Comment on Gollin and Rosser

Gollin and Rosser (Journal of Experimental Child Psychology, 1974, 17, 539–544) misunderstand the way in which we use the term mediation. We agree with them in the need to reject inferences about cognitive processes from what children do not do, and we also agree in the need for age × task experimental designs.     

Yablifying the Rosser Sentence

In a recent paper (Cie?liński and Urbaniak 2012), Urbaniak and Cie?liński describe an analogue of the Yablo Paradox, in the domain of formal provability. Just as the infinite sequence of Yablo sentences inherit the paradoxical behavior of the liar sentence, an infinite sequence of sentences can be constructed that inherit the distinctive behavior of the Gödel sentence. This phenomenon—the transfer of the properties of self-referential sentences of formal mathematics to their “unwindings” into infinite sequences of sentences—suggests a number of interesting logical questions. The purpose of this paper is to give a precise statement of a conjecture from Cie?liński and Urbaniak (2012) regarding the unwinding of the Rosser sentence, and to demonstrate that this precise statement is false. We begin with some preliminary motivation, introduce the conjecture against the background of some related results, and finally, in the last section, move on to the proof, which adapts a method used by Solovay and Guaspari.     

Rosser Provability and Normal Modal Logics

Studia Logica - In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose...     

Generic generalized Rosser fixed points

To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical (Solovay-type) completeness theorem with respect to PA is obtained for LR. This paper supersedes: Franco Montagna, Extremely undecidable sentences and generic generalized Rosser's fixed points, Rapporto Matematico, No. 95, Siena, 1983.     

Acquiring complex partial orderings in comparison with acquiring similar-sized linear orderings

The acquisition of a 14-term partial ordering was compared with the acquisition of a 14-term linear ordering. Learning the partial ordering was found to be more difficult because of two factors: (1) Subjects do not appear to have in their knowledge systems a prototype or rule for representing a long list of comparative relations as a partial ordering structure, and (2) the partial ordering must be presented so that some of the adjacently presented premises do not contain a common element. When these two factors were controlled, the partial ordering was as easy to learn as the linear ordering. It was also found that subjects learning the partial ordering showed no evidence of a serial position learning curve, whereas subjects learning the linear ordering showed some tendency to produce a serial position learning curve.     

Relationships between personality variables and bizarreness effects in free recall

Relationships between 6 personality variables and each of 3 different measures of recall for bizarre and common sentences were examined. The personality variables investigated included measures of sensation seeking, novelty experiencing, desire for novelty, arousal-seeking tendency, social potency, and conservatism. Recall was measured in terms of sentences accessed, target words recovered per accessed sentence, and misplaced target words. The results indicated the typical pattern of bizarreness effects on recall and significant relationships between personality variables and these effects. Arousal seeking and conservatism were positively related to a bizarreness advantage in sentences accessed. Additionally, high social potency was related to the recovery of more details from common than bizarre sentences, and high desire for novelty was related to a greater bizarre misplacement effect. The results are discussed in terms of orienting and defensive responses to bizarreness.     

Positional discriminability in linear orderings

Subjects were taught two eight-term linear orders of the form “A taller than B taller than C ….,” They were then asked to choose the “taller” term in all possible pairwise combinations within each series, and reaction time was measured for each pair. In addition, subjects performed a further task in which they judged whether or not two terms were adjacent in the ordering. In subsequent sessions, subjects were told that the “shortest” term on one list was taller than the “tallest” term on the other, so that the two lists were merged into a single 16-term series. They were then required to choose the “taller” term for both within-groups and between-groups pairs. Subjects did not appear to use the initial groupings in performing this task, even when given training on differential categorical codes (“tall” vs. “short”) for the two sublists. Rather, subjects in all tasks appeared to represent the items as ordered positions along an internal array, so that comparison times depended largely on the differential discriminability of the item positions. In each task decisions were made more quickly if the terms being compared were near the ends of the ordering, rather than near the middle.     

On L-decomposability of random orderings

In the setting of random orderings, we study conditional independence properties related to L-decomposability. We show that if a random ordering satisfies L-decomposability for any labelling of the ranks, then it is quasi-independent, provided the number of alternatives is at least 4, and each ordering has a positive probability.     

The effect of encoding variables on the free recall of words and action events

On the basis of previously determined properties, it was hypothesized that, whereas the free recall of words is a strategic memory test, the free recall of action events in the form of tasks performed by the subject is nonstrategic. This hypothesis was tested in two experiments. In Experiment 1, some presented items were designated as being more important to remember than others. Although this encoding manipulation had a marked effect on word recall, its effect on task recall was minimal In Experiment 2, the encoding impact of events was measured by having subjects rate, during presentation, how likely they were to recall each event on a subsequent test. The ratings showed good predictive power for the recall of words, but not of action events. These results were accepted as further support for a strategic/nonstrategic distinction between word and task recall.     

A coding of the countable linear orderings

Associate to any linear ordering on the integers the mapping whose value on n is the cardinality of {k<n; kn}: a purely combinatorial characterization for the mappings associated to the well-orderings is established.This work was partially supported by a CNRS grant PRC mathématiques and informatique.     

The effects of instructional variables on young children's organization and free recall

The relationship between verbal instruction and children's free recall was examined. Sixty first-grade children (mean age 84 months) were trained to categorize pictures of common objects. Half were given organizational instructions emphasizing categorization, and half were given instructions emphasizing the individual objects. Similarly, half were given detailed verbal instructions and encouraged to listen, and half were taught using a question type format that encouraged participation. Immediately and after a 2- to 3-week delay, children who were shown an organizational strategy categorized the materials during study and attained higher recall scores than children who were shown a specific strategy. After the 2- to 3-week delay, children in the explanation groups out-performed children in the question-asking groups. The child's competence as measured by intelligence and performance in school did not contribute to amount recalled when subjects were given instructions emphasizing the general strategy. However, children of high competence benefited more from specific training by recalling more items than did children of low competence. These findings point to the importance of instructional variables in studying changes in children's processing capabilities.     

On the existence of extremal cones and comparative probability orderings

We study the recently discovered phenomenon [Conder, M. D. E., & Slinko, A. M. (2004). A counterexample to Fishburn's conjecture. Journal of Mathematical Psychology, 48(6), 425-431] of existence of comparative probability orderings on finite sets that violate the Fishburn hypothesis [Fishburn, P. C. (1996). Finite linear qualitative probability. Journal of Mathematical Psychology, 40, 64-77; Fishburn, P. C. (1997). Failure of cancellation conditions for additive linear orders. Journal of Combinatorial Designs, 5, 353-365]—we call such orderings and the discrete cones associated with them extremal. Conder and Slinko constructed an extremal discrete cone on a set of n=7 elements and showed that no extremal cones exist on a set of n?6 elements. In this paper we construct an extremal cone on a finite set of prime cardinality p if p satisfies a certain number theoretical condition. This condition has been computationally checked to hold for 1725 of the 1842 primes between 132 and 16,000, hence for all these primes extremal cones exist.     

Making sense of (in)determinate truth: the semantics of free variables

It is argued that truth value of a sentence containing free variables in a context of use (or the truth value of the proposition it expresses in a context of use), just as the reference of the free variables concerned, depends on the assumptions and posits given by the context. However, context may under-determine the reference of a free variable and the truth value of sentences in which it occurs. It is argued that in such cases a free variable has indeterminate reference and a sentence in which it occurs may have indeterminate truth value. On letting, say, x be such that \(x^2=4\), the sentence ‘Either \(x=2\) or \(x=-2\)’ is true but the sentence ‘\(x=2\)’ has an indeterminate truth value: it is determinate that the variable x refers to either 2 or \(-2\), but it is indeterminate which of the two it refers to, as a result ‘\(x=2\)’ has a truth value but its truth value is indeterminate. The semantic indeterminacy is analysed in a ‘radically’ supervaluational (or plurivaluational) semantic framework closely analogous to the treatment of vagueness in McGee and McLaughlin (South J Philos 33:203–251, 1994, Linguist Philos 27:123–136, 2004) and Smith (Vagueness and degrees of truth, Oxford University Press, Oxford, 2008), which saves bivalence, the T-schema and the truth-functional analysis of the boolean connectives. It is shown that on such an analysis the modality ‘determinately’ is quite clearly not an epistemic modality, avoiding a potential objection raised by Williamson (Vagueness, Routledge, London, 1994) against such ‘radically’ supervaluational treatments of vagueness, and that determinate truth (rather than truth simpliciter) is the semantic value preserved in classically valid arguments. The analysis is contrasted with the epistemicist proposal of Breckenridge and Magidor (Philos Stud 158:377–400, 2012) which implies that (in the given context) ‘\(x=2\)’ has a determinate but unknowable truth value.     

The role of imagery in the representation of linear orderings

Two experiments were carried out to investigate whether visual imagery was used in representing transitive linear ordering relationships. Subjects were presented with passages describing either a linear ordering or a set inclusion relationship, while being subjected to either visual or verbal interference. Performance was tested by asking subjects to judge the truth or falsity of statements concerning both the information presented in the passages and inferences that could be drawn from this information. In neither experiment was there any evidence for the linear ordering material being selectively disrupted by the visual interference task, as would have been predicted by the imagery theory. Thus it is concluded that linear orderings are probably not represented as visual images.