these notions in natural language have developed they have taken on a life of their own, resulting in refinements and generalizations of the classical quantifiers as well as the discovery of new (...) types of quantification which exceed the expressive capacity of the classical quantifiers. We refer the reader to Keenan and Westerståhl (1997) for an overview of results in this area. Here, we focus on one property of quantification in natural language?its inherently sortal nature?which distinguishes it from quantification in classical logic. (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.
The papers in this volume are updated versions of talks that were presented at the workshop QP structure, Nominalizations, and the role of DP that we organized at Saarland University, Germany, in December 2005. Although the connection between QP structure and definiteness, on the one hand, and nominalizations and definiteness on the other, were long observed in the literature, there has never been an attempt to bring the three together, and our aim at the workshop was to do exactly this: (...) to address recent developments in the area of quantifier phrase structure, nominalizations, and the linking definite determiner D. We invited discussions among the central approaches in syntax, morphology, semantics, and typology, paving the way towards a more comprehensive understanding of how quantification, definiteness and nominalizations are encoded in the grammar. (shrink)
Epistemic modal predicate logic raises conceptual problems not faced in the case of alethic modal predicate logic: Frege’s “Hesperus-Phosphorus” problem—how to make sense of ascribing to agents ignorance of necessarily true identity statements—and the related “Hintikka-Kripke” problem—how to set up a logical system combining epistemic and alethic modalities, as well as others problems, such as Quine’s “Double Vision” problem and problems of self-knowledge. In this paper, we lay out a philosophical approach to epistemic predicate logic, implemented formally in Melvin Fitting’s (...) First-Order Intensional Logic, that we argue solves these and other conceptual problems. Topics covered include: Quine on the “collapse” of modal distinctions; the rigidity of names; belief reports and unarticulated constituents; epistemic roles; counterfactual attitudes; representational vs. interpretational semantics; ignorance of co-reference vs. ignorance of identity; two-dimensional epistemic models; quantification into epistemic contexts; and an approach to multi-agent epistemic logic based on centered worlds and hybrid logic. (shrink)
While the control of cell migration by biochemical and biophysical factors is largely documented, a precise quantification of cell migration parameters in different experimental contexts is still questionable. Indeed, these phenomenological parameters can be evaluated from data obtained either at the cell population level or at the individual cell level. However, the range within which both characterizations of cell migration are equivalent remains unclear. We analyse here to which extent both sources of data could be integrated within a unified (...) description of cell migration by considering the motility of the endothelial cell line EAhy926. Using time-lapse video-microscopy and associated analysis of digital image time series, we quantified EAhy926 random motility coefficient, migration speed and trajectory persistence time in two different migration assays: the in vitro wound healing assay, and the cell-populated agarose drop assay. In order to analyse the agreement between independent quantifications of cell motility based either on individual cell analysis or cell population dynamic analysis, a theoretical multi-agents cellular model was developed and discussed as a possible theoretical framework able to unify these multi-scale data. Model simulations especially reveal the potential bias induced by cell proliferation and cell-cell adhesion when cell migration parameters are estimated from the extensively used in vitro wound healing assay. (shrink)
. 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)
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)
This article studies the monotonicity behavior of plural determinersthat quantify over collections. Following previous work, we describe thecollective interpretation of determiners such as all, some andmost using generalized quantifiers of a higher type that areobtained systematically by applying a type shifting operator to thestandard meanings of determiners in Generalized Quantifier Theory. Twoprocesses of counting and existential quantification thatappear with plural quantifiers are unified into a single determinerfitting operator, which, unlike previous proposals, both capturesexistential quantification with plural determiners and (...) respects theirmonotonicity properties. However, some previously unnoticed factsindicate that monotonicity of plural determiners is not always preservedwhen they apply to collective predicates. We show that the proposedoperator describes this behavior correctly, and characterize themonotonicity of the collective determiners it derives. It is proved thatdeterminer fitting always preserves monotonicity properties ofdeterminers in their second argument, but monotonicity in the firstargument of a determiner is preserved if and only if it is monotonic inthe same direction in the second argument. We argue that this asymmetryfollows from the conservativity of generalized quantifiers innatural language. (shrink)
The second printing of Principia Mathematica in 1925 offered Russell an occasion to assess some criticisms of the Principia and make some suggestions for possible improvements. In Appendix A, Russell offered *8 as a new quantification theory to replace *9 of the original text. As Russell explained in the new introduction to the second edition, the system of *8 sets out quantification theory without free variables. Unfortunately, the system has not been well understood. This paper shows that Russell (...) successfully antedates Quine's system of quantification theory without free variables. It is shown as well, that as with Quine's system, a slight modification yields a quantification theory inclusive of the empty domain. (shrink)
This paper shows that the semantics of shenme ‘what’ in Chinese bare conditionals may exhibit a phenomenon of double quantification. I argue that such double quantification can be nicely accounted for if one adopts Carlson's (1977a, b) semantics of bare plurals and verb meanings as well as the following two assumptions: (i) shenme ‘what’ can be a proform of bare NPs and hence has the same kind of denotation as bare NPs, and (ii) Chinese bare NPs are names (...) of kinds of things. This analysis of Chinese bare conditionals lends support to Carlson's approach to bare plurals despite Wilkinson's (1991) criticisms. I also show that an extension of Heim's (1987) analysis of what as ‘something of kind x’ to Chinese shenme ‘what’ encounters problems when shenme ‘what’ is a shared constituent of a predicate which applies to kinds and another predicate which applies to objects. (shrink)
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. (shrink)
The standard response is illustrated by E, J. Lemmon's claim that if all objects in a given universe had names and there were only finitely many of them, then we could always replace a universal proposition about that universe by a complex proposition. It is because these two requirements are not always met that we need universal quantification. This paper is partly in agreement with Lemmon and partly in disagreement. From the point of view of syntax and semantics we (...) can replace a universal proposition about any universe (finite or infinite, countable or uncountable) by a complex proposition (= sentence built up from atomic sentences and the connectives). But from the point of view of communication such a replacement is not possible if the universe is infinite. (shrink)
We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, φ, built up from propositional variables (p,q,r,...) and falsity $(\perp)$ using conjunction $(\wedge)$ , disjunction (∨) and implication (→). Write $\vdash\phi$ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula φ there exists a formula Apφ (effectively computable from φ), containing only variables not equal to p which occur in φ, and such that for (...) all formulas ψ not involving $p, \vdash \psi \rightarrow A_p\phi$ if and only if $\vdash \psi \rightarrow \phi$ . Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions. An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra. (shrink)
This influence of accent has been taken as evidence that adverbial quantification is focus sensitive (cf. Rooth (1985)) or presupposition sensitive (cf. von Fintel (1994), Rooth (1995)). I will discuss a problem that has been identified by von Fintel and Rooth, the requantifiation problem. Roughly stated, standard accounts of indefinites as NPs that introduce new discourse referents are at odds with standard accounts of the focus sensitivity or presupposition sensitivity of (1), which force us to assume that indefinites may (...) pick up existing discourse referents and “requantify” over them. I will argue for a special class of indefinites that pick up existing discourse referents, which I will call non-novel indefinites, to explain the nature.. (shrink)
This paper examines the quantification theory of *9 of Principia Mathematica. The focus of the discussion is not the philosophical role that section *9 plays in Principia's full ramified type-theory. Rather, the paper assesses the system of *9 as a quantificational theory for the ordinary predicate calculus. The quantifier-free part of the system of *9 is examined and some misunderstandings of it are corrected. A flaw in the system of *9 is discovered, but it is shown that with a (...) minor repair the system is semantically complete. Finally, the system is contrasted with the system of *8 of Principia's second edition. (shrink)
In ??28-31 of his Grundgesetze der Arithmetik, Frege forwards a demonstration that every correctly formed name of his formal language has a reference. Examination of this demonstration, it is here argued, reveals an incompleteness in a procedure of contextual definition. At the heart of this incompleteness is a difference between Frege?s criteria of referentiality and the possession of reference as it is ordinarily conceived. This difference relates to the distinction between objectual and substitutional quantification and Frege?s vacillation between (...) the two. (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)
This is part two of our discussion of discourses involving anaphora to and by quantificational expressions of various types. In part one (March 8), we focused on quantification over individuals ("Nominal quantification as top-level anaphora"). In part two (March 22-29), we show that the proposed analysis of quantification, as anaphoric discourse reference to top-ranked sets, automatically generalizes to temporal quantifiers (over times, events, or states).
We describe an extension to our quantifier-free computational logic to provide the expressive power and convenience of bounded quantifiers and partial functions. By quantifier we mean a formal construct which introduces a bound or indicial variable whose scope is some subexpression of the quantifier expression. A familiar quantifier is the Σ operator which sums the values of an expression over some range of values on the bound variable. Our method is to represent expressions of the logic as objects in the (...) logic, to define an interpreter for such expressions as a function in the logic, and then define quantifiers as "mapping functions." The novelty of our approach lies in the formalization of the interpreter and its interaction with the underlying logic. Our method has several advantages over other formal systems that provide quantifiers and partial functions in a logical setting. The most important advantage is that proofs not involving quantification or partial recursive functions are not complicated by such notions as "capturing," "bottom," or "continuity." Naturally enough, our formalization of the partial functions is nonconstructive. The theorem prover for the logic has been modified to support these new features. We describe the modifications. The system has proved many theorems that could not previously be stated in our logic. Among them are. (shrink)
Summary This paper will attempt to integrate (1) some new reflections on the implications for ontology of Monistic interpretations of formulae in quantification theory, with (2) a review of earlier material that I have published on such implications, and with (3) a sketch of several points made by others which bear on related issues.
Adding branching quantification to a first-order language increases the expressive power of the language,without adding to its ontology. The present paper is a defense of this claim against Quine (1970) and Patton (1991).
This book attempts to marry truth-conditional semantics with cognitive linguistics in the church of computational neuroscience. To this end, it examines the truth-conditional meanings of coordinators, quantifiers, and collective predicates as neurophysiological phenomena that are amenable to a neurocomputational analysis. Drawing inspiration from work on visual processing, and especially the simple/complex cell distinction in early vision (V1), we claim that a similar two-layer architecture is sufficient to learn the truth-conditional meanings of the logical coordinators and logical quantifiers. As a prerequisite, (...) much discussion is given over to what a neurologically plausible representation of the meanings of these items would look like. We eventually settle on a representation in terms of correlation, so that, for instance, the semantic input to the universal operators (e.g. and, all)is represented as maximally correlated, while the semantic input to the universal negative operators (e.g. nor, no)is represented as maximally anticorrelated. On the basis this representation, the hypothesis can be offered that the function of the logical operators is to extract an invariant feature from natural situations, that of degree of correlation between parts of the situation. This result sets up an elegant formal analogy to recent models of visual processing, which argue that the function of early vision is to reduce the redundancy inherent in natural images. Computational simulations are designed in which the logical operators are learned by associating their phonological form with some degree of correlation in the inputs, so that the overall function of the system is as a simple kind of pattern recognition. Several learning rules are assayed, especially those of the Hebbian sort, which are the ones with the most neurological support. Learning vector quantization (LVQ) is shown to be a perspicuous and efficient means of learning the patterns that are of interest. We draw a formal parallelism between the initial, competitive layer of LVQ and the simple cell layer in V1, and between the final, linear layer of LVQ and the complex cell layer in V1, in that the initial layers are both selective, while the final layers both generalize. It is also shown how the representations argued for can be used to draw the traditionally-recognized inferences arising from coordination and quantification, and why the inference of subalternacy breaks down for collective predicates. Finally, the analogies between early vision and the logical operators allow us to advance the claim of cognitive linguistics that language is not processed by proprietary algorithms, but rather by algorithms that are general to the entire brain. Thus in the debate between objectivist and experiential metaphysics, this book falls squarely into the camp of the latter. Yet it does so by means of a rigorous formal, mathematical, and neurological exposition – in contradiction of the experiential claim that formal analysis has no place in the understanding of cognition. To make our own counter-claim as explicit as possible, we present a sketch of the LVQ structure in terms of mereotopology, in which the initial layer of the network performs topological operations, while the final layer performs mereological operations. The book is meant to be self-contained, in the sense that it does not assume any prior knowledge of any of the many areas that are touched upon. It therefore contains mini-summaries of biological visual processing, especially the retinocortical and ventral /what?/ parvocellular pathways computational models of neural signaling, and in particular the reduction of the Hodgkin-Huxley equations to the connectionist and integrate-and-fire neurons Hebbian learning rules and the elaboration of learning vector quantization the linguistic pathway in the left hemisphere memory and the hippocampus truth-conditional vs. image-schematic semantics objectivist vs. experiential metaphysics and mereotopology. All of the simulations are implemented in MATLAB, and the code is available from the book’s website. • The discovery of several algorithmic similarities between visison and semantics. • The support of all of this by means of simulations, and the packaging of all of this in a coherent theoretical framework. (shrink)
We distinguish three different readings of the intuitionistic notions of validity, soundness, and completeness with respect to the quantification occurring in the notion of validity, and we establish certain relations between the different readings. For each of the meta-logical notions considered we suggest that the most natural reading (which is not the same for all cases) is precisely the one which is required by the recent intuitionistic completeness theorems for IPC.
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)
Traction forces developed by most cell types play a significant role in the spatial organisation of biological tissues. However, due to the complexity of cell-extracellular matrix interactions, these forces are quantitatively difficult to estimate without explicitly considering cell properties and extracellular mechanical matrix responses. Recent experimental devices elaborated for measuring cell traction on extracellular matrix use cell deposits on a piece of gel placed between one fixed and one moving holder. We formulate here a mathematical model describing the dynamic behaviour (...) of the cell-gel medium in such devices. This model is based on a mechanical force balance quantification of the gel visco-elastic response to the traction forces exerted by the diffusing cells. Thus, we theoretically analyzed and simulated the displacement of the free moving boundary of the system under various conditions for cells and gel concentrations. This modelis then used as the theoretical basis of an experimental device where endothelial cells are seeded on a rectangular biogel of fibrin cast between two floating holders, one fixed and the other linked to a force sensor. From a comparison of displacement of the gel moving boundary simulated by the model and the experimental data recorded from the moving holder displacement, the magnitude of the traction forces exerted by the endothelial cell on the fibrin gel was estimated for different experimental situations. Different analytical expressions for the cell traction term are proposed and the corresponding force quantifications are compared to the traction force measurements reported for various kind of cells with the use of similar or different experimental devices. (shrink)
In his Trust in Numbers: The Pursuit of Objectivity in Science and Public Life, Ted Porter asks how to account for the prestige and power of quantitative methods in the modern world. His answer involves two theses. One reverses a standard claim by asserting that quantification in basic sciences can often be driven by quantification in more applied areas such as government and business. The second thesis, which I call judgment replacement, asserts that quantification overcomes lack of (...) trust in humans by replacing human judgment in scientific communities and public life. Some aspects of the latter thesis are insightful and convincing. However, as a general claim, the judgment replacement thesis says that quantification and objectivity imply shallowness, superficiality and lack of subtlety. I examine one of Porter's key examples and show that as a general proposition the judgment replacement thesis gives a warped account of governmental decisions that involve a great deal of scientific input, an activity that colleagues and I have called mandated science. I show that Porter obfuscates the very features of mandated science that need the most clarification. The quantitative mentality can be superficial but it can also be complex and profound, and quantification can actually increase human judgment. The virtues of quantitative methods help account for their prestige and power. (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.
Some natural language expressions –namely, determiners like every, some, most, etc.— introduce quantification over individuals (or, in other words, they express relations between sets of individuals). For example, the truth conditions of a sentence like (1a) are represented in Predicate Logic (PrL) by binding the..
In this paper, a response to Ed Levy's discussion of medical quantification, I reflect on the ambitions of my book Trust in Numbers. I explore the idealized method of randomized clinical trials, revealed in his case study, as a social technology, one endowed with a persuasive scientific rationale but shaped also by political and social demands. The scholarly study of quantification requires not a choice between blind admiration and sweeping rejection, but a nuanced understanding. This should take into (...) account not only the cognitive aspects of science, but also its role in relation to institutions and customs, examined with some specificity. While history is narrowed and distorted when it is written to support a position on some present issue, historical and social studies of science should at least provide tools of criticism. For this, the historian of science must look beyond narrow communities of specialists, and seek a wider perspective on science as an administrative tool and a bearer of cultural and political values. (shrink)
In my paper, I present two competing perspectives on the foundational problem (as opposed to the descriptive problem) of quantifier domain restriction: the objective perspective on context (OPC) and the intentional perspective on context (IPC). According to OPC, the relevant domain for a quantified sentence is determined by objective facts of the context of utterance. In contrast, according to IPC, we must consider certain features of the speaker’s intention in order to determine the proposition expressed. My goal is to offer (...) a plausible and fair reconstruction of IPC. Drawing a parallel between quantifier domain restriction and standard cases of context dependence as indexicality, I argue that the speaker’s intentions can play a semantic role only if they satisfy an Availability Constraint: an intention must be made available or communicated to the addressee, and for that purpose the speaker can exploit any feature of the objective context (words, gestures, relevance or uniqueness of either the quantificational domain or of the referent in the context of utterance). An intention satisfying the Availability Constraint must be something that a hearer in normal circumstances is able to work out by relying on the physical surroundings of the utterance situation, on utterances exchanged during the previous conversation, and on background knowledge shared by speaker and addressee. (shrink)
Spade 1988 suggests that there are actually two theories to address this question to, an early one and a later one.[ii] Most of the present paper is a development of this idea. I suggest that early work by Sherwood and others was a study of quantifiers: their semantics and the effects of context on inferences that can be made from quantified terms. Later, in the hands of Burley and others, it changed into a study of something else, a study of (...) what I call global quantificational effect. In section 1, I explain what these two options are. (shrink)
This paper follows up a suggestion by Paul Vincent Spade that there were two Medieval theories of the modes of personal supposition. I suggest that early work by Sherwood and others was a study of quantifiers: their semantics and the effects of context on inferences that can be made from quantified terms. Later, in the hands of Burley and others, it changed into a study of something else, a study of what I call global quantificational effect. For example, although the (...) quantifier in ¬xPx is universal, it can be seen globally as having an existential effect; this is because the formula containing it is equivalent to x¬Px. The notion of global effect can be explained in terms of the modern theory of normal forms. I suggest that early authors were studying quantifiers, and the terminology of the theory of personal supposition is a classification of kinds of quantifiers. In this theory, to say that a term has distributive supposition is to say, roughly, that it is quantified by a universal quantifying sign. Later authors turned this into a theory of global quantificational effect. In the later theory, to say that a term has distributive supposition is to say that the overall effect is as if the term were universally quantified with a quantifier taking (relatively) wide scope. The difference between these two approaches is illustrated by the fact that the term man is classified as having distributive ("universal") supposition in Not every man is running in the earlier theory, whereas in the later theory that term does not have distributive supposition; it has determinate ("existential") supposition. In the paper I explain these options, and I argue from several texts that the earlier and later medieval theories actually worked like this. In an appendix I make further efforts to clarify the obscure early accounts, as well as the nineteenth century "doctrine of distribution". The last section of the paper discusses the "purpose(s)" of supposition theory. (shrink)
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. (shrink)
In this paper, I propose that the debate in epistemology concerning the nature and value of understanding can shed light on the role of scientific idealizations in producing scientific understanding. In philosophy of science, the received view seems to be that understanding is a species of knowledge. On this view, understanding is factive just as knowledge is, i.e., if S knows that p, then p is true. Epistemologists, however, distinguish between different kinds of understanding. Among epistemologists, there are those who (...) think that a certain kind of understanding—objectual understanding—is not factive, and those who think that objectual understanding is quasi-factive. Those who think that understanding is not factive argue that scientific idealizations constitute cognitive success, which we then consider as instances of understanding, and yet they are not true. This paper is an attempt to draw lessons from this debate as they pertain to the role of idealizations in producing scientific understanding. I argue that scientific understanding is quasi-factive. (shrink)
Necessitism is the view that necessarily everything is necessarily something; contingentism is the negation of necessitism. The dispute between them is reminiscent of, but clearer than, the more familiar one between possibilism and actualism. A mapping often used to ‘translate’ actualist discourse into possibilist discourse is adapted to map every sentence of a first-order modal language to a sentence the contingentist (but not the necessitist) may regard as equivalent to it but which is neutral in the dispute. This mapping enables (...) the necessitist to extract a ‘cash value’ from what the contingentist says. Similarly, a mapping often used to ‘translate’ possibilist discourse into actualist discourse is adapted to map every sentence of the language to a sentence the necessitist (but not the contingentist) may regard as equivalent to it but which is neutral in the dispute. This mapping enables the contingentist to extract a ‘cash value’ from what the necessitist says. Neither mapping is a translation in the usual sense, since necessitists and contingentists use the same language with the same meanings. The former mapping is extended to a second-order modal language under a plural interpretation of the second-order variables. It is proved that the latter mapping cannot be. Thus although the necessitist can extract a ‘cash value’ from what the contingentist says in the second-order language, the contingentist cannot extract a ‘cash value’ from some of what the necessitist says, even when it raises significant questions. This poses contingentism a serious challenge. (shrink)
Whether or not there are non-existent objects seems to be one of the more mysterious and speculative issues in ontology.1 To affirm that there are non-existent objects is to affirm that reality consists of two kinds of things, the existing and the non-existing. The existing contains all of what is in our space-time world, plus all abstract objects, if there are any. Most people, it seems fair to say, would think that this is all there is. For them the only (...) real question in ontology can be what kinds of existing things there are. However, followers of Meinong maintain that this isn’t all there is. There is also another kind of things, those that do not exist. And to say this, the Meinongians continue, is to accept that reality is divided into two basic kinds of things, the existing and the non-existing. Whether or not reality contains two basic categories of things, existing and non-existing, or only one, existing, is what the debate about non-existent objects is all about. And as such it seems to be the most speculative of the debates in ontology. How could we human beings possibly decide it? One might think that to find out whether or not there are abstract objects is hard to decide, since they are not in space and time, causally inaccessible, unobservable, etc.. But whatever difficulty there might be to answer the question whether or not there are abstract objects, it has to be even harder to decide whether or not there are non-existent objects. Abstract objects, if there are any, at least.. (shrink)
Quine said that the ontological question can be asked in three words, ‘What is there?’, and answered in one, ‘everything’. He was wrong. We need an extra word to ask the ontological question: it is ‘What is there, really?’; and it cannot be answered truthfully with ‘everything’ because there are some things that exist but which don’t really exist (and maybe even some things that really exist but which don’t exist).
This paper argues for the thesis that, roughly put, it is impossible to talk about absolutely everything. To put the thesis more precisely, there is a particular sense in which, as a matter of semantics, quantifiers always range over domains that are in principle extensible, and so cannot count as really being ‘absolutely everything’. The paper presents an argument for this thesis, and considers some important objections to the argument and to the formulation of the thesis. The paper also offers (...) an assessment of just how implausible the thesis really is. It argues that the intuitions against the thesis come down to a few special cases, which can be given special treatment. Finally, the paper considers some metaphysical ideas that might surround the thesis. Particularly, it might be maintained that an important variety of realism is incompatible with the thesis. The paper argues that this is not the case. (shrink)
This chapter begins with a discussion of Kant's theory of judgment-forms. It argues that it is not true in Kant's logic that assertoric or apodeictic judgments imply problematic ones, in the manner in which necessity and truth imply possibility in even the weakest systems of modern modal logic. The chapter then discusses theories of judgment-form after Kant, the theory of quantification, Frege's Begriffsschrift, C. I. Lewis and the beginnings of modern modal logic, the proof-theoretic approach to modal logic, possible (...) world semantics, correspondence theory, and modality and quantification. (shrink)
We hardly ever mean exactly what we say. I don’t mean that we generally speak figuratively or that we’re generally insincere. Rather, I mean that we generally speak loosely, omitting words that could have made what we meant more explicit and letting our audience fill in the gaps. Language works far more efficiently when we do that. Literalism can have its virtues, as when we’re drawing up a contract, programming a computer, or writing a philosophy paper, but we generally opt (...) for efficiency over explicitness. In.. (shrink)
