On the axiomatization of finiteK-frames

We find a short way to construct a formula which axiomatizes a given finite frame of the modal logicK, in the sense that for each finite frameA, we construct a formula ωA which holds in those and only those frames in which every formula true inA holds. To obtain this result we find, for each finite model \(\mathfrak{A}\) and each natural numbern, a formula ω \(\mathfrak{A}\) which holds in those and only those models in which every formula true in \(\mathfrak{A}\) , and involving the firstn propositional letters, holds.     

Dependence and Independence

We introduce an atomic formula ${\vec{y} \bot_{\vec{x}}\vec{z}}$ intuitively saying that the variables ${\vec{y}}$ are independent from the variables ${\vec{z}}$ if the variables ${\vec{x}}$ are kept constant. We contrast this with dependence logic ${\mathcal{D}}$ based on the atomic formula = ${(\vec{x}, \vec{y})}$ , actually equivalent to ${\vec{y} \bot_{\vec{x}}\vec{y}}$ , saying that the variables ${\vec{y}}$ are totally determined by the variables ${\vec{x}}$ . We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ can be used to give partially ordered quantifiers and IF-logic an alternative interpretation without some of the shortcomings related to so called signaling that interpretations using = ${(\vec{x}, \vec{y})}$ have.     

The epistemology of absence-based inference

Our main aim in this paper is to contribute towards a better understanding of the epistemology of absence-based inferences. Many absence-based inferences are classified as fallacies. There are exceptions, however. We investigate what features make absence-based inferences epistemically good or reliable. In Section 2 we present Sanford Goldberg’s account of the reliability of absence-based inference, introducing the central notion of epistemic coverage. In Section 3 we approach the idea of epistemic coverage through a comparison of alethic and evidential principles. The Equivalence Schema–a well-known alethic principle–says that it is true that $p$ if and only if $p$ . We take epistemic coverage to underwrite a suitably qualified evidential analogue of the Equivalence Schema: for a high proportion of values of $p$ , subject $S$ has evidence that $p$ due to her reliance on source $S^{*}$ if and only if $p$ . We show how this evidential version of the Equivalence Schema suffices for the reliability of certain absence-based inferences. Section 4 is dedicated to exploring consequences of the Evidential Equivalence Schema. The slogan ‘absence of evidence is evidence of absence’ has received a lot of bad press. More elaborately, what has received a lot of bad press is something like the following idea: absence of evidence sufficiently good to justify belief in $p$ is evidence sufficiently good to justify belief in $\sim p$ . A striking consequence of the Evidential Equivalence Schema is that absence of evidence sufficiently good to justify belief in p is evidence sufficiently good to justify belief in $\sim p$ . We establish this claim in Section 4 and show how this supports the reliability of an additional type of absence-based inference. Section 4 immediately raises the following question: how can we make philosophically good sense of the idea that absence of evidence is evidence of absence? We address this question in Section 5. Section 6 contains some summary remarks.     

Completeness in Hybrid Type Theory

We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret \(@_i\) in propositional and first-order hybrid logic. This means: interpret \(@_i\alpha _a\) , where \(\alpha _a\) is an expression of any type \(a\) , as an expression of type \(a\) that rigidly returns the value that \(\alpha_a\) receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and (as is usual inhybrid logic) we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logicover Henkin’s logic.     

Inverse Images of Box Formulas in Modal Logic

We investigate, for several modal logics but concentrating on KT, KD45, S4 and S5, the set of formulas B for which ${\square B}$ is provably equivalent to ${\square A}$ for a selected formula A (such as p, a sentence letter). In the exceptional case in which a modal logic is closed under the (‘cancellation’) rule taking us from ${\square C \leftrightarrow \square D}$ to ${C \leftrightarrow D}$ , there is only one formula B, to within equivalence, in this inverse image, as we shall call it, of ${\square A}$ (relative to the logic concerned); for logics for which the intended reading of “ ${\square}$ ” is epistemic or doxastic, failure to be closed under this rule indicates that from the proposition expressed by a knowledge- or belief-attribution, the propositional object of the attitude in question cannot be recovered: arguably, a somewhat disconcerting situation. More generally, the inverse image of ${\square A}$ may comprise a range of non-equivalent formulas, all those provably implied by one fixed formula and provably implying another—though we shall see that for several choices of logic and of the formula A, there is not even such an ‘interval characterization’ of the inverse image (of ${\square A}$ ) to be found.     

Probability Propagation in Generalized Inference Forms

Probabilistic inference forms lead from point probabilities of the premises to interval probabilities of the conclusion. The probabilistic version of Modus Ponens, for example, licenses the inference from \({P(A) = \alpha}\) and \({P(B|A) = \beta}\) to \({P(B)\in [\alpha\beta, \alpha\beta + 1 - \alpha]}\) . We study generalized inference forms with three or more premises. The generalized Modus Ponens, for example, leads from \({P(A_{1}) = \alpha_{1}, \ldots, P(A_{n})= \alpha_{n}}\) and \({P(B|A_{1} \wedge \cdots \wedge A_{n}) = \beta}\) to an according interval for P(B). We present the probability intervals for the conclusions of the generalized versions of Cut, Cautious Monotonicity, Modus Tollens, Bayes’ Theorem, and some SYSTEM O rules. Recently, Gilio has shown that generalized inference forms “degrade”—more premises lead to less precise conclusions, i.e., to wider probability intervals of the conclusion. We also study Adam’s probability preservation properties in generalized inference forms. Special attention is devoted to zero probabilities of the conditioning events. These zero probabilities often lead to different intervals in the coherence and the Kolmogorov approach.     

How schoolchildren’s acceptance of self and others relate to their attitudes to victims of bullying

Previous research has indicated that harm to the mental health of children who are repeatedly victimised by their peers at school can be ameliorated through social or emotional support provided by other students. In this study we examined whether student provictim attitudes are related to more basic attitudes to self and attitudes to others. Measures of these attitudes were developed for use with children, and shown to be internally consistent and possess concurrent validity. Relations between these variables were then examined with a sample of Australian school children ( $N = 212$ ) aged between 9 and 14 years. Provictim attitudes correlated significantly ( $r = .55$ ) with acceptance of others, but not with acceptance of self ( $r = -.04$ ). Controlling for age and gender, multiple regression analyses confirmed these findings. Attitudes to victims were less positive for older students ( $r = -.22$ ). Gender differences, however, were not significant. These findings suggest that among schoolchildren self-acceptance is unrelated to positive attitudes towards victims, whereas acceptance of others has a moderately strong relation with provictim attitude. Implications for improving attitudes to victims among school children were examined.     

Relation Formulas for Protoalgebraic Equality Free Quasivarieties; Pałasińska’s Theorem Revisited

We provide a new proof of the following Pa?asińska's theorem: Every finitely generated protoalgebraic relation distributive equality free quasivariety is finitely axiomatizable. The main tool we use are ${\mathcal{Q}}$ Q -relation formulas for a protoalgebraic equality free quasivariety ${\mathcal{Q}}$ Q . They are the counterparts of the congruence formulas used for describing the generation of congruences in algebras. Having this tool in hand, we prove a finite axiomatization theorem for ${\mathcal{Q}}$ Q when it has definable principal ${\mathcal{Q}}$ Q -subrelations. This is a property obtained by carrying over the definability of principal subcongruences, invented by Baker and Wang for varieties, and which holds for finitely generated protoalgebraic relation distributive equality free quasivarieties.     

Propositional Reasoning that Tracks Probabilistic Reasoning

This paper concerns the extent to which uncertain propositional reasoning can track probabilistic reasoning, and addresses kinematic problems that extend the familiar Lottery paradox. An acceptance rule assigns to each Bayesian credal state p a propositional belief revision method ${\sf B}_{p}$ , which specifies an initial belief state ${\sf B}_{p}(\top)$ that is revised to the new propositional belief state ${\sf B}(E)$ upon receipt of information E. An acceptance rule tracks Bayesian conditioning when ${\sf B}_{p}(E) = {\sf B}_{p|_{E}}(\top)$ , for every E such that p(E)?>?0; namely, when acceptance by propositional belief revision equals Bayesian conditioning followed by acceptance. Standard proposals for uncertain acceptance and belief revision do not track Bayesian conditioning. The ??Lockean?? rule that accepts propositions above a probability threshold is subject to the familiar lottery paradox (Kyburg 1961), and we show that it is also subject to new and more stubborn paradoxes when the tracking property is taken into account. Moreover, we show that the familiar AGM approach to belief revision (Harper, Synthese 30(1?C2):221?C262, 1975; Alchourrón et al., J Symb Log 50:510?C530, 1985) cannot be realized in a sensible way by any uncertain acceptance rule that tracks Bayesian conditioning. Finally, we present a plausible, alternative approach that tracks Bayesian conditioning and avoids all of the paradoxes. It combines an odds-based acceptance rule proposed originally by Levi (1996) with a non-AGM belief revision method proposed originally by Shoham (1987).     

Reconstructing an Open Order from Its Closure, with Applications to Space-Time Physics and to Logic

In his logical papers, Leo Esakia studied corresponding ordered topological spaces and order-preserving mappings. Similar spaces and mappings appear in many other application areas such the analysis of causality in space-time. It is known that under reasonable conditions, both the topology and the original order relation ${\preccurlyeq}$ can be uniquely reconstructed if we know the “interior” ${\prec}$ of the order relation. It is also known that in some cases, we can uniquely reconstruct ${\prec}$ (and hence, topology) from ${\preccurlyeq}$ . In this paper, we show that, in general, under reasonable conditions, the open order ${\prec}$ (and hence, the corresponding topology) can be uniquely determined from its closure ${\preccurlyeq}$ .     

The variance of d′ estimates obtained in yes—no and two-interval forced choice procedures

In studies of detection and discrimination, data are often obtained in the form of a 2 × 2 matrix and then converted to an estimate of d′, based on the assumptions that the underlying decision distributions are Gaussian and equal in variance. The statistical properties of the estimate of d′, $\hat d'$ , are well understood for data obtained using the yes—no procedure, but less effort has been devoted to the more commonly used two-interval forced choice (2IFC) procedure. The variance associated with $\hat d'$ is a function of trued′ in both procedures, but for small values of trued′, the variance of $\hat d'$ obtained using the 2IFC procedure is predicted to be less than the variance of $\hat d'$ obtained using yes—no; for large values of trued′, the variance of $\hat d'$ obtained using the 2IFC procedure is predicted to be greater than the variance of $\hat d'$ from yes—no. These results follow from standard assumptions about the relationship between the two procedures. The present paper reviews the statistical properties of $\hat d'$ obtained using the two standard procedures and compares estimates of the variance of $\hat d'$ as a function of trued′ with the variance observed in values of $\hat d'$ obtained with a 2IFC procedure.     

Information closure and the sceptical objection

In this article, I define and then defend the principle of information closure (pic) against a sceptical objection similar to the one discussed by Dretske in relation to the principle of epistemic closure. If I am successful, given that pic is equivalent to the axiom of distribution and that the latter is one of the conditions that discriminate between normal and non-normal modal logics, a main result of such a defence is that one potentially good reason to look for a formalization of the logic of “ $S$ is informed that $p$ ” among the non-normal modal logics, which reject the axiom, is also removed. This is not to argue that the logic of “ $S$ is informed that $p$ ” should be a normal modal logic, but that it could still be insofar as the objection that it could not be, based on the sceptical objection against pic, has been removed. In other word, I shall argue that the sceptical objection against pic fails, so such an objection provides no ground to abandon the normal modal logic B (also known as KTB) as a formalization of “ $S$ is informed that $p$ ”, which remains plausible insofar as this specific obstacle is concerned.     

The sampling distribution ofd′

The distribution of sample $\hat d's$ , although mathematically intractable, can be tabulated readily by computer. Such tabulations reveal a number of interesting properties of this distribution, including: (1) sample $\hat d's$ are biased, with an expected value that can be higher or lower than the true value, depending on the sample size, the true value itself, and the convention adopted for handling cases in which the sample $\hat d'$ is undefined; (2) the variance of $\hat d'$ also depends on the convention adopted for handling cases in which the sample $\hat d'$ is undefined and is in some cases poorly approximated by the standard approximation formula, (3) the standard formula for a confidence interval for $\hat d'$ is quite accurate with at least 50–100 trials per condition, but more accurate intervals can be obtained by direct computation with smaller samples.     

A multi-dimensional model of the origins of attitude certainty: teachers’ attitudes toward attention-deficit/hyperactivity disorder

This research tested and extended a laboratory-derived model of the origins of attitude certainty using a real attitude object: Teaching children with attention-deficit/hyperactivity disorder (ADHD). In Study 1, an experiment manipulated the amount of information, thought, and consistency of information delivered to pre-service teachers ( $n = 224$ ) with no prior experience with ADHD. Structural equation modelling (SEM) of Study 1 did not support the original model, in which relationships between attitude certainty and objective knowledge, thought, and consistency were mediated by perceived knowledge, thought, and ambivalence. Instead, objective amount of information, thought, and consistency interacted in their effect on attitude certainty. Study 2 ( $n = 368$ ) used a survey to test whether experiences with ADHD (personal, direct and indirect) among in-service and pre-service teachers were antecedents of attitude certainty, and to test perceived accessibility as a mediator. SEM supported both these hypotheses. Perceived accessibility and perceived knowledge mediated the relationship between attitude certainty and prior experiences with ADHD, and between attitude certainty and objective knowledge. Together, the results suggest that psychological processes underlying strong attitude certainty differ according to the familiarity and personal relevance of the attitude object, and the context and stage of attitude formation. The results have practical utility for teacher training at pre-service and in-service levels.     

Decidability of the Equational Theory of the Continuous Geometry CG(\Bbb {F})

For $\Bbb {F}$ the field of real or complex numbers, let $CG(\Bbb {F})$ be the continuous geometry constructed by von Neumann as a limit of finite dimensional projective geometries over $\Bbb {F}$ . Our purpose here is to show the equational theory of $CG(\Bbb {F})$ is decidable.     

Dynamics of lying

We propose a dynamic logic of lying, wherein a ‘lie that $\varphi $ ’ (where $\varphi $ is a formula in the logic) is an action in the sense of dynamic modal logic, that is interpreted as a state transformer relative to the formula $\varphi $ . The states that are being transformed are pointed Kripke models encoding the uncertainty of agents about their beliefs. Lies can be about factual propositions but also about modal formulas, such as the beliefs of other agents or the belief consequences of the lies of other agents. We distinguish two speaker perspectives: (Obs) an outside observer who is lying to an agent that is modelled in the system, and (Ag) an agent who is lying to another agent, and where both are modelled in the system. We distinguish three addressee perspectives: (Cred) the credulous agent who believes everything that it is told (even at the price of inconsistency), (Skep) the skeptical agent who only believes what it is told if that is consistent with its current beliefs, and (Rev) the belief revising agent who believes everything that it is told by consistently revising its current, possibly conflicting, beliefs. The logics have complete axiomatizations, which can most elegantly be shown by way of their embedding in what is known as action model logic or in the extension of that logic to belief revision.     

Actuality in Propositional Modal Logic

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula ${\phi}$ in the propositional modal language with A, there is a formula ${\psi}$ not containing A such that ${\phi}$ and ${\psi}$ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5.     

On Endomorphisms of Ockham Algebras with Pseudocomplementation

A pO-algebra ${(L; f, \, ^{\star})}$ is an algebra in which (L; f) is an Ockham algebra, ${(L; \, ^{\star})}$ is a p-algebra, and the unary operations f and ${^{\star}}$ commute. Here we consider the endomorphism monoid of such an algebra. If ${(L; f, \, ^{\star})}$ is a subdirectly irreducible pK _1,1- algebra then every endomorphism ${\vartheta}$ is a monomorphism or ${\vartheta^3 = \vartheta}$ . When L is finite the endomorphism monoid of L is regular, and we determine precisely when it is a Clifford monoid.