The Logic and Meaning of Plurals. Part II

In this sequel to “The logic and meaning of plurals. Part I”, I continue to present an account of logic and language that acknowledges limitations of singular constructions of natural languages and recognizes plural constructions as their peers. To this end, I present a non-reductive account of plural constructions that results from the conception of plurals as devices for talking about the many. In this paper, I give an informal semantics of plurals, formulate a formal characterization of truth for the regimented languages that results from augmenting elementary languages with refinements of basic plural constructions of natural languages, and account for the logic of plural constructions by characterizing the logic of those regimented languages.

The Logic and Meaning of Plurals. Part I

Contemporary accounts of logic and language cannot give proper treatments of plural constructions of natural languages. They assume that plural constructions are redundant devices used to abbreviate singular constructions. This paper and its sequel, “The logic and meaning of plurals, II”, aim to develop an account of logic and language that acknowledges limitations of singular constructions and recognizes plural constructions as their peers. To do so, the papers present natural accounts of the logic and meaning of plural constructions that result from the view that plural constructions are, by and large, devices for talking about many things (as such). The account of logic presented in the papers surpasses contemporary Fregean accounts in its scope. This extension of the scope of logic results from extending the range of languages that logic can directly relate to. Underlying the view of language that makes room for this is a perspective on reality that locates in the world what plural constructions can relate to. The papers suggest that reflections on plural constructions point to a broader framework for understanding logic, language, and reality that can replace the contemporary Fregean framework as this has replaced its Aristotelian ancestor.     

Modality effects in compounding with English inflectional morphology

The treatment of plural morphemes in English noun–noun compounds is significant because it provides a test case for competing theories of language acquisition and representation. Even when the first noun in a compound refers to plural items, native speakers frequently use the singular form ( Murphy, 2000 ). Sometimes, they will use the irregular plural form (‘mice chaser’) but very rarely are regular plurals (‘rats chaser’) used as the first noun in a compound. This effect has been found with native English‐ speaking children ( Gordon, 1985 ; Nicoladis, 2000 ; Oetting & Rice, 1993 ; van der Lely & Christian, 2000 ); native English‐speaking teenagers ( van der Lely & Christian, 2000 ); and native English‐speaking adults ( Lardiere & Schwartz, 1997 ; Murphy, 2000 ). The apparent dissociation between regular and irregular plurals (i.e. that irregular plurals are included before a second noun but regular plurals are almost never included before a second noun) is thought to be due to innate morphological constraints ( Marcus, Brinkmann, Clahsen, Weise, & Pinker, 1995 ). Such constraints predict that all items of regular morphology should be treated differently from all items of irregular morphology by language users in all situations. However, if external factors such as input and response modality affect the number of plurals included in compounds, then this questions the internal constraint‐based explanations of compounding and encourages investigation of how external factors might influence the number of plurals included in compounds.     

Plural quantifiers: a modal interpretation

One of the standard views on plural quantification is that its use commits one to the existence of abstract objects–sets. On this view claims like ‘some logicians admire only each other’ involve ineliminable quantification over subsets of a salient domain. The main motivation for this view is that plural quantification has to be given some sort of semantics, and among the two main candidates—substitutional and set-theoretic—only the latter can provide the language of plurals with the desired expressive power (given that the nominalist seems committed to the assumption that there can be at most countably many names). To counter this approach I develop a modal-substitutional semantics of plural quantification (on which plural variables, roughly speaking, range over ways names could be) and argue for its nominalistic acceptability.     

An analysis of individual differences in generalization between receptive and productive language in retarded children

Retarded subjects were taught generative pluralization rules concurrently in both the receptive and productive modalities of language. Receptive training established correct pointing to either one or a pair of objects, in response to a spoken singular or plural label of the object(s); productive training established correct spoken labels of one or a pair of objects presented visually. However, these pluralization rules were established in each modality only for a specific class of plurals: those ending in -s for one modality, those ending in -es for the other modality. This training was successful in establishing generative, or rule-governed behaviors, such that untrained examples of singulars and plurals were usually responded to correctly. Nevertheless, despite this concurrent, generative behavior, probes revealed little generalization between modalities: three of four subjects did not generalize clearly from receptive training with one class of plurals to correct productive use of that class, nor did they generalize from productive training of the other class of plurals to correct receptive response to that class. The fourth subject, however, did show strong generalization of both these types. It was concluded that automatic generalization between receptive and productive language is not necessarily an inevitable result of language training in such subjects, and therefore may require explicit, if temporary, programming, such as by direct reinforcement.     

How to Lewis a Kripke–Hintikka

It has been argued that a combination of game-theoretic semantics and independence-friendly (IF) languages can provide a novel approach to the conceptual foundations of mathematics and the sciences. I introduce and motivate an IF first-order modal language endowed with a game-theoretic semantics of perfect information. The resulting interpretive independence-friendly logic (IIF) allows to formulate some basic model-theoretic notions that are inexpressible in the ordinary quantified modal logic. Moreover, I argue that some key concepts of Kripke’s new theory of reference are adequately modeled within IIF. Finally, I compare the logic IIF to David Lewis counterpart theory, drawing some morals concerning the interrelation between metaphysical and semantic issues in possible-world semantics.     

Mass Nouns in a Logic of Classes as Many

A semantic analysis of mass nouns is given in terms of a logic of classes as many. In previous work it was shown that plural reference and predication for count nouns can be interpreted within this logic of classes as many in terms of the subclasses of the classes that are the extensions of those count nouns. A brief review of that account of plurals is given here and it is then shown how the same kind of interpretation can also be given for mass nouns.     

Regular and irregular inflection in the acquisition of German noun plurals.

In this paper we study the acquisition of German noun plurals in relation to the question of how children represent regular and irregular inflection. Pinker and Prince (1992) have demonstrated several dissociations between regular and irregular inflection in the English past tense system. However, in English, the default status of -ed is confounded with its high frequency; therefore inflectional systems other than English past tense formation must be examined. The noun plural system in German is particularly interesting, because most nouns have irregular plurals in German and the regular (default) plural is less frequent than several of the irregular plurals. Thus it is unclear how a language learner determines whether German even has a regular plural, and if so what form it takes. Based on longitudinal data from impaired and unimpaired monolingual German-speaking children, we find a striking, statistically significant correlation: plural affixes that are used in overregularizations, namely -n or -s, are left out within compounds. This correlation shows that even impaired children are sensitive to the distinction between regular and irregular morphology. We propose a linguistic analysis of the correlation in terms of Kiparsky's (1982, 1985) level-ordering model plus an additional ordering condition on affixes: default (regular) affixes cannot serve as input to compounding processes.     

Learning language from the input: why innate constraints can't explain noun compounding

Do the production and interpretation of patterns of plural forms in noun-noun compounds reveal the workings of innate constraints that govern morphological processing? The results of previous studies on compounding have been taken to support a number of important theoretical claims: first, that there are fundamental differences in the way that children and adults learn and process regular and irregular plurals, second, that these differences reflect formal constraints that govern the way the way regular and irregular plurals are processed in language, and third, that these constraints are unlikely to be the product of learning. In a series of seven experiments, we critically assess the evidence that is cited in support of these arguments. The results of our experiments provide little support for the idea that substantively different factors govern the patterns of acquisition, production and interpretation patterns of regular and irregular plural forms in compounds. Once frequency differences between regular and irregular plurals are accounted for, we find no evidence of any qualitative difference in the patterns of interpretation and production of regular and irregular plural nouns in compounds, in either adults or children. Accordingly, we suggest that the pattern of acquisition of both regular and irregular plurals in compounds is consistent with a simple account, in which children learn the conventions that govern plural compounding using evidence that is readily available in the distribution patterns of adult speech.     

Composition as a Kind of Identity

Composition as identity, as I understand it, is a theory of the composite structure of reality. The theory’s underlying logic is irreducibly plural; its fundamental primitive is a generalized identity relation that takes either plural or singular arguments. Strong versions of the theory that incorporate a generalized version of the indiscernibility of identicals are incompatible with the framework of plural logic, and should be rejected. Weak versions of the theory that are based on the idea that composition is merely analogous to (one-one) identity are too weak to be interesting, lacking in metaphysical consequence. I defend a moderate version according to which composition is a kind of identity, and argue that the difference is metaphysically substantial, not merely terminological. I then consider whether the notion of generalized identity, though fundamental, can be elucidated in modal terms by reverse engineering Hume’s Dictum. Unfortunately, for realists about possible worlds, such as myself, who understand Hume’s Dictum in terms of duplicates, the elucidation never gets off the ground; for modal primitivists, the elucidation may succeed in capturing some general notion of identity, but it’s characterization is too general to target the particular notion of identity, tied to classical mereology, that is the object of my theorizing. In the end, I have little to offer the skeptic who claims not to understand the relevant notion of generalized identity, or not to take it to be compulsory in providing an account of the structure of reality.     

What "mice trap" tells us about the mental lexicon

Level ordering has proven inadequate as a morphological theory, leaving unexplained the experimental results taken to support it as a component of innate grammar-young children's acceptance of irregular plurals in English compounds. The present study demonstrates that these results can be explained by slower access to the grammatically preferred singulars of irregular nouns when compounds are created on-line from plural stimuli. Experiments on English noun-noun compound production and on production of either singular or plural forms from the same or opposite form confirmed that more irregular than regular plurals were used in compounds, and showed that producing irregular singulars from plurals was slower than producing regular singulars. Plural responses were also slower when cue and required response number differed.     

Does Science Influence the Logic we Ought to Use: A Reflection on the Quantum Logic Controversy

In this article I argue that there is a sense in which logic is empirical, and hence open to influence from science. One of the roles of logic is the modelling and extending of natural language reasoning. It does so by providing a formal system which succeeds in modelling the structure of a paradigmatic set of our natural language inferences and which then permits us to extend this structure to novel cases with relative ease. In choosing the best system of those that succeed in this, we seek certain virtues of such structures such as simplicity and naturalness (which will be explained). Science can influence logic by bringing us, as in the case of quantum mechanics, to make natural language inferences about new kinds of systems and thereby extend the set of paradigmatic cases that our formal logic ought to model as simply and naturally as possible. This can alter which structures ought to be used to provide semantics for such models. I show why such a revolution could have led us to reject one logic for another through explaining why complex claims about quantum mechanical systems failed to lead us to reject classical logic for quantum logic.     

走向模型论的模态逻辑

增加特定的基数量词,扩张一阶语言,就可以导致实质性地增强语言的表达能力,这样许多超出一阶逻辑范围的数学概念就能得到处理。由于在模型的层次上基本模态逻辑可以看作一阶逻辑的互模拟不变片断,显然它不能处理这些数学概念。因此,增加说明后继状态类上基数概念的模态词,原则上我们就能以模态的方式处理所有基数。我们把讨论各种模型论逻辑的方式转移到模态方面。     

From Modal Discourse to Possible Worlds

The possible-worlds semantics for modality says that a sentence is possibly true if it is true in some possible world. Given classical prepositional logic, one can easily prove that every consistent set of propositions can be embedded in a ‘maximal consistent set’, which in a sense represents a possible world. However the construction depends on the fact that standard modal logics are finitary, and it seems false that an infinite collection of sets of sentences each finite subset of which is intuitively ‘possible’ in natural language has the property that the whole set is possible. The argument of the paper is that the principles needed to shew that natural language possibility sentences involve quantification over worlds are analogous to those used in infinitary modal logic.     

What “Mice Trap” tells us about the mental lexicon

Level ordering has proven inadequate as a morphological theory, leaving unexplained the experimental results taken to support it as a component of innate grammar—young children’s acceptance of irregular plurals in English compounds. The present study demonstrates that these results can be explained by slower access to the grammatically preferred singulars of irregular nouns when compounds are created on-line from plural stimuli. Experiments on English noun–noun compound production and on production of either singular or plural forms from the same or opposite form confirmed that more irregular than regular plurals were used in compounds, and showed that producing irregular singulars from plurals was slower than producing regular singulars. Plural responses were also slower when cue and required response number differed.     

Noun Imageability Facilitates the Acquisition of Plurals: Survival Analysis of Plural Emergence in Children

Some research in child language suggests that semantically general verbs appear in grammatical structures earlier than semantically complex, specific ones. The present study examines whether this was the case in nouns, using imageability as a proxy measure of semantic generality. Longitudinal corpus data from 12 children from the Manchester corpus in CHILDES were used to obtain information on the first occurrence of plurals. A total of 3,560 uninflected nouns were identified in the corpora, of which 1,202 were observed in the plural. Survival analyses indicated that the chance of observing a plural form increases with the imageability rating of the noun, even after accounting for the age of acquisition of the uninflected noun, maternal input frequency, and word length. Noun imageability thus facilitates the acquisition of plural forms. This finding contradicts the observations from verbs, and indicates that the acquisition of grammar is facilitated by high imageability rather than semantic generality.     

Sets and Plural Comprehension

The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. This leads to paradoxes analogous to those that afflict naïve set theory. Here I suggest that they are the very same paradoxes, because sets can be reduced to states of affairs. However, to obtain a consistent theoretical reduction we must restrict the usual axiom scheme of Comprehension for plural logic to ‘stratified’ formulas, to avoid viciously circular definitions. I prove that with this modification to the background plural logic, the theory of states of affairs is consistent; moreover, it yields the axioms of the familiar set theory NFU.     

A logic for epistemic two-dimensional semantics

Epistemic two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. While this theory is usually presented in an informal manner, I take some steps in formalizing it in this paper. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that captures the relevant ideas of epistemic two-dimensional semantics. I also describe some properties of the logic that are interesting from a philosophical perspective, and apply it to the so-called nesting problem.     

A functional analysis of receptive language and productive speech: acquisition of the plural morpheme

Operant conditioning procedures were applied to two retardates to establish receptive auditory plurals: correct pointing to single or paired objects was reinforced after hearing singular or plural labels. This training proceeded until an errorless (generative) criterion of correct performance was achieved. Unreinforced probes measuring expressive use of singulars and plurals were interspersed in this receptive training. Neither subject generalized from this receptive training to expressive plurals, in that each used singulars when labeling pairs. Then, both subjects were directly trained in conventional expressive plurals to an errorless (generative) criterion. The previous design was then repeated, but the receptive repertoire was reversed: pointing at pairs in response to singular labels was reinforced, and vice-versa. Unreinforced probes of expressive plural usage again showed its independence of the current receptive repertoire in that conventional (unreversed) plural usage was displayed. Thus, the independence of the expressive repertoire (even when unreinforced) from the reinforced patterns of the receptive repertoire was demonstrated.