The Analytic Truth and Falsity of Disjunctions

Disjunctive inferences are difficult. According to the theory of mental models, it is because of the alternative possibilities to which disjunctions refer. Three experiments corroborated further predictions of the mental model theory. Participants judged that disjunctions, such as Either this year is a leap year or it is a common year are true. Given a disjunction such as Either A or B, they tended to evaluate the four cases in its ‘partition’: A and B, A and not‐B, not‐A and B, not‐A and not‐B, as ‘possible’ or ‘impossible’ in ways that bore out the difference between inclusive disjunctions (‘or both’) and exclusive disjunctions (‘but not both’). Knowledge usually concerns what is true, and so when participants judge that a disjunction is false, or contingent, and evaluate the cases in its partition, they depend on inferences that yield predictable errors. They tended to judge that disjunctions, such as follows: Either the food is cold or else it is tepid, but not both, are true, though in fact they could be false. They tended to infer ‘mirror‐image’ evaluations that yield the same possibilities for false disjunctions as those for true disjunctions. The article considers the implications of these results for alternative theories based on classical logic or on the probability calculus.     

Reasoning with realistic disjunctives

The research reported here investigated how people interpret and reason with disjunctive statements. Experiment 1 demonstrated a general preference for exclusive over inclusive interpretations, although this varied as a function of the context in which the statement occurred, and there were some contexts in which other, quite different, interpretations were possible. Experiments 2 and 3 showed that the logical inferences subjects would draw from disjunctive statements also varied with context. It is concluded that or permits a number of different interpretations, which are determined by contextual cues. Previous research may have underestimated disjunctive reasoning performance by using contexts that lead to non-logical responses; certainly in the present experiments reasoning was generally accurate given the interpretation adopted.     

The modulation of disjunctive assertions

The theory of mental models postulates that disjunctions of the sort, A or B, where A and B are sensible everyday clauses, have a core meaning that allows an inclusive interpretation, referring to three possibilities: A and not-B, not-A and B, and A and B. The meaning of the clauses and knowledge can modulate this meaning by blocking the construction of at least one model of a possibility—for example, “Rui is playing tennis or he is surfing” blocks the model of Rui doing both activities. This theory is implemented in a computer program. Three experiments investigated the core interpretation and interpretations in which the contents of the clauses should block the model of A and B (as in the preceding example), the model of A and not-B, or the model of not-A and B. In Experiment 1, the participants listed the possibilities for each of the four sorts of disjunction. The results corroborated the predicted modulations. In Experiment 2, these predicted interpretations governed the conclusions that participants accepted from disjunctions and categorical premises. In Experiment 3, the predicted interpretations yielded reliable effects on the conclusions that the participants drew for themselves. We relate these results to theories of reasoning.     

Making disjunctions exclusive

This work examines how people interpret the sentential connective "or", which can be viewed either inclusively (A or B or both) or exclusively (A or B but not both). Following up on prior work concerning quantifiers (Bott & Noveck, 2004; Noveck, 2001; Noveck & Posada, 2003), which shows that the common pragmatic interpretation of "some", some but not all, is conveyed as part of an effortful step, we investigate how extra effort applied to disjunctive statements leads to a pragmatic interpretation of "or", or but not both. Experiment 1 compelled participants to wait for three seconds before answering, hence giving them the opportunity to process the utterance more deeply. Experiments 2 and 3 emphasized "or", either by visual means ("OR") or by prosodic means (contrastive stress) as another way to encourage participants to apply more effort. Following a relevance-theoretic line of argument, we hypothesized that conditions encouraging more processing effort would give rise to more pragmatic inferences and hence to more exclusive interpretations of the disjunction. This prediction was confirmed in the three experiments.     

Making disjunctions exclusive

This work examines how people interpret the sentential connective “or”, which can be viewed either inclusively (A or B or both) or exclusively (A or B but not both). Following up on prior work concerning quantifiers (Bott & Noveck, 2004; Noveck, 2001; Noveck & Posada, 2003), which shows that the common pragmatic interpretation of “some”, some but not all, is conveyed as part of an effortful step, we investigate how extra effort applied to disjunctive statements leads to a pragmatic interpretation of “or”, or but not both. Experiment 1 compelled participants to wait for three seconds before answering, hence giving them the opportunity to process the utterance more deeply. Experiments 2 and 3 emphasized “or”, either by visual means (“OR”) or by prosodic means (contrastive stress) as another way to encourage participants to apply more effort. Following a relevance-theoretic line of argument, we hypothesized that conditions encouraging more processing effort would give rise to more pragmatic inferences and hence to more exclusive interpretations of the disjunction. This prediction was confirmed in the three experiments.     

Reasoning with exclusive disjunction arguments

An experiment was performed to determine the effects of negation in the major premise and/or conclusion of exclusive disjunction arguments on adults' abilities to apply two basic principles of exclusive disjunction. The component of the disjunctive premise that was referred to in the minor premise was varied systematically so that the influence of this factor could also be explored. The results indicated that arguments involving the confirmation of one of the components of the disjunctive premise followed by the denial of the other component of this premise were easier to evaluate than those in which this confirmation/denial sequence was reversed. In addition, arguments in which the disjunctive premise had two affirmative components were easier than those in which the disjunctive premise contained at least one negative component, but arguments with affirmative conclusions were more difficult than those with negative conclusions. Most importantly, an examination of subjects' errors and their response patterns revealed primitive response tendencies that paralleled those reported in previous studies of adults' propositional reasoning abilities.     

On Disjunctive Rights

This article examines the idea of disjunctive rights—an idea first suggested by Joel Feinberg and more recently advocated by Richard Arneson. Using a hypothetical scenario to bring forward a conflict between two rights that cannot be simultaneously fulfilled, the suggestion that the conflict can be solved by describing the right‐holders as holding disjunctive rights—rights that involve, in a significant way, a disjunction—is scrutinized. Several interpretations of the idea of disjunctive rights are examined from the perspectives of the interest theory and will theory of rights. Ultimately, the idea of disjunctive rights fails to provide an acceptable solution to the problem at hand, as each interpretation has unacceptable implications. This conclusion challenges the compossibilist thesis, according to which moral rights do not, ultimately, conflict. Alternatively, if one wishes to keep the possibility of compossibility and disjunctive rights, the mainstream theories of rights must be revised or rejected.     

Reasoning from connectives and relations between entities

This article reports investigations of inferences that depend both on connectives between clauses, such as or else, and on relations between entities, such as in the same place as. Participants made more valid inferences from biconditionals—for instance, Ann is taller than Beth if and only if Beth is taller than Cath—than from exclusive disjunctions (Exp. 1). They made more valid transitive inferences from a biconditional when a categorical premise affirmed rather than denied one of its clauses, but they made more valid transitive inferences from an exclusive disjunction when a categorical premise denied rather than affirmed one of its clauses (Exp. 2). From exclusive disjunctions, such as Either Ann is not in the same place as Beth or else Beth is not in the same place as Cath, individuals tended to infer that all three individuals could be in different places, whereas in fact this was impossible (Exps. 3a and 3b). The theory of mental models predicts all of these results.     

The consistency of disjunctive assertions

In two experiments, we established a new phenomenon in reasoning from disjunctions of the grammatical form either A or else B, where A and B are clauses. When individuals have to assess whether pairs of assertions can be true at the same time, they tend to focus on the truth of each clause of an exclusive disjunction (and ignore the concurrent falsity of the other clause). Hence, they succumb to illusions of consistency and of inconsistency with pairs consisting of a disjunction and a conjunction (Experiment 1), and with simpler problems consisting of pairs of disjunctions, such as eIther there is a pie or else there is a cake and Either there isn't a pie or else there is a cake (Experiment 2), that appear to be consistent with one another, but in fact are not. These results corroborate the theory that reasoning depends on envisaging models of possibilities.     

Effects of content on inclusive disjunction reasoning

An experiment is reported which establishes that inclusive disjunction arguments embedded in concrete content are not always easier to reason with than those involving abstract content. The subjects had to assess conclusions drawn from pairs of premises such as “Either Joan is intelligent or she is rich (or both); Joan is intelligent” or “Either Joan is intelligent or she is rich (or both); Joan is not intelligent”. The terms in the disjunctive premise were varied systematically across three content dimensions (i.e. compatible, abstract and contradictory). An analysis of variance revealed significant differences according to both principle of inference and type of content, and a significant interaction between these factors. The results demonstrated that semantically incompatible premise content had a marked influence on comprehension of inclusive disjunction reasoning schemes. The response patterns suggest that these reasoning schemes invited erroneous judgments based on other logical connectives.     

Propositional reasoning by model.

This article describes a new theory of propositional reasoning, that is, deductions depending on if, or, and, and not. The theory proposes that reasoning is a semantic process based on mental models. It assumes that people are able to maintain models of only a limited number of alternative states of affairs, and they accordingly use models representing as much information as possible in an implicit way. They represent a disjunctive proposition, such as "There is a circle or there is a triangle," by imagining initially 2 alternative possibilities: one in which there is a circle and the other in which there is a triangle. This representation can, if necessary, be fleshed out to yield an explicit representation of an exclusive or an inclusive disjunction. The theory elucidates all the robust phenomena of propositional reasoning. It also makes several novel predictions, which were corroborated by the results of 4 experiments.     

Similarity between hypotheses and evidence

We explore two novel consequences of similarity-based likelihood judgment. In Section I, we distinguish between the evidence on which judgments are based and the hypotheses that serve as the objects of judgment. The location of a feature, whether in the evidence or the hypotheses, influences the perceived similarity between evidence and hypotheses and consequently yields judgments that are inconsistent with the requirements of probability theory. In Section II, we examine judgment of disjunctive hypotheses. For certain types of disjunctions, the assessment of similarity produces consistent nonmonotonicities: the support of a disjunction is smaller than that of one of its components. Finally, we discuss the implications of our findings in terms of support theory and the principle of context independence.     

Two-year-olds' recognition of hierarchies: Evidence from their interpretation of the semantic relation between object labels

The present studies investigated 2-year-olds' recognition of hierarchies by examining under various conditions the relative frequency with which children interpreted novel and familiar labels for objects as mutually exclusive. Two-year-olds were taught novel labels in one of three ways: (a) inclusive input: “[novel label] is a kind of [familiar label]”; (b) exclusive input: “[novel label] is not a [familiar label]”; or (c) (Study 2) no relational input. The referents of the novel and familiar labels were taxonomically either: (a) strongly related (e.g., a fighter airplane and a passenger airplane), or (b) weakly related (e.g., a paint brush and a toothbrush). Children were less likely to interpret the labels as picking out mutually exclusive categories when: (a) the labels were introduced with the inclusive input; and (b) the referents of the labels were taxonomically strongly related. This modulation of mutual exclusivity interpretations in response to the various hierarchical relations instantiated in the stimuli and input provides evidence for 2-year-olds' emerging capacity to recognize hierarchical relations.     

The Proof by Cases Property and its Variants in Structural Consequence Relations

This paper is a contribution to the study of the rôle of disjunction in Abstract Algebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this paper), and to show several interesting applications allowed by the presence of a suitable disjunction in a given logic.     

Hypocrisy as Either Deception or Akrasia

The intuitive, folk concept of hypocrisy is not a unified moral category. While many theorists hold that all cases of hypocrisy involve some form of deception, I argue that this is not the case. Instead, I argue for a disjunctive account of hypocrisy whereby all cases of “hypocrisy” involve either the deceiving of others about the sincerity of an agent's beliefs or the lack of will to carry through with the demands of an agent's sincere beliefs. Thus, all cases of hypocrisy can be described either as cases of deception or as cases of akrasia. If this analysis correct, then I suggest further that the moral status of all instances of hypocrisy must be reduced either to the moral blameworthiness of deception or to the moral blameworthiness of akrasia. There can be no unified account of the moral wrongness of “hypocrisy” that holds across the disjunction.     

Typicality effects in logically defined categories

Subjects learned to categorize geometric designs by some rule combining two features, x and y. When xy stimuli (stimuli with both features) are always positive, the rule (concept) is inclusive disjunctive; when xy stimuli are always negative, the rule is exclusive disjunctive. In three other experimental conditions, the probability of xy stimuli being positive during acquisition was .25, .50, and .75. In a variety of postacquisition tests, xy stimuli were chosen as prototypical despite their less than consistent occurrence in the positive category. Stimuli with one relevant feature (x or y), although consistently assigned to the positive category, were often evaluated as poorer examples of the concept. These results are interpreted in terms of a schema model in which information acquired during learning is organized according to probability density functions over feature dimensions. This theory appears to be general enough to accommodate the evidence from laboratory studies of both logical and natural concepts.     

Socrates and Superiority

I propose an alternative interpretation of the Crito. The arguments that are typically taken to be Socrates' primary arguments against escape are actually supplementary arguments that rely on what I call the Superiority Thesis, the thesis that the state and its citizens are members of a moral hierarchy where those below are tied by bonds of obligation to those above. I provide evidence that Socrates holds this thesis, demonstrate how it resolves a number of apparent difficulties, and show why my interpretation is preferable to competing interpretations.     

对相容和不相容析取关系下析取句与条件句之间推理的比较

研究析取句和条件句之间的推理能检验推理心理学中的主要理论解释。一个实验考察了析取句和条件句中两个基本成分之间的关系(相容析取与不相容析取)和推理方向(由析取句推条件句与由条件句推析取句)对析取句与其等值的条件句之间推理的影响。结果发现,推理方向的影响取决于条件句和析取句中基本成分间的关系,在不相容析取关系下的推理成绩好于在相容析取关系下的推理成绩。现有的推理理论不能解释被试的整体反应模式,作者提出的特殊知识启发式与一般的句法意义对比启发式的竞争能统一解释被试的主导反应模式。     

Differences in Scope Assignments for Child and Adult Speakers of Mandarin

The present study used a Truth Value Judgment Task to investigate whether changes in sentence structure lead to corresponding changes in the assignment of scope relations by Mandarin-speaking children and adults. In one condition, participants were presented with ordinary negative sentences containing disjunction; this condition was designed to verify the existing claim that disjunction is a positive polarity item for adult speakers of Mandarin, but not for child speakers. In a second condition, participants were presented with negative sentences where the disjunction phrase was preposed from object position; this condition was designed to examine the extent to which changes in sentence structure can result in changes in scope assignments to negated disjunctions. The results indicate that the preposed disjunction phrase undergoes reconstruction for children, whereas reconstruction is blocked for adults. This finding also suggests that Mandarin-speaking children and adults exhibit different scope preferences for negated disjunctions, regardless of where the disjunction phrase appears in the surface syntax.     

Inference processes in learning well-defined concepts

In an adaptation of usual experimental techniques for studying the performance of a subject who has to discover a well-defined concept, two theories were independently tested. It was found that Bourne's set of inference operations (Bourne 1974; Salatas and Bourne 1974) predicted fairly well the relative difficulty for the connectives studied (inclusive disjunctive, alternative denial, joint denial, exclusive, exclusive disjunctive) for a stimulus population based on three-valued dimensions but not for a four-valued stimulus population. Reaction times gathered after the subject made no further classification errors fitted the decision-tree structures developed by Hunt et al. (1966). Inspection of the hypotheses subjects made about the rule, revealed two possible major reasoning obstacles: the introduction of a negative attribute and the notion of exception i.e. when the assignment of an attribute in one class depends on another attribute whereas in another class it does not.