Expressive power and semantic completeness: Boolean connectives in modal logic

We illustrate, with three examples, the interaction between boolean and modal connectives by looking at the role of truth-functional reasoning in the provision of completeness proofs for normal modal logics. The first example (§ 1) is of a logic (more accurately: range of logics) which is incomplete in the sense of being determined by no class of Kripke frames, where the incompleteness is entirely due to the lack of boolean negation amongst the underlying non-modal connectives. The second example (§ 2) focusses on the breakdown, in the absence of boolean disjunction, of the usual canonical model argument for the logic of dense Kripke frames, though a proof of incompleteness with respect to the Kripke semantics is not offered. An alternative semantic account is developed, in terms of which a completeness proof can be given, and this is used (§ 3) in the discussion of the third example, a bimodal logic which is, as with the first example, provably incomplete in terms of the Kripke semantics, the incompleteness being due to the lack of disjunction (as a primitive or defined boolean connective).     

Some kinds of modal completeness

In the modal literature various notions of completeness have been studied for normal modal logics. Four of these are defined here, viz. (plain) completeness, first-order completeness, canonicity and possession of the finite model property — and their connections are studied. Up to one important exception, all possible inclusion relations are either proved or disproved. Hopefully, this helps to establish some order in the jungle of concepts concerning modal logics. In the course of the exposition, the interesting properties of first-order definability and preservation under ultrafilter extensions are introduced and studied as well.     

Syntactical results on the arithmetical completeness of modal logic

In this paper the PA-completeness of modal logic is studied by syntactical and constructive methods. The main results are theorems on the structure of the PA-proofs of suitable arithmetical interpretationsS of a modal sequentS, which allow the transformation of PA-proofs ofS into proof-trees similar to modal proof-trees. As an application of such theorems, a proof of Solovay's theorem on arithmetical completeness of the modal system G is presented for the class of modal sequents of Boolean combinations of formulas of the form p _i,m _i=0, 1, 2, ... The paper is the preliminary step for a forthcoming global syntactical resolution of the PA-completeness problem for modal logic.     

On the completeness of first degree weakly aggregative modal logics

This paper extends David Lewis result that all first degree modal logics are complete to weakly aggregative modal logic by providing a filtration-theoretic version of the canonical model construction of Apostoli and Brown. The completeness and decidability of all first-degree weakly aggregative modal logics is obtained, with Lewiss result for Kripkean logics recovered in the case k=1.     

Expressive and receptive language skills of temperamentally shy preschoolers

Although shy children speak less in social situations, the extent to which their language skills fall behind those of their more outgoing peers remains unclear. We selected 22 temperamentally shy and 22 non‐shy children from a larger group of 400 4‐year‐old children who were prescreened for temperamental shyness by maternal report, using the Colorado Childhood Temperament Inventory (CCTI). We then compared the two groups on widely used measures that index expressive and receptive language skills. We found that, although the temperamentally shy children scored lower on both expressive and receptive language skills compared with their non‐shy counterparts, they were nonetheless performing at their age equivalency. The non‐shy children, however, were performing significantly above their age level on expressive and receptive language skills. These findings suggest that the development of normal language skills is not compromised in temperamentally shy preschoolers. Copyright © 2004 John Wiley & Sons, Ltd.     

Arithmetical completeness versus relative completeness

In this paper we study the status of the arithmetical completeness of dynamic logic. We prove that for finitistic proof systems for dynamic logic results beyond arithmetical completeness are very unlikely. The role of the set of natural numbers is carefully analyzed.     

Expressive language and prosocial behaviour in early childhood: Longitudinal associations in the UK Millennium Cohort Study

Background: Early childhood is a crucial period for language development and building social skills. While distinct, these two processes may impact upon each other.

Aims: The current study aimed to identify the directional associations between expressive language ability and prosocial behaviour between three and five years of age.

Methods: Participants included 14, 004 children and their families enrolled in the UK Millennium Cohort Study. Children’s expressive language and prosocial behaviour were assessed at three and five years of age utilizing standardized assessments and parent reports. Cross-lagged models were used for data analysis.

Results: Better expressive language at three years was associated with increased prosocial behaviour by five years. No support for the inverse direction of association was found.

Conclusions: Children’s early ability to effectively express themselves with others may help in building better social relationships by entry into formal schooling. Programming efforts that are tailored towards enhancing positive behavioural growth and social skills in the toddler years are likely to be effective when expressive language is also a targeted component of the toddler’s skill development.     

Spatial metaphor in language can promote the development of cross‐modal mappings in children

Pitch is often described metaphorically: for example, Farsi and Turkish speakers use a ‘thickness’ metaphor (low sounds are ‘thick’ and high sounds are ‘thin’), while German and English speakers use a height metaphor (‘low’, ‘high’). This study examines how child and adult speakers of Farsi, Turkish, and German map pitch and thickness using a cross‐modal association task. All groups, except for German children, performed significantly better than chance. German‐speaking adults’ success suggests the pitch‐to‐thickness association can be learned by experience. But the fact that German children were at chance indicates that this learning takes time. Intriguingly, Farsi and Turkish children's performance suggests that learning cross‐modal associations can be boosted through experience with consistent metaphorical mappings in the input language.     

Algorithmic correspondence and completeness in modal logic. V. Recursive extensions of SQEMA

The previously introduced algorithm SQEMA computes first-order frame equivalents for modal formulae and also proves their canonicity. Here we extend SQEMA with an additional rule based on a recursive version of Ackermann's lemma, which enables the algorithm to compute local frame equivalents of modal formulae in the extension of first-order logic with monadic least fixed-points FOμ. This computation operates by transforming input formulae into locally frame equivalent ones in the pure fragment of the hybrid μ-calculus. In particular, we prove that the recursive extension of SQEMA succeeds on the class of ‘recursive formulae’. We also show that a certain version of this algorithm guarantees the canonicity of the formulae on which it succeeds.     

Presuppositional completeness

Some notions of the logic of questions (presupposition of a question, validation, entailment) are used for defining certain kinds of completeness of elementary theories. Presuppositional completeness, closely related to -completeness ([3], [6]), is shown to be fulfilled by strong elementary theories like Peano arithmetic.     

Expressive Art Therapy

No abstract available for this article.     

Intermodal Expressive Arts in Group Supervision

Group supervision is an integral part of developing counseling skills and case conceptualization. Group supervision can also be used as a supervision intervention to facilitate the development of supervisory skills when group members are supervising counselors and the focus of the group is on supervision of supervision. As with any group, group supervision members can often become stagnated when group trust and cohesiveness have not been well developed, thus hindering the group supervision process. The authors describe how an intermodal expressive arts technique was employed to develop trust and cohesiveness in a supervision group that was experiencing ineffective group dynamics.     

The completeness of S

The subsystem S of Parry's AI [10] (obtained by omitting modus ponens for the material conditional) is axiomatized and shown to be strongly complete for a class of three valued Kripke style models. It is proved that S is weakly complete for the class of consistent models, and therefore that Ackermann's rule is admissible in S. It also happens that S is decidable and contains the Lewis system S4 on translation — though these results are not presented here. S is arguably the most relevant relevant logic known at this time to be decidable.I wish to thank Jill Pipher and Professora Nuel D. Belnap, Jr. and David Kaplan for their helpful comments on an earlier version of this paper.     

Illusions in modal reasoning

According to the mental model theory, models represent what is true, but not what is false. One unexpected consequence is that certain inferences should have compelling, but invalid, conclusions. Three experiments corroborated the occurrence of such illusions in reasoning about possibilities. When problems had the heading "Only one of the premises is true," the participants considered the truth of each premise in turn, but neglected the fact that when one premise is true, the others are false. When two-premise problems had the heading "One of the premises is true and one is false," the participants still neglected the falsity of one of the premises. As predicted, however, the illusions were reduced when reasoners were told to check their conclusions against the constraint that only one of the premises was true. We discuss alternative explanations for illusory inferences and their implications for current theories of reasoning.     

Probabilities on Sentences in an Expressive Logic

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter)examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic.