We formulate and explore two basic axiomatic systems of type-free subjective probability. One of them explicates a notion of finitely additive probability. The other explicates a concept of infinitely additive probability. It is argued that the first of these systems is a suitable background theory for formally investigating controversial principles about type-free subjective probability.

In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study of certain principles of self-referential truth.

We explore the problems that confront any attempt to explain or explicate exactly what a primitive logical rule of inference is, or consists in. We arrive at a proposed solution that places a surprisingly heavy load on the prospect of being able to understand and deal with specifications of rules that are essentially self-referring. That is, any rule $\rho $ is to be understood via a specification that involves, embedded within it, reference to rule $\rho $ itself. Just how we arrive at this position is explained by reference to familiar rules as well as less familiar ones with unusual features. An inquiry of this kind is surprisingly absent from the foundations of inferentialism—the view that meanings of expressions (especially logical ones) are to be characterized by the rules of inference that govern them.

Previous work on aesthetic experience suggests that aesthetic judgments are self-referential. The self-reference effect (SRE) is the tendency for individuals to show improved memory for items that are judged in relation to themselves. The current study sought to understand if the SRE exists for aesthetic judgments of music. Participants heard musical excerpts (classical, jazz, and electronic) and rated either a) how much they liked the music (Self condition), b) how much a close relative or friend would like the music (Other condition), or c) the genre of the music (Genre condition). After a retention interval, participants completed a recognition memory task for the musical excerpts. Participants did not show improved memory for musical excerpts encoded in the Self condition. These results extend the concept of the SRE into the domain of aesthetic judgments, but do not provide support for a memory advantage when making aesthetic judgments in relation to the self.

In several psychiatric disorders, key symptoms are associated with aspects of an individual, which are usually referred to as the “self”. For example in schizophrenia, it has been suggested that the activity of the self and the distinction between self and others are impaired. However, such models of the self and its dysfunction have been developed among Western societies and may not easily be transferred into different cultural settings, which can be characterized by alternative concepts of a person's self. This study compares traditional Western concepts of the self and its dysfunction with self-concepts developed in Caribbean, African and South-East-Asian societies.

This review demonstrates that “the self” is a fluid concept. Social function and dysfunction of such a self-concepts depend on a given cultural context. We argue that the cursive concept of the self is culturally constructed around cursive experiences which are shared by all human beings. Such universal experiences may include the prereflective access to individual thoughts and feelings, an automatic knowledge that (at least in non-pathological states) these emotions and cognitions belong to my self. Conscious self-reflection and its narrative articulation, on the other hand, is necessarily imbued with social and cultural norms, images and events, often of conflicting nature.

A cornerstone of modern mathematical logic is the diagonal lemma of Gödel and Carnap. It is used in, for example, the classical proofs of the theorems of Gödel, Rosser, and Tarski. From its first explication in 1934, just essentially one proof has appeared for the diagonal lemma in the literature; a proof that is so tricky and hard to relate that many authors have tried to avoid the lemma altogether. As a result, some so-called diagonal-free proofs have been given for the above-mentioned fundamental theorems of logic. In this paper, we provide new proofs for the semantic formulation of the diagonal lemma, and for a weak version of the syntactic formulation of it.

Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel’s first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of self-reference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference is actually required. These notions have been so far elusive and are surrounded by an aura of scepticism that has kept most philosophers away. In this paper I suggest we shouldn’t give up all hope. First, I introduce the reader to these issues. Second, I discuss the conditions a good notion of reference in arithmetic must satisfy. Accordingly, I then introduce adequate notions of reference for the language of first-order arithmetic, which I show to be fruitful for addressing the aforementioned issues in metamathematics.

We introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook [5, 6] and Rabern, Rabern, and Macauley [16]. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s [14] notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth values. It is shown that this notion of paradoxicality coincides with that of Kripke [13]. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo [21]), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs.

The author takes the paradox of omnipotence faced by lawyers in the context of constitutional change as a starting point to explore the relationship between formal law, logic, and the “pragmatics” that inform legal reasoning. Self-reference in constitutions appears problematic because it has no representation in basic, first-order logic. But self-reference in the context of legal change effectively represents a time dimension that is essential to the practice of law. The dissolution of the paradox is then used to illuminate the relationship between formal law and the context in which it is embedded. The author concludes with a nuanced understanding of law as a semi-open system in which judges play the key role of translating fundamental constitutional change into the law’s “internal” point of view.