Vagueness and Quantification

This paper deals with the question of what it is for a quantifier expression to be vague. First it draws a distinction between two senses in which quantifier expressions may be said to be vague, and provides an account of the distinction which rests on independently grounded assumptions. Then it suggests that, if some further assumptions are granted, the difference between the two senses considered can be represented at the formal level. Finally, it outlines some implications of the account provided which bear on three debated issues concerning quantification.     

Quantification and existence

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”.     

Quantification and Ontology

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.     

Quantification and Realism*

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.     

Quantification and semantic memory

Two models are considered for how people verify explicitly quantified sentences, such as All fathers are parents and Some fathers are parents. The models share the same second stage, but have different first-stage mechanisms. In the Predicate Intersection Model, suggested by Meyer (1970), the first stage involves a serial, self-terminating search among names of categories that intersect the predicate category. In the Feature Comparison Model (Smith, Shoben & Rips, 1974a), the first stage involves evaluating the overall relatedness between the subject and predicate categories by comparing their semantic features. To test the models, three reaction time experiments required subjects to verify statements quantified by Some or All. In the first experiment, the semantic relatedness of categories in false Some-statements was varied. Contrary to predictions of the Predicate Intersection Model, related categories increased reaction time for both true and false Some-statements. While the first experiment revealed that All-statements took longer to verify than comparable Some-statements, the second experiment demonstrated that the All-Some difference can be eliminated by presenting both statement types in the same block of trials, also disconfirming the Predicate Intersection Model. Finally, Experiment III examined the meaning of Some-statements in more detail by having subjects interpret the quantifier Some as “some but not all.” With this interpretation, Some-statements took longer to verify than All-statements. Overall the results support the Feature Comparison Model.     

The Tyranny of Taxonomy Sexuality and Anomaly

Human sexuality is not binary: this, although counter-intuitive initially, is a medical fact. Homo-sexuality was an anomaly under a M/F taxonomy, and so ‘unnatural’ and ‘an abomination’. It is a mere statistical anomaly: it is a fact of Nature, nevertheless. Doctrines of Natural Law must recognize that even if Nature is stable, the notion/word ‘Nature’ is a shifter. As medical and other sciences amend our understanding of Nature, the idea of ‘Nature’ shifts. Natural Law theory is – and must continue to be – based on Nature: the contents of the idea of ‘Nature’ change progressively. So must cultural attitudes.     

Truth and Generalized Quantification*

Kripke [1975] gives a formal theory of truth based on Kleene's strong evaluation scheme. It is probably the most important and influential that has yet been given—at least since Tarski. However, it has been argued that this theory has a problem with generalized quantifiers such as All(?, ψ)—that is, All ?s are ψ—or Most(?, ψ). Specifically, it has been argued that such quantifiers preclude the existence of just the sort of language that Kripke aims to deliver—one that contains its own truth predicate. In this paper I solve the problem by showing how Kleene's strong scheme, and Kripke's theory based on it, can in a natural way be extended to accommodate the full range of generalized quantifiers.     

Relative Necessity and Propositional Quantification

Following Smiley’s (The Journal of Symbolic Logic, 28, 113–134 1963) influential proposal, it has become standard practice to characterise notions of relative necessity in terms of simple strict conditionals. However, Humberstone (Reports on Mathematical Logic, 13, 33–42 1981) and others have highlighted various flaws with Smiley’s now standard account of relative necessity. In their recent article, Hale and Leech (Journal of Philosophical Logic, 46, 1–26 2017) propose a novel account of relative necessity designed to overcome the problems facing the standard account. Nevertheless, the current article argues that Hale & Leech’s account suffers from its own defects, some of which Hale & Leech are aware of but underplay. To supplement this criticism, the article offers an alternative account of relative necessity which overcomes these defects. This alternative account is developed in a quantified modal propositional logic and is shown model-theoretically to meet several desiderata of an account of relative necessity.

     

Quine on Quantification and Existence

Quine's justly famous paper "On What There Is" introduced a criterion of ontological commitment which has been almost universally accepted by analytic philosophers ever since.In this paper I try to unpack some of the substantive and controversial philosophical commitments that are presupposed by this criterion.The aim is not to show that the criterion is incorrect,but merely that it is not as obvious as it is taken to be by many,and that we might have reasons to explore alternative ways of thinking about ontological commitments.     

Quantification of Electroencephalography

In Experiment 1, a study by Messer was replicated with use of 18 children aged 40 to 67 months and nonsense words in which the phonemes of the consonant onsets of permissible and impermissible words were equated for frequency of occurrence. The children did not choose significantly more permissible words at any age, indicating that they did not use phonological rules as a basis for choice. In Experiment 2, similar materials were used with 14 children aged 56 to 74 months. By means of an operant conditioning procedure and a learning set paradigm, an experimental group (five children) was reinforced for choosing the permissible member of 10 permissible/impermissible pairs of nonsense words to a criterion of 80 percent correct responses. On transfer to a test task (16 problems given at a rate of 12 trials per problem) the group reached a mean performance level of 76 percent correct responses, which was significantly greater than that attained by either of two control groups (nine children). It was therefore concluded that some evidence for the existence of implicit phonological rules had been obtained, although the conclusion had to be qualified because (a) the mean level of performance of the experimental group during the test task was relatively low, and (b) because only children with high 1Qs performed at a level better than chance.     

Fregean Quantification Theory

Frege’s system of first-order logic is presented in a contemporary framework. The system described is distinguished by economy of expression and an unusual syntax.     

Relevant Restricted Quantification

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.     

Cut-Elimination and Quantification in Canonical Systems

Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a “canonical system” to first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with unary quantifiers and show that it remains constructive. Then we provide semantics for such canonical systems using 2-valued non-deterministic matrices extended to languages with quantifiers, and prove that the following properties are equivalent for a canonical system G: (1) G admits Cut-Elimination, (2) G is coherent, and (3) G has a characteristic 2-valued non-deterministic matrix.     

Anomaly Detection: Eye Movement Patterns

The symptom of a garden path in sentence processing is an apparent anomaly in the input string. This anomaly signals to the parser that an error has occurred, and provides cues for how to repair it. Anomaly detection is thus an important aspect of sentence processing. In the present study, we investigated how the parser responds to unambiguous sentences that contain syntactic anomalies and pragmatic anomalies, examining records of eye movement during reading. While sensitivity to the two kinds of anomaly was very rapid and essentially simultaneous, qualitative differences existed in the patterns of first-pass reading times and eye regressions. The results are compatible with the proposal that syntactic information and pragmatic information are used differently in garden-path recovery.