The prevalent interpretation of Gödel’s Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of the underlying mathematical theorem depends on several arbitrary formalisation choices. In this paper I examine the theorem’s dependency regarding Gödel numberings. I introducedeviantnumberings, yielding provability predicates satisfying Löb’s conditions, which result in provable consistency sentences. According to the main result of this paper however, these (...) “counterexamples” do not refute the theorem’s prevalent interpretation, since once a natural class ofadmissiblenumberings is singled out, invariance is maintained. (shrink)

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. (shrink)

On business ethics of Papuan peoples in Indonesia.

There is a well-known gap between metamathematical theorems and their philosophical interpretations. Take Tarski’s Theorem. According to its prevalent interpretation, the collection of all arithmetical truths is not arithmetically definable. However, the underlying metamathematical theorem merely establishes the arithmetical undefinability of a set of specific Gödel codes of certain artefactual entities, such as infix strings, which are true in the standard model. That is, as opposed to its philosophical reading, the metamathematical theorem is formulated (and proved) relative to a specific (...) choice of the Gödel numbering and the notation system. A similar observation applies to Gödel’s and Church’s theorems, which are commonly taken to impose severe limitations on what can be proved and computed using the resources of certain formalisms. The philosophical force of these limitative results heavily relies on the belief that these theorems do not depend on contingencies regarding the underlying formalisation choices. The main aim of this paper is to provide metamathematical facts which support this belief. While employing a fixed notation system, I showed in previous work (_Review of Symbolic Logic_, 2021, 14(1):51–84) how to abstract away from the choice of the Gödel numbering. In the present paper, I extend this work by establishing versions of Tarski’s, Gödel’s and Church’s theorems which are invariant regarding _both_ the notation system and the numbering. This paper thus provides a further step towards absolute versions of metamathematical results which do not rely on contingent formalisation choices. (shrink)

This paper investigates the conditions under which diagonal sentences can be taken to constitute paradigmatic cases of self-reference. We put forward well-motivated constraints on the diagonal operator and the coding apparatus which separate paradigmatic self-referential sentences, for instance obtained via Gödel’s diagonalization method, from accidental diagonal sentences. In particular, we show that these constraints successfully exclude refutable Henkin sentences, as constructed by Kreisel.

Ce livre est la brillante synthèse d'une œuvre qui figure au premier plan de la pensée catholique au XXe siècle. L'élan qui anime tous les écrits de Hans Urs von Balthasar ne s'affirme dans aucun livre mieux que dans cette Théologie de l'histoire. Celle-ci est parfaitement centrée sur le temps du Christ : Dieu toujours présent parmi les hommes. Depuis l'incarnation, l'histoire se déroule dans ce temps nouveau ; il est la norme de l'évolution humaine ; c'est lui qui (...) mesure le cours de l'existence, le progrès vertical et horizontal de l'humanité. Une fresque extraordinaire se déploie sous les yeux du lecteur l'histoire s'intériorise en un destin personnel, la vocation de chacun se situe dans un destin historique universel. Cette vision eschatologique, d'un réalisme éblouissant et pathétique, nous montre que la fin des temps et les temps nouveaux sont déjà commencés. (shrink)

v. 1. Truth of the world -- v. 2 Truth of God -- v. 3. The spirit of truth.

Religionskritik als Chance und Herausforderung für den christlichen Glauben - Selbstverantwortung und religiöse Mündigkeit - Dialog zwischen Religionen.

Cognitive Science, Volume 45, Issue 11, November 2021.

Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of (...) arithmetic. In doing so, we provide set-theoretical facts which, we believe, are crucial for informed assessment of reductionism. (shrink)

Language’s intentional nature has been highlighted as a crucial feature distinguishing it from other communication systems. Specifically, language is often thought to depend on highly structured intentional action and mutual mindreading by a communicator and recipient. Whilst similar abilities in animals can shed light on the evolution of intentionality, they remain challenging to detect unambiguously. We revisit animal intentional communication and suggest that progress in identifying analogous capacities has been complicated by (i) the assumption that intentional (that is, voluntary) production (...) of communicative acts requires mental-state attribution, and (ii) variation in approaches investigating communication across sensory modalities. To move forward, we argue that a framework fusing research across modalities and species is required. We structure intentional communication into a series of requirements, each of which can be operationalised, investigated empirically, and must be met for purposive, intentionally communicative acts to be demonstrated. Our unified approach helps elucidate the distribution of animal intentional communication and subsequently serves to clarify what is meant by attributions of intentional communication in animals and humans. (shrink)

The language comprehension system preferentially assumes that agents come first during incremental processing. While this might reflect a biologically fixed bias, shared with other domains and other species, the evidence is limited to languages that place agents first, and so the bias could also be learned from usage frequency. Here, we probe the bias with electroencephalography (EEG) in Äiwoo, a language that by default places patients first, but where sentence-initial nouns are still locally ambiguous between patient or agent roles. Comprehenders (...) transiently interpreted nonhuman nouns as patients, eliciting a negativity when disambiguation was toward the less common agent-initial order. By contrast and against frequencies, human nouns were transiently interpreted as agents, eliciting an N400-like negativity when the disambiguation was toward patient-initial order. Consistent with the notion of a fixed property, the agent bias is robust against usage frequency for human referents. However, this bias can be reversed by frequency experience for nonhuman referents. (shrink)

Cognitive Science, Volume 46, Issue 2, February 2022.

Balthasar Hubmaier is often called ‘the theologian of the Anabaptists’ for he was the only early Anabaptist leader with an earned doctorate. The former Catholic priest embraced the reforming thought of Erasmus, Zwingli, and eventually Zwingli’s former pupils and led the Moravian city of Nikolsburg to become a bastion of Anabaptist thought and practice. The multi-dimensional religious landscape both afforded Hubmaier the opportunity and compelled him to author the first Anabaptist catechism. Through the work, Hubmaier articulated a clear and (...) succinct portrayal of Anabaptist theology and ecclesiology summed up in the Erasmian tenet of the love of God and neighbor. (shrink)

Through a survey of the discussions of the decline of the sixteenth- and seventeenth-century witch-craze of Hugh Trevor-Roper, Keith Thomas and Brian Easlea, the role and impact of Balthasar Bekker, a seventeenth-century Dutch Cartesian, is shown to have been under-estimated, and not inconsiderable.

L'A. étudie l'usage que Balthasar fait de la pensée grecque post-chalcédonienne. Le théologien allemand attire l'attention sur le style symphonique de la pensée grecque, sa concentration christologique et son caractère trinitaire. La première partie de ce travail est consacrée aux problèmes que soulève l'extension de l'usage de la pensée grecque post-chalcédonienne : peut-on mettre sur le même plan Maxime le Confesseur et le Pseudo-Denys ? L'A. montre ensuite que, si pour Balthasar le mérite de cette théologie tient à (...) sa propension à l'esthétique, cela n'exclut pas une doctrine christologique et trinitaire fouillée. La troisième partie est axée sur l'emploi critique que fait Balthasar de ses sources. (shrink)

The volume and diversity of Balthasar's output makes it hard to grasp his theological purpose. This article considers the aims of Herrlichkeit in particular. To take the beautiful seriously is ontologically important but evangelically more urgent still. The Gospel requires sensitivity to the form of Jesus Christ. If the anxieties of those who fear the aesthetic will displace the ethical and even the religious can be overcome, this approach permits a wonderful integration of apologetics and dogma, fundamental and systematic (...) theology. This primary aim of Herrlichkeit is supported by showing it is not unknown to the history of Christian thought. To gain a proper idea of its scope, however, requires exploration of pre‐Christian paganism's estimate of the beauty of being, and of the human capacity to register it. then, under the impact of revelation, it is possible to locate the climax of ontological thinking about beauty in the work of Aquinas—which integrated revealed and natural sources in an account of being as glorious because given away: supremely on the Cross, in self‐diffusion of the Trinitarian life. Balthasar's purpose now shifts. He will show how Western metaphysics declined from this high‐point of perception—but in such a way that the glory of being took refuge in the holy folly of the saints. Finally, he aims to show how re‐reading Scripture in the light of the above can generate the conviction that the message of the Bible is ultimately about the self‐giving divine glory as universal foundation of history and the world. (shrink)