Arithmetic knowledge from the spontaneous attention to relations

Children's knowledge of arithmetic principles is a key aspect of early mathematics knowledge. Knowledge of arithmetic principles predicts how children approach solving arithmetic problems and the likelihood of their success. Prior work has begun to address how children might learn arithmetic principles in a classroom setting. Understanding of arithmetic principles involves understanding how numbers in arithmetic equations relate to another. For example, the Relation to Operands (RO) principle is that for subtracting natural numbers (A ? B = C), the difference (C) must be smaller than the minuend (A). In the current study we evaluate if individual differences in arithmetic principle knowledge (APK) can be predicted by the learners' spontaneous attention to relations (SAR) and if feedback can increase their attention to relations. Results suggest that participants’ Spontaneous Attention to Number (SAN) does not predict their knowledge of the RO principle for symbolic arithmetic. Feedback regarding the attention to relations did not show a significant effect on SAR or participants’ APK. We also did not find significant relations between reports of parent talk and the home environment with individual differences in SAN. The amount of parent's talk about relations was not significantly associated with learner's SAR and APK. We conclude that children's SAR with non‐symbolic number does not generalize to attention to relations with symbolic arithmetic.     

Is everyday causation deterministic or probabilistic?

One view of causation is deterministic: A causes B means that whenever A occurs, B occurs. An alternative view is that causation is probabilistic: the assertion means that given A, the probability of B is greater than some criterion, such as the probability of B given not-A. Evidence about the induction of causal relations cannot readily decide between these alternative accounts, and so we examined how people refute causal assertions. In four experiments most participants judged that a single counterexample of A and not-B refuted assertions of the form, A causes B. And, as a deterministic theory based on mental models predicted, participants were more likely to request multiple refutations for assertions of the form, A enables B. Similarly, refutations of the form not-A and B were more frequent for enabling than causal assertions. Causation in daily life seems to be a deterministic concept.     

Conformity of two sociometric relations

The problem of comparing two sociometric relations or measurements (A andB) recorded in distinct sociomatrices was originally discussed by Katz and Powell in the early 1950's and Hubert and Baker in the late 1970's. The problem is considered again using a probabilistic model designed specifically for discrete-valued network measurements. The model allows for the presence of various structural tendencies, such as reciprocity and differential popularity. A parameter that isolates the tendency for actors to choose other actors on both relations simultaneously is introduced, and estimated conditional on the presence of other parameters that reflect additional important network properties. The parameter is presented as a symmetric index but is also generalized to the predictive (A onB orB onA) situation. This approach to the problem is illustrated with the same data used by the earlier solutions, and the unique nature of the two relations in the data set (A = received choices,B = perceived choices), as it affects the modeling, is discussed. Significance tests for the parameter and related parameters are described, as well as an extension to more than two relations.Research support provided by National Science Grant #SES84-08626 to the University of Illinois at Urbana-Champaign. I am grateful to Dawn Iacobucci and Sheila Weaver for assistance with the research reported here, and to Carolyn Anderson, James Green, David Holtgrave, and four anonymous referees for comments on the paper.     

Against Probabilistic Measures of Coherence

It is shown that the probabilistic theories of coherence proposed up to now produce a number of counter-intuitive results. The last section provides some reasons for believing that no probabilistic measure will ever be able to adequately capture coherence. First, there can be no function whose arguments are nothing but tuples of probabilities, and which assigns different values to pairs of propositions {A, B} and {A, C} if A implies both B and C, or their negations, and if P(B)=P(C). But such sets may indeed differ in their degree of coherence. Second, coherence is sensitive to explanatory relations between the propositions in question. Explanation, however, can hardly be captured solely in terms of probability.     

Rough approximation of pairwise comparisons described by multi‐attribute stochastic dominance

Given a finite set A of actions evaluated by a set of attributes, preferential information is considered in the form of a pairwise comparison table including pairs of actions from subset B⊂A described by stochastic dominance relations on particular attributes and a total order on the decision attribute. Using a rough sets approach for the analysis of the subset of preference relations, a set of decision rules is obtained, and these are applied to a set A\B of potential actions. The rough sets approach of looking for the reduction of the set of attributes gives us the possibility of operating on a multi‐attribute stochastic dominance for a reduced number of attributes. Copyright © 1999 John Wiley & Sons, Ltd.     

Introspection,mindreading, and the transparency of belief

This paper explores the nature of self‐knowledge of beliefs by investigating the relationship between self‐knowledge of beliefs and one's knowledge of other people's beliefs. It introduces and defends a new account of self‐knowledge of beliefs according to which this type of knowledge is developmentally interconnected with and dependent on resources already used for acquiring knowledge of other people's beliefs, which is inferential in nature. But when these resources are applied to oneself, one attains and subsequently frequently uses a method for acquiring knowledge of beliefs that is non‐inferential in nature. The paper argues that this account is preferable to some of the most common empirically motivated theories of self‐knowledge of beliefs and explains the origin of the widely discussed phenomenon that our own beliefs are often transparent to us in that we can determine whether we believe that p simply by settling whether p is the case.     

Answer Sets and Qualitative Decision Making

Logic programs under answer set semantics have become popular as a knowledge representation formalism in Artificial Intelligence. In this paper we investigate the possibility of using answer sets for qualitative decision making. Our approach is based on an extension of the formalism, called logic programs with ordered disjunction (LPODs). These programs contain a new connective called ordered disjunction. The new connective allows us to represent alternative, ranked options for problem solutions in the heads of rules: A × B intuitively means: if possible A, but if A is not possible then at least B. The semantics of logic programs with ordered disjunction is based on a preference relation on answer sets. We show that LPODs can serve as a basis for qualitative decision making.     

On Permutation in Simplified Semantics

This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Restall, J Philos Logic 22(5):481–511, 1993) concerning the modelling conditions for the axioms of assertion A → ((A → B) → B) (there called c6) and permutation (A → (B → C)) → (B → (A → C)) (there called c7). We show that the modelling conditions for assertion and permutation proposed in ‘Simplified Semantics’ overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the “base point” is inconsistent. This problem is not restricted to ‘Simplified Semantics.’ The techniques of that paper are used in Graham Priest’s textbook An Introduction to Non-Classical Logic (Priest, 2001), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose two different corrections.     

Are students' beliefs about knowledge and learning associated with their reported use of learning strategies?

Background. Although considerable research has examined beliefs and learning outcomes (e.g. Schommer, 1990, 1993a, 1993b ; Schommer & Dunnell, 1997 ), little has looked at the relationship between beliefs and the actual learning process. Aims. This research examines the relationship between beliefs about learning and knowledge, and reports of learning strategy‐use relevant for successful text comprehension. Sample. Participants were 81 Norwegian university students who had studied from 1 to 4 years in a range of disciplines. Method. Students' beliefs about knowledge and learning were measured with the Schommer Epistemological Questionnaire (SEQ; Schommer, 1998b ). Learning strategies particularly useful for text‐based learning were measured with the Motivated Strategies for Learning Questionnaire (MSLQ; Pintrich, Smith, Garcia, & McKeachie, 1991 ). A correlational analysis between measures and full regression analyses of how beliefs influence strategy selection were performed. Results. Beliefs about how thoroughly knowledge is integrated in networks (simple) and how fixed the ability to learn is from birth (fixed) contributed significantly to reported strategy use: Simple to rehearsal and organizational strategies, fixed to elaboration and critical thinking strategies, and a combination of simple and fixed to strategies relevant to the thoughtful monitoring of learning tasks. Beliefs about how certain knowledge is (certain) and how quickly learning can be expected to occur (quick) were not found to contribute to reported learning‐ strategy use in any significant way. Conclusion. Some, but not all, beliefs about knowledge and learning offer insight into students' reported use of learning strategies relevant for reading course literature.     

Marital Status and Health Beliefs: Different Relations for Men and Women

Although relations between marital status and health have been substantiated, the results of relatively few studies suggest how or why marriage is associated with health. To understand how marriage and health are associated, this study was designed to examine the role of health beliefs. Two thousand two hundred and six (2,206) adults who participated in the New Jersey Family Health Survey provided information about their marital status, proactive health beliefs, and proactive health behaviors. Results indicated that being married (vs. single) was positively associated with men's proactive health beliefs, whereas marriage did not appear to influence women's proactive health beliefs positively. Significant relations between participants' reports of proactive health beliefs and proactive health behaviors were found. Findings are discussed in terms of the importance of understanding the complex nature of associations between social relationships and health.     

Replacement in Logic

We study a range of issues connected with the idea of replacing one formula by another in a fixed (linguistic) context. The replacement core of a consequence relation ? is the relation holding between a set of formulas {A ₁, ..., A _m , ...} and a formula B when for every context C(·), we have C(A ₁), ..., C(A _m ), ...???C(B). Section 1 looks at some differences between which inferences are lost on passing to the replacement cores of the classical and intuitionistic consequence relations. For example, we find that while the inference from A and B to $A \land B$ , sanctioned by both these initial consequence relations, is retained on passage to the replacement core in the classical case, it is lost in the intuitionistic case. Further discussion of these two (and some other) logics occupies Sections 3 and 4. Section 2 looks at the m?=?1 case, describing A as replaceable by B according to ? when B is a consequence of A by the replacement core of ?, and inquiring as to which choices of ? render this induced replaceability relation symmetric. Section 5 investigates further conceptual refinements— such as a contrast between horizontal and vertical replaceability—suggested by some work of R. B. Angell and R. Harrop (and a comment on the latter by T. J. Smiley) in the 1950s and 1960s. Appendix 1 examines a related aspect of term-for-term replacement in connection with identity in predicate logic. Appendix 2 is a repository for proofs which would otherwise clutter up Section 3.     

Distinct Pathways From Parental Beliefs and Practices to Children’s Numeric Skills

It is imperative to identify contextual factors contributing to the development of early math skills, considering their role in later academic achievement. To pursue this goal, the present study investigated the paths connecting parental beliefs and practices during the preschool years to children’s numeric skills at the end of kindergarten (N = 98). Results were consistent with theoretical predictions of specific relations between particular types of parental input and different aspects of number knowledge. Direct math learning activities mediated the relation between parental beliefs and children’s number identification skills. Daily activities involving quantitative components mediated the relation between parental beliefs and children’s numerical magnitude understanding. Both types of activities predicted arithmetic skills that integrate the basic aspects of symbolic number knowledge. These findings contribute to developmental theory by specifying how characteristics of children’s environments are related to particular aspects of their development, which is critical for informing intervention work to improve early math skills.     

Inference,explanation, and asymmetry

Explanation is asymmetric: if A explains B, then B does not explain A. Traditionally, the asymmetry of explanation was thought to favor causal accounts of explanation over their rivals, such as those that take explanations to be inferences. In this paper, we develop a new inferential approach to explanation that outperforms causal approaches in accounting for the asymmetry of explanation.

     

Warrant without truth?

This paper advances the debate over the question whether false beliefs may nevertheless have warrant, the property that yields knowledge when conjoined with true belief. The paper’s first main part—which spans Sections 2–4—assesses the best argument for Warrant Infallibilism, the view that only true beliefs can have warrant. I show that this argument’s key premise conflicts with an extremely plausible claim about warrant. Sections 5–6 constitute the paper’s second main part. Section 5 presents an overlooked puzzle about warrant, and uses that puzzle to generate a new argument for Warrant Fallibilism, the view that false beliefs can have warrant. Section 6 evaluates this pro-Fallibilism argument, finding ultimately that it defeats itself in a surprising way. I conclude that neither Infallibilism nor Fallibilism should now constrain theorizing about warrant.     

Recovery Recovered

The most controversial condition that the AGM theory of rational belief change places on belief contraction is the recovery condition. The condition is controversial because of a series of putative counterexamples due (separately) to I. Levi and S. O. Hansson. In this paper we show that the conflicts that Levi and Hansson arrange between AGM contraction and our intuitions about how to give up beliefs are merely apparent. We argue that these conflicts disappear once we attend more closely to the identification of the beliefs contracted away in each counterexample case. Since, on our view, speakers" belief contraction intentions are often more complicated than they may first appear, we are led to develop apparatus for thinking about the communication and identification of those intentions. Our argument refocuses attention on the difficult question of how to apply the AGM theory to particular cases.     

The larger the cause,the larger the effect: evidence of speed judgment biases in causal scenarios

ABSTRACT

When two motions appear to be causally related, the spatiotemporal features of motions are sometimes distorted in order to increase the consistency with causal impressions. Here, in four experiments, we tested if varying the speed of an object A could affect the judged speed of an object B that appeared to be causally related to A. Participants were presented with classic launching stimuli (Experiment 1), a variant of launching stimuli in which A could move with uniformly accelerated or decelerated motion (Experiment 2), non-launching stimuli that elicited a causal impression (Experiment 3), and stimuli showing a three-object launching event (Experiment 4). Main results showed that the judged speed of B was systematically biased towards the speed of A, and moreover that the judged speed of B depended on the average speed of A, rather than on the speed of A at the moment of collision as it would be predicted by Newtonian mechanics. The results are consistent with the hypothesis that internal representations of causal events based on property transmission (for instance, impetus) can affect judgments of the low-level properties of causal scenarios.     

Levi on Causal Decision Theory and the Possibility of Predicting One's Own Actions

Isaac Levi has long criticized causal decisiontheory on the grounds that it requiresdeliberating agents to make predictions abouttheir own actions. A rational agent cannot, heclaims, see herself as free to choose an actwhile simultaneously making a prediction abouther likelihood of performing it. Levi is wrongon both points. First, nothing in causaldecision theory forces agents to makepredictions about their own acts. Second,Levi's arguments for the ``deliberation crowdsout prediction thesis' rely on a flawed modelof the measurement of belief. Moreover, theability of agents to adopt beliefs about theirown acts during deliberation is essentialto any plausible account of human agency andfreedom. Though these beliefs play no part inthe rationalization of actions, they arerequired to account for the causalgenesis of behavior. To explain the causes ofactions we must recognize that (a) an agentcannot see herself as entirely free in thematter of A unless she believes herdecision to perform A will cause A,and (b) she cannot come to a deliberatedecision about A unless she adoptsbeliefs about her decisions. FollowingElizabeth Anscombe and David Velleman, I arguethat an agent's beliefs about her own decisionsare self-fulfilling, and that this can beused to explain away the seeming paradoxicalfeatures of act probabilities.     

A Nelsonian Response to ‘the Most Embarrassing of All Twelfth-century Arguments’

Alberic of Paris put forward an argument, ‘the most embarrassing of all twelfth-century arguments’ according to Christopher Martin, which shows that the connexive principles contradict some other logical principles that have become deeply entrenched in our most widely accepted logical theories. Building upon some of Everett Nelson’s ideas, we will show that the steps in Alberic of Paris’ argument that should be rejected are precisely the ones that presuppose the validity of schemas that are nowadays taken as some of the most trivial logical truths: (A∧B) →_AB A and (A∧B) →_AB B, i.e. Simplification.     

Betting on transitivity in probabilistic causal chains

Causal reasoning is crucial to people’s decision making in probabilistic environments. It may rely directly on data about covariation between variables (correspondence) or on inferences based on reasonable constraints if larger causal models are constructed based on local relations (coherence). For causal chains an often assumed constraint is transitivity. For probabilistic causal relations, mismatches between such transitive inferences and direct empirical evidence may lead to distortions of empirical evidence. Previous work has shown that people may use the generative local causal relations A → B and B → C to infer a positive indirect relation between events A and C, despite data showing that these events are actually independent (von Sydow et al. in Proceedings of the thirty-first annual conference of the cognitive science society. Cognitive Science Society, Austin, 2009, Proceedings of the 32nd annual conference of the cognitive science society. Cognitive Science Society, Austin, 2010, Mem Cogn 44(3):469–487, 2016). Here we used a sequential learning scenario to investigate how transitive reasoning in intransitive situations with negatively related distal events may relate to betting behavior. In three experiments participants bet as if they were influenced by a transitivity assumption, even when the data strongly contradicted transitivity.     

Effect of information on reducing inappropriate expectations and requests for antibiotics

People often expect antibiotics when they are clinically inappropriate (e.g., for viral infections). This contributes significantly to physicians’ decisions to prescribe antibiotics when they are clinically inappropriate, causing harm to the individual and to society. In two pre-registered studies employing UK general population samples (n₁ = 402; n₂ = 190), we evaluated the relationship between knowledge and beliefs with antibiotic expectations, and the effects of information provision on such expectations. We conducted a correlational study (study 1), in which we examined the role of antibiotic knowledge and beliefs and an experiment (study 2) in which we assessed the causal effect of information provision on antibiotic expectations. In study 1, we found that both knowledge and beliefs about antibiotics predicted antibiotic expectations. In study 2, a 2 (viral information: present vs. absent) × 2 (antibiotic information: present vs. absent) experimental between-subjects design, information about antibiotic efficacy significantly reduced expectations for antibiotics, but viral aetiology information did not. Providing antibiotic information substantially diminishes inappropriate expectations of antibiotics. Health campaigns might also aim to change social attitudes and normative beliefs, since more complex sociocognitive processes underpin inappropriate expectations for antibiotics.