Finite axiomatization for some intermediate logics

LetN. be the set of all natural numbers (except zero), and letD _n ^* = {k ∈N ∶k|n} ∪ {0} wherek¦n if and only ifn=k.x f or somex∈N. Then, an ordered setD _n ^* = 〈D _n ^* , ? _n , wherex? _ny iffx¦y for anyx, y∈D _n ^* , can easily be seen to be a pseudo-boolean algebra. In [5], V.A. Jankov has proved that the class of algebras {D _n ^* ∶n∈B}, whereB =,{k ∈N∶ ? \(\mathop \exists \limits_{n \in N} \) (n > 1 ≧n ² k)is finitely axiomatizable. The present paper aims at showing that the class of all algebras {D _n ^* ∶n∈B} is also finitely axiomatizable. First, we prove that an intermediate logic defined as follows: $$LD = Cn(INT \cup \{ p_3 \vee [p_3 \to (p_1 \to p_2 ) \vee (p_2 \to p_1 )]\} )$$ finitely approximatizable. Then, defining, after Kripke, a model as a non-empty ordered setH = 〈K, ?〉, and making use of the set of formulas true in this model, we show that any finite strongly compact pseudo-boolean algebra ? is identical with. the set of formulas true in the Kripke modelH _B = 〈P(?), ?〉 (whereP(?) stands for the family of all prime filters in the algebra ?). Furthermore, the concept of a structure of divisors is defined, and the structure is shown to beH D _n ^* = 〈P (D _n ^* ), ?〉for anyn∈N. Finally, it is proved that for any strongly compact pseudo-boolean algebraU satisfying the axiomp ₃∨ [p ₃→(p₁→p₂)∨(p₂→p₁)] there is a structure of divisorsD _* ⁿ such that it is possible to define a strong homomorphism froomiH D _n ^* ontoH D _U . Exploiting, among others, this property, it turns out to be relatively easy to show that \(LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )\) .     

Sensing sociality in dogs: what may make an interactive robot social?

This study investigated whether dogs would engage in social interactions with an unfamiliar robot, utilize the communicative signals it provides and to examine whether the level of sociality shown by the robot affects the dogs’ performance. We hypothesized that dogs would react to the communicative signals of a robot more successfully if the robot showed interactive social behaviour in general (towards both humans and dogs) than if it behaved in a machinelike, asocial way. The experiment consisted of an interactive phase followed by a pointing session, both with a human and a robotic experimenter. In the interaction phase, dogs witnessed a 6-min interaction episode between the owner and a human experimenter and another 6-min interaction episode between the owner and the robot. Each interaction episode was followed by the pointing phase in which the human/robot experimenter indicated the location of hidden food by using pointing gestures (two-way choice test). The results showed that in the interaction phase, the dogs’ behaviour towards the robot was affected by the differential exposure. Dogs spent more time staying near the robot experimenter as compared to the human experimenter, with this difference being even more pronounced when the robot behaved socially. Similarly, dogs spent more time gazing at the head of the robot experimenter when the situation was social. Dogs achieved a significantly lower level of performance (finding the hidden food) with the pointing robot than with the pointing human; however, separate analysis of the robot sessions suggested that gestures of the socially behaving robot were easier for the dogs to comprehend than gestures of the asocially behaving robot. Thus, the level of sociality shown by the robot was not enough to elicit the same set of social behaviours from the dogs as was possible with humans, although sociality had a positive effect on dog–robot interactions.     

Socratic Trees

The method of Socratic proofs (SP-method) simulates the solving of logical problem by pure questioning. An outcome of an application of the SP-method is a sequence of questions, called a Socratic transformation. Our aim is to give a method of translation of Socratic transformations into trees. We address this issue both conceptually and by providing certain algorithms. We show that the trees which correspond to successful Socratic transformations—that is, to Socratic proofs—may be regarded, after a slight modification, as Gentzen-style proofs. Thus proof-search for some Gentzen-style calculi can be performed by means of the SP-method. At the same time the method seems promising as a foundation for automated deduction.     

Agency,communion and entitlement

In this study, the relationship between agency, communion, and the active, passive, and revenge forms of entitlement is examined. Results indicate that active entitlement was positively related to agency, negatively to communion (Study 1), and unrelated to unmitigated agency and communion (Study 2). Passive entitlement was positively related to communion (in regular and unmitigated forms) and negatively related to agency (in both forms). Revenge entitlement was positively related to agency (unmitigated and regular), and negatively related to both regular and unmitigated communal orientations. Detected relationships were independent from self‐esteem (Study 1). The findings are discussed in relation to distinctions between narcissistic and healthy entitlement, and within the context of the three‐dimensional model of entitlement.     

Distribution of Cr atoms in a strained and strain-relaxed Fe89.15Cr10.75 alloy: a Mössbauer effect study

An Fe_89.15Cr_10.75 alloy in a heavily strained (by cold rolling) state and in strain-relaxed states was studied by means of conversion electrons Mössbauer spectroscopy. Analysis of the spectra in terms of a two-shell model revealed significant differences between the studied samples, particularly in values of the hyperfine field and the distribution of Cr atoms within the first two neighbour shells. The latter is quantified in terms of short-range order parameters.