Consistent Truth

Modern Logic has generated a lot of problems for itself through inattention to natural forms of speech. In particular it has had difficulties with a large group of ‘logical paradoxes’ through its preoccupation with the Predicate Calculus and related structures to the exclusion of other formal structures that represent natural language more fully, and thereby escape these paradoxes. In natural speech the unrecognized forms involved are principally individual referring terms with a non‐specific or fictional reference. For, under the influence of the Logical Empiricists, the focus in formal logic has been on individual terms with a specific, factual reference. But a similar point arises with certain predicates, and indeed even the proper concept of a predicate has been lost, producing the most notorious of the standard puzzles: Russell's Paradox. The paper inevitably makes reference to contemporary logical symbolism in places, but its purpose is to show that the only way forward is to use a symbolism that is abbreviatory only, i.e. maps directly onto natural speech.     

Comparing pluralities

What does it mean to compare sets of objects along a scale, for example by saying “the men are taller than the women”? We explore comparison of pluralities in two experiments, eliciting comparison judgments while varying the properties of the members of each set. We find that a plurality is judged as “bigger” when the mean size of its members is larger than the mean size of the competing plurality. These results are incompatible with previous accounts, in which plural comparison is inferred from many instances of singular comparison between the members of the sets (Matushansky &; Ruys, 2006). Our results suggest the need for a type of predication that ascribes properties to plural entities, not just individuals, based on aggregate statistics of their members. More generally, these results support the idea that sets and their properties are actively represented as single units.     

Recollection is a continuous process: Evidence from plurality memory receiver operating characteristics

Is recollection a continuous/graded process or a threshold/all-or-none process? Receiver operating characteristic (ROC) analysis can answer this question as the continuous model and the threshold model predict curved and linear recollection ROCs, respectively. As memory for plurality, an item's previous singular or plural form, is assumed to rely on recollection, the nature of recollection can be investigated by evaluating plurality memory ROCs. The present study consisted of four experiments. During encoding, words (singular or plural) or objects (single/singular or duplicate/plural) were presented. During retrieval, old items with the same plurality or different plurality were presented. For each item, participants made a confidence rating ranging from “very sure old”, which was correct for same plurality items, to “very sure new”, which was correct for different plurality items. Each plurality memory ROC was the proportion of same versus different plurality items classified as “old” (i.e., hits versus false alarms). Chi-squared analysis revealed that all of the plurality memory ROCs were adequately fit by the continuous unequal variance model, whereas none of the ROCs were adequately fit by the two-high threshold model. These plurality memory ROC results indicate recollection is a continuous process, which complements previous source memory and associative memory ROC findings.     

To Be is to Be the Object of a Possible Act of Choice

Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference. Plural quantification will be then explained in terms of plural reference. As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom of infinite à la Dedekind, as the unique non-logical axiom. We will also sketch a virtual interpretation of the classical continuum involving no other infinite than a countable plurality of individuals.     

The class-inclusion task: Question form and distributive comparisons

The class-inclusion task is regarded by Piaget as a measure of the child's mastery of the structure of hierarchical classification. Class-inclusion was improved by changing the wording of the question to conform to standard English usage. A theoretical argument is offered that the child's difficulties with this task derive from confusion of collective comparisons, in which properties of classes are compared, and distributive comparisons, in which properties of elements are compared. A grammatical constraint on expression of distributive comparisons—an element of a class cannot be compared to an element of an included subclass—is hypothesized to be overgeneralized to expressions referring to collective comparisons such as the class-inclusion task. This hypothesis accounts for the improvement in class-inclusion performance with changes in wording of the question and for the finding that young children's responses to class-inclusion questions and to ungrammatical requests for comparison of an element of a class and an element of an included subclass are similar: the children respond readily but understand wrongly that the comparison involves coordinate classes.This work was supported by MH Grant 23505 and HD Grant 12821.     

Plural self-awareness

It has been claimed in the literature that collective intentionality and group attitudes presuppose some “sense of ‘us’” among the participants (other labels sometimes used are “sense of community,” “communal awareness,” “shared point of view,” or “we-perspective”). While this seems plausible enough on an intuitive level, little attention has been paid so far to the question of what the nature and role of this mysterious “sense of ‘us’” might be. This paper states (and argues for) the following five claims: (1) it is neither the case that the sense in question has the community (or “us”) in its content or as its object nor does the attitude in question presuppose a preexistent community (or “us”) as its subject. (2) The “sense of ‘us’” is plural pre-reflective self-awareness. (3) Plural pre-reflective self-awareness plays the same role in the constitution of a common mind that singular pre-reflective self-awareness plays in the individual mind. (4) The most important conceptions of plural subjects, collective persons, or group agents in the current literature fail to recognize the nature and role of plural self-awareness, and therefore fall short in important respects. (5) In spite of the striking similarities between the plural and the singular mind, there are important differences to consider. The authority of the singular first person point of view has no equivalent in the plural case.     

Is two a plural marker in early child language?

Is two ever a plural marker in child language? By some accounts, children bootstrap the distinction between the words one and two by observing their use with singular-plural marking (one ball/two balls). Others argue that the numeral two marks plurality before children begin using numerals to denote precise quantities. We tested the relation between numerals and singular-plural marking in English-speaking 2- and 3-year-olds by asking them to label sets of objects. When children were not prompted to use numerals they hardly ever did so, although they did frequently use plural marking. Thus, it does not appear that children spontaneously use numerals like two as plural markers. Also, children who used numerals when labeling sets were significantly more likely to use a plural marker than children who did not use numerals, suggesting that most children view plural marking as obligatory when numerals are used, rather than viewing the 2 forms as alternative markers of plurality. Finally, two was no more likely than other numerals to be used with unmarked nouns (i.e., as an alternative to the plural), suggesting that it does not have a special status as a plural marker. We conclude that two is not a plural marker in early child language.     

First–Person Plural Legislature: Political Reflexivity and Representation*

Abstract

In the Social Contract Rousseau gives what could be called a philosophical rule of recognition for law in Modernity: a law is law if and only if ‘the whole people rules over the whole people’. Thus, he defines self-legislation as, at bottom, collective intentional action. I will first map out the speech act structure [LEX] underlying self-legislation on this account. In particular, I argue for a first person plural counterpart of the reflexive structure inherent to intentions generally: the notion of a collective self. Then I take issue with Bratman's analysis of shared intentional activity in terms of mutuality, submitting that it misses out on the specifically political presupposition involved in ‘doing something together’. I will show why ‘mutuality’ requires representation of the unity of a polity, and how this representation can take form without either external authority or mutual responsiveness.     

Structural Injustice and Massively Shared Obligations

It is often argued that our obligations to address structural injustice are collective in character. But what exactly does it mean for ‘ordinary citizens’ to have collective obligations vis‐à‐vis large‐scale injustice? In this article, I propose to pay closer attention to the different kinds of collective action needed in addressing some of these structural injustices and the extent to which these are available to large, unorganised groups of people. I argue that large, dispersed, and unorganised groups of people are often in a position to perform distributive collective actions. As such, ordinary citizens can have massively shared obligations to address structural injustice through distributive action, but, ultimately, such obligations are ‘collective’ only in a fairly weak sense.     

Conceptual Factors Influence Children's Distributivity Bias

Sentences such as “The girls read a book” can lead to multiple interpretations: They could be reading the same book (collective) or they could each be reading their own book (distributive). In ambiguous contexts, adults prefer the collective interpretation, and preschool children show a slight bias for a distributive interpretation. The current research investigated whether conceptual factors (number of actors) influences interpretations of these kinds of predicates. The data with children show a stronger collective bias when there are four actors compared to two actors. This may be because children are more likely to conceptualize groups of four (vs. two) as sets rather than collections of individuals. These data are discussed with reference to research on visual and conceptual integration abilities, showing that children lag behind in their ability to combine components into a whole in comparison to parsing objects into their component parts.     

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.     

Reference to numbers in natural language

A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather ‘aspects’ of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted.     

One or more: psycholinguistic evidence for divergence of numerosity and grammatical number assignment.

Naturalistic slips of the tongue in Finnish were studied in order to reveal the production of grammatical number. It appears that speakers may consider plural forms referring to single events as singulars and collective nouns as plurals. These result in violations in number agreement. It was found that errors in numerosity are both structural and semantic in nature and that one type can be traced to the level of morphological encoding and the other type to message formulation.     

Improving the Accuracy of Eyewitnesses in the Presence of Misinformation with the Plurality Option

Misinformation has negative effects in the accuracy of eyewitnesses' reports. We investigated whether those negative effects could be reduced when participants are allowed to decide how many answers to include in their answers, that is, to regulate the plurality option. Participants watched a slideshow and received misinformation through a narrative. In the recognition memory test, participants had to select, out of five alternatives, one (single) and then three alternatives (plural answer) and to indicate which one they preferred to report if they were in a courtroom. Perceived likelihood ratings were also collected. Results showed that the regulation of the plurality option increased accuracy even in the presence of misinformation and highlighted the importance of the perceived likelihood ratings in the decision to select a single or plural answer. In general, the results suggest that better testimonies could be obtained if witnesses are given some control over their answers. Copyright © 2012 John Wiley & Sons, Ltd.     

Unintentional collective action

In this paper, I examine the manner in which analyses of the action of single agents have been pressed into service for constructing accounts of collective action. Specifically, I argue that the best analogy to collective action is a class of individual action that Carl Ginet has called ‘aggregate action.’ Furthermore, once we use aggregate action as a model of collective action, then we see that existing accounts of collective action have failed to accommodate an important class of (what I shall call) ‘unintentional collective actions.’     

Taking on semantic commitments, II: Collective versus distributive readings

In earlier work, Frazier and Rayner (1990) provided evidence for a processing principle termed the Minimal Semantic Commitment (MSC)hypothesis. In the present study, we used the MSC hypothesis as a starting point in addressing the issue of when to treat mental representations as vague versus determinate and ambiguous. Given ambiguous representations, the MSC hypothesis predicts that the processor will commit to one interpretation (the grammatical ambiguity hypothesis). On the other hand, given a single underspecified representation, the MSC hypothesis predicts that the processor will await disambiguating information before fully committing to an interpretation (the vagueness hypothesis). In an experiment designed to evaluate these hypotheses with respect to the representation of distributivity, participants' eye movements were recorded as they read sentences containing distributive or collective predicates that were either disambiguated by a preceding adverb or left locally ambiguous by delaying the disambiguating adverb until the end of the predicate. The results suggested that a semantic commitment is made in locally indeterminate cases as evidenced by a significant interaction of ambiguity and distributivity in first pass times, total times, and regressions. If the difficulty of distributives simply reflected the difficulty of postulating a distributive operator when evidence warranting it is encountered, then no interaction would be expected. Hence we argue that the distributive/collective distinction is treated as a matter of ambiguity rather than as one of vagueness. In the absence of evidence for a distributive reading, the processor commits itself to a collective reading sometime during the processing of the predicate (before the disambiguation in our late disambiguation examples). The findings are discussed in relation to recent linguistic work on the representation of distributivity.     

Regulation of memory accuracy with multiple answers: the plurality option

We report two experiments that investigated the regulation of memory accuracy with a new regulatory mechanism: the plurality option. This mechanism is closely related to the grain-size option but involves control over the number of alternatives contained in an answer rather than the quantitative boundaries of a single answer. Participants were presented with a slideshow depicting a robbery (Experiment 1) or a murder (Experiment 2), and their memory was tested with five-alternative multiple-choice questions. For each question, participants were asked to generate two answers: a single answer consisting of one alternative and a plural answer consisting of the single answer and two other alternatives. Each answer was rated for confidence (Experiment 1) or for the likelihood of being correct (Experiment 2), and one of the answers was selected for reporting. Results showed that participants used the plurality option to regulate accuracy, selecting single answers when their accuracy and confidence were high, but opting for plural answers when they were low. Although accuracy was higher for selected plural than for selected single answers, the opposite pattern was evident for confidence or likelihood ratings. This dissociation between confidence and accuracy for selected answers was the result of marked overconfidence in single answers coupled with underconfidence in plural answers. We hypothesize that these results can be attributed to overly dichotomous metacognitive beliefs about personal knowledge states that cause subjective confidence to be extreme.     

Can there be a feature-placing language?

The aim of this article is to argue against the real possibility of languages without subject-predicate structure, so-called feature-placing languages. They were first introduced by Strawson (1959/1990), later given formal expression through Quine's Predicate Functor Logic (Quine, 1960, Quine, 1971/Quine, 1976, Quine, 1992), and further elaboration in (Hawthorne & Cortens, 1995). I argue that, on the presumption that feature-placing languages are not mere notational variants on first-order languages, the idea of such languages is incoherent. The argument for this view rests on two highly plausible views about the conditions for the possibility of sentential meaning in natural languages. If accepted, these two views jointly imply that the sentences of feature-placing languages lack sufficient internal structure to meet the conditions for sentential meaning. Feature-placing languages cannot then count as genuine languages.