Algebras of Intervals and a Logic of Conditional Assertions

Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, ukasiewicz's three-valued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates.     

A Revenge-Immune Solution to the Semantic Paradoxes

The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema True(A)A, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in ordinary contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(A) with A within the language.The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are defective. We can in fact define a hierarchy of defectiveness predicates within the language. Contrary to claims that any solution to the paradoxes just breeds further paradoxes (revenge problems) involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various levels of defectiveness can all be made coherent together within a single object language.     

Confidentiality and paradox: the location of ethical space

Implicit in Jung's alchemical metaphor of the vas bene clausum is the idea of an analytic frame with a space inside it for something vital, a relationship between two selves, to develop. For this to happen, analysts must respect their patients' rights to confidentiality. The paper explores the analyst's state of mind and the intrapsychic processes that are constellated when struggling with ethical dilemmas about issues of confidentiality in analytic work. The author suggests that at times when analysts consider breaking confidentiality, there is often a disturbing inner conflict between their moral principles (codes of ethics) and their internal personal ethical attitude. At these difficult moments, the mutuality of the work and the vas bene clausum can be significantly disrupted. The analyst tries to find a third position, a mental and emotional ethical space where the subjective and the objective, the ethical and the unethical can become more companionable bedfellows.     

The Logic of Instance Ontology

An ontology's theory of ontic predication has implications for the concomitant predicate logic. Remarkable in its analytic power for both ontology and logic is the here developed Particularized Predicate Logic (PPL), the logic inherent in the realist version of the doctrine of unit or individuated predicates. PPL, as axiomatized and proven consistent below, is a three-sorted impredicative intensional logic with identity, having variables ranging over individuals x, intensions R, and instances of intensions Ri. The power of PPL is illustrated by its clarification of the self-referential nature of impredicative definitions and its distinguishing between legitimate and illegitimate forms. With a well-motivated refinement on the axiom of comprehension, PPL is, in effect, a higher-order logic without a forced stratification of predicates into types or the use of other ad hoc restrictions. The Russell–Priest characterization of the classic self-referential paradoxes is used to show how PPL diagnosis and solves these antimonies. A direct application of PPL is made to Grelling's Paradox. Also shown is how PPL can distinguish between identity and indiscernibility.     

What the Liar Taught Achilles

Zeno's paradoxes of motion and the semantic paradoxes of the Liar have long been thought to have metaphorical affinities. There are, in fact, isomorphisms between variations of Zeno's paradoxes and variations of the Liar paradox in infinite-valued logic. Representing these paradoxes in dynamical systems theory reveals fractal images and provides other geometric ways of visualizing and conceptualizing the paradoxes.     

Logical Pluralism from a Pragmatic Perspective

This paper presents a new view of logical pluralism. This pluralism takes into account how the logical connectives shift, depending on the context in which they occur. Using the Question-Under-Discussion Framework as formulated by Craige Roberts, I identify the contextual factor that is responsible for this shift. I then provide an account of the meanings of the logical connectives which can accommodate this factor. Finally, I suggest that this new pluralism has a certain Carnapian flavour. Questions about the meanings of the connectives or the best logic outside of a specified context are not legitimate questions.     

Two Envelopes and Binding

This paper describes a way of defending a modification of Eckhardt's [2013] solution to the Two Envelopes Paradox. The defence is based on ideas from Arntzenius, Elga, and Hawthorne [2004].     

Observing a client's grieving process: bringing logical positivism into qualitative grief counselling research

This positional paper originates from our need as researcher/practitioners to establish a meaningful epistemological framework for research into bereaved people's journey through loss and grief over time. We describe how the field of grief research has a long and established biological basis, in keeping with a positivist epistemology. However, there has been a diminution of the influence of logical positivism in twenty-first-century counselling research. We argue that in grief counselling research, naturalistic observation of the grieving process within a logical positivist paradigm, remains a valid and valuable construct. We posit an observational protocol for the grief counselling process which minimises the intrusion of research method into the therapeutic process. We offer this as a means of conducting qualitative research within a bereavement counselling service. Further, we suggest that the development of an observational protocol for a client's grieving process has potential implications for developing good practice in grief work.     

Better than ever? Employee reactions to ethical failures in organizations,and the ethical recovery paradox

This research examines organizational attempts to recover internally from ethical failures witnessed by employees. Drawing on research on service failure recovery, relationship repair, and behavioral ethics, we investigate how witnessing unethical acts in an organization impacts employees and their relationship with their organization. In two studies—one in the lab and one in the field—we examine the extent to which it is possible for organizations to recover fully from these ethical lapses. Results reveal an ethical recovery paradox, in which exemplary organizational efforts to recover internally from ethical failure may enhance employee perceptions of the organization to a more positive level than if no ethical failure had occurred.     

IN PRAISE OF A LOGIC OF DEFINITIONS THAT TOLERATES Ω‐INCONSISTENCY

I argue that a general logic of definitions must tolerate ω‐inconsistency. I present a semantical scheme, , under which some definitions imply ω‐inconsistent sets of sentences. I draw attention to attractive features of this scheme, and I argue that yields the minimal general logic of definitions. I conclude that any acceptable general logic should permit definitions that generate ω‐inconsistency. This conclusion gains support from the application of to the theory of truth.