Outliers and Influential Observations in Exponential Random Graph Models

We discuss measuring and detecting influential observations and outliers in the context of exponential family random graph (ERG) models for social networks. We focus on the level of the nodes of the network and consider those nodes whose removal would result in changes to the model as extreme or “central” with respect to the structural features that “matter”. We construe removal in terms of two case-deletion strategies: the tie-variables of an actor are assumed to be unobserved, or the node is removed resulting in the induced subgraph. We define the difference in inferred model resulting from case deletion from the perspective of information theory and difference in estimates, in both the natural and mean-value parameterisation, representing varying degrees of approximation. We arrive at several measures of influence and propose the use of two that do not require refitting of the model and lend themselves to routine application in the ERGM fitting procedure. MCMC p values are obtained for testing how extreme each node is with respect to the network structure. The influence measures are applied to two well-known data sets to illustrate the information they provide. From a network perspective, the proposed statistics offer an indication of which actors are most distinctive in the network structure, in terms of not abiding by the structural norms present across other actors.     

Religious Beliefs,Religious Contexts,and Perceived Interactions with Ex-Offenders

Religious institutions can provide a means of support and resources for many people, particularly, individuals who have criminal histories, specifically parolees, sex offenders, and insanity acquittees. When these individuals are released into the community, they often face difficulties in acquiring shelter, work, and social support. Out of their own volition or as a result of faith-based prison programs, they might turn to religious institutions for help reintegrating into the community. However, whether congregation members are more or less accepting of these individuals is unaddressed in the current literature. Two studies assess the relationship between religious beliefs, religious contexts, legal attitudes, and perceived social interactions with parolees, sex offenders, and insanity acquittees. Results suggest that religious beliefs, legal attitudes, and religious contexts play an important role in individuals’ perceived interactions with these individuals. Religious contexts, specifically, might facilitate more positive social interactions with parolees. Implications are discussed.     

Engaging with Buddhism

In his new book, Jay Garfield invites philosophers of all persuasions to engage with Buddhist philosophy. In part I of this paper, I raise some questions on behalf of the philosopher working in the analytic tradition about the way in which Buddhist philosophy understands itself. I then turn, in part II, to look at what Orthodox Buddhism has to say about the self. I examine the debate between the Buddhist position discussed and endorsed by Garfield and that of a lesser-known school that he mentions only briefly, the Pudgalavāda (“Personalists”). I suggest that the views of the Pudgalavādins are strikingly similar to a position held, in the twentieth century analytic philosophy, by Peter Strawson.     

Propositional Epistemic Logics with Quantification Over Agents of Knowledge

The paper presents a family of propositional epistemic logics such that languages of these logics are extended by quantification over modal (epistemic) operators or over agents of knowledge and extended by predicate symbols that take modal (epistemic) operators (or agents) as arguments. Denote this family by \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\). There exist epistemic logics whose languages have the above mentioned properties (see, for example Corsi and Orlandelli in Stud Log 101:1159–1183, 2013; Fitting et al. in Stud Log 69:133–169, 2001; Grove in Artif Intell 74(2):311–350, 1995; Lomuscio and Colombetti in Proceedings of ATAL 1996. Lecture Notes in Computer Science (LNCS), vol 1193, pp 71–85, 1996). But these logics are obtained from first-order modal logics, while a logic of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) can be regarded as a propositional multi-modal logic whose language includes quantifiers over modal (epistemic) operators and predicate symbols that take modal (epistemic) operators as arguments. Among the logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) there are logics with a syntactical distinction between two readings of epistemic sentences: de dicto and de re (between ‘knowing that’ and ‘knowing of’). We show the decidability of logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) with the help of the loosely guarded fragment (LGF) of first-order logic. Namely, we generalize LGF to a higher-order decidable loosely guarded fragment. The latter fragment allows us to construct various decidable propositional epistemic logics with quantification over modal (epistemic) operators. The family of this logics coincides with \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\). There are decidable propositional logics such that these logics implicitly contain quantification over agents of knowledge, but languages of these logics are usual propositional epistemic languages without quantifiers and predicate symbols (see Grove and Halpern in J Log Comput 3(4):345–378, 1993). Some logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) can be regarded as counterparts of logics defined in Grove and Halpern (J Log Comput 3(4):345–378, 1993). We prove that the satisfiability problem for these logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) is Pspace-complete using their counterparts in Grove and Halpern (J Log Comput 3(4):345–378, 1993).     

Interpolation in 16-Valued Trilattice Logics

In a recent paper we have defined an analytic tableau calculus \({{\mathbf {\mathsf{{PL}}}}}_{\mathbf {16}}\) for a functionally complete extension of Shramko and Wansing’s logic based on the trilattice \({SIXTEEN}_3\). This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic—such as the relations Open image in new window , Open image in new window , and Open image in new window that each correspond to a lattice order in \({SIXTEEN}_3\); and Open image in new window , the intersection of Open image in new window and Open image in new window . It turns out that our method of characterising these semantic relations—as intersections of auxiliary relations that can be captured with the help of a single calculus—lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that Open image in new window , when restricted to \(\mathcal {L}_{tf}\), the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano.     

Two Kinds of Consequential Implication

The first section of the paper establishes the minimal properties of so-called consequential implication and shows that they are satisfied by at least two different operators of decreasing strength (symbolized by \(\rightarrow \) and \(\Rightarrow \)). Only the former has been analyzed in recent literature, so the paper focuses essentially on the latter. Both operators may be axiomatized in systems which are shown to be translatable into standard systems of normal modal logic. The central result of the paper is that the minimal consequential system for \(\Rightarrow \), CI\(\Rightarrow \), is definitionally equivalent to the deontic system KD and is intertranslatable with the minimal consequential system for \(\rightarrow \), CI. The main drawback ot the weaker operator \(\Rightarrow \) is that it lacks unrestricted contraposition, but the final section of the paper argues that \(\Rightarrow \) has some properties which make it a valuable alternative to \(\rightarrow \), turning out especially plausible as a basis for the definition of operators representing synthetic (i.e. context-dependent) conditionals.