The representation of negative numbers: exploring the effects of mode of processing and notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.     

The representation of negative numbers: Exploring the effects of mode of processing and notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.     

The adverbial theory of numbers: some clarifications

In a forthcoming paper in this journal, entitled “Bad company objection to Joongol Kim’s adverbial theory of numbers”, Namjoong Kim presents an ingenious Russell-style paradox based on an analogue of Kim’s definition of the number 1, and argues that Kim’s theory needs to provide a criterion of demarcation between acceptable and unacceptable definitions of adverbial entities. This paper addresses this ‘bad company’ objection and some other related issues concerning Kim’s adverbial theory by clarifying the purposes and uses of the formal framework (ALA) of the theory.

     

Belief revision,epistemic conditionals and the Ramsey test

Epistemic conditionals have often been thought to satisfy the Ramsey test (RT): If A, then B is acceptable in a belief state G if and only if B should be accepted upon revising G with A. But as Peter Gärdenfors has shown, RT conflicts with the intuitively plausible condition of Preservation on belief revision. We investigate what happens if (a) RT is retained while Preservation is weakened, or (b) vice versa. We also generalize Gärdenfors' approach by treating belief revision as a relation rather than as a function.In our semantic approach, the same relation is used to model belief revision and to give truth-conditions for conditionals. The approach validates a weak version of the Ramsey Test (WRR) — essentially, a restriction of RT to maximally consistent belief states.We prove that alternatives (a) and (b) are both consistent, but argue that (b) is philosophically more promising. Gärdenfors' belief-revision axioms are compatible with WRR together with RT from left to right: the only direction of the test that is defensible on intuitive grounds.An earlier version of this paper was presented at the conference on the dynamics of knowledge and belief at Lund University, 24–26 August 1989. We wish to thank Sven Danielsson, Peter Gärdenfors, Sören Halldén, David Makinson, Hugh Mellor, Michael Morreau, Nils-Eric Sahlin and Brian Skyrms for their very helpful suggestions and remarks. We are also grateful for thought-provoking criticism and comments from two anonymous referees.     

Moore-paradoxical belief, conscious belief and the epistemic Ramsey test

Chalmers and Hájek argue that on an epistemic reading of Ramsey??s test for the rational acceptability of conditionals, it is faulty. They claim that applying the test to each of a certain pair of conditionals requires one to think that one is omniscient or infallible, unless one forms irrational Moore-paradoxical beliefs. I show that this claim is false. The epistemic Ramsey test is indeed faulty. Applying it requires that one think of anyone as all-believing and if one is rational, to think of anyone as infallible-if-rational. But this is not because of Moore-paradoxical beliefs. Rather it is because applying the test requires a certain supposition about conscious belief. It is important to understand the nature of this supposition.     

Orienting to counterpredictive gaze and arrow cues

In separate experiments, counterpredictive arrow, eye gaze, or abrupt-onset cues were used to test the hypothesis that individual differences in voluntary control influence involuntary orienting. In contrast with previous findings (Friesen, Ristic, & Kingstone, 2004), involuntary orienting effects were found for arrow cues.Furthermore, for both eye gaze and arrow cues, individual differences in voluntarycontrol were associatedwith involuntary orienting: Involuntary orienting effects were larger for participants who were more effective at using the cue to reorient attention, and also for participants who reported greater ability to control attention. Orienting to abrupt-onset cues was not correlated to individual differences in self-reported attentional control. The findings show that eye gaze and arrow cues instigate similar involuntary and voluntary effects and that involuntary orienting to symbolic cues is linked to individual differences in voluntary control.     

An introduction to state notation and SKED

SKED is a minicomputer operating system based on the functional use of state notation as a programming language. An overall view of the operating system, state notation, and the hardware necessary to simultaneously control as many as 12 experimental stations is provided.     

张行成先天数学初探--三论中国数学派

南宋易学家张行成是继邵雍之后的又一位数学派哲学家,在"象"与"数"的内涵和关系问题上,认为"象"为卦爻象,"数"为奇偶数;先数后象,由数生象。在"理"与"数"问题上,主张数生于理。提出"理数"的概念,以"理数"为宇宙万物的本体。通过对"太极"的阐释,进一步表明"数本论"思想。认为"太极"是兼包理气、虚实、动静之"数"。通过《易》《玄》的比较说明"先天图"为浑天象,"太玄图"为盖天象。先天图实际上反映的是先天数。由"象"而推及"数",由"数"而推及"理",由"理数"而说明天道规律和宇宙本体,是张行成象数哲学的最大特色。     

Arbitrary reference,numbers, and propositions

Reductionist realist accounts of certain entities, such as the natural numbers and propositions, have been taken to be fatally undermined by what we may call the problem of arbitrary identification. The problem is that there are multiple and equally adequate reductions of the natural numbers to sets (see Benacerraf, 1965), as well as of propositions to unstructured or structured entities (see, e.g., Bealer, 1998; King, Soames, & Speaks, 2014; Melia, 1992). This paper sets out to solve the problem by canvassing what we may call the arbitrary reference strategy. The main claims of such strategy are 2. First, we do not know which objects are the referents of proposition and numerical terms since their reference is fixed arbitrarily. Second, our ignorance of which object is picked out as the referent does not entail that no object is referred to by the relevant expression. Different articulations of the strategy are assessed, and a new one is defended.     

Attentional effects of counterpredictive gaze and arrow cues

The authors used counterpredictive cues to examine reflexive and volitional orienting to eyes and arrows. Experiment 1 investigated the effects of eyes with a novel design that allowed for a comparison of gazed-at (cued) target locations and likely (predicted) target locations against baseline locations that were not cued and not predicted. Attention shifted reflexively to the cued location and volitionally to the predicted location, and these 2 forms of orienting overlapped in time. Experiment 2 discovered that another well-learned directional stimulus, an arrow, produced a different effect: Attention was shifted only volitionally to the predicted location. The authors suggest that because there is a neural architecture specialized for processing eyes, gaze-triggered attention is more strongly reflexive than orienting to arrows.     

Gaze and arrow distractors influence saccade trajectories similarly

Perceiving someone's averted eye-gaze is thought to result in an automatic shift of attention and in the preparation of an oculomotor response in the direction of perceived gaze. Although gaze cues have been regarded as being special in this respect, recent studies have found evidence for automatic attention shifts with nonsocial stimuli, such as arrow cues. Here, we directly compared the effects of social and nonsocial cues on eye movement preparation by examining the modulation of saccade trajectories made in the presence of eye-gaze, arrows, or peripheral distractors. At a short stimulus onset asynchrony (SOA) between the distractor and the target, saccades deviated towards the direction of centrally presented arrow distractors, but away from the peripheral distractors. No significant trajectory deviations were found for gaze distractors. At the longer SOA, saccades deviated away from the direction of the distractor for all three distractor types, but deviations were smaller for the centrally presented gaze and arrow distractors. These effects were independent of whether line-drawings or photos of faces were used and could not be explained by differences in the spatial properties of the peripheral distractor. The results suggest that all three types of distractors (gaze, arrow, peripheral) can induce the automatic programming of an eye movement. Moreover, the findings suggest that gaze and arrow distractors affect oculomotor preparation similarly, whereas peripheral distractors, which are classically regarded as eliciting an automatic shift of attention and an oculomotor response, induce a stronger and faster acting influence on response preparation and the corresponding inhibition of that response.