An applicative theory for FPH

In this paper we introduce an applicative theory which characterizes the polynomial hierarchy of time.Comment: In Proceedings CL&C 2010, arXiv:1101.520

A Logician\u27s Sidelong Glance at Irony

In Irony as Expression (of a Sense of the Absurd) Mitchell Green is presenting an interesting account of communicative irony where ``we express a sense of a situation\u27s absurdity (wackiness, goofiness, etc.).\u27\u27 In this line of argument, he is questioning the adequateness of irony as meaning-inversion and irony as conversational implicature. In this note, we would like to take the idea of absurdity a little bit further, considering it in its logical sense. As a consequence we can offer a possibility to defend, at least partially, irony as meaning-inversion and conversational implicature

Structured belief bases

In this paper we discuss a formal approach to belief representation which stores proof-theoretic information together with formulae. It is illustrated how this additional information can be used in the context of belief revision. The general aims of this paper are the following three: First, we would like to give a descriptive approach to belief revision, in contrast to a normative one. Secondly, the given theory should avoid (the consequences of) logical omniscience of beliefs. Finally, from a broader point of view, the presented approach can be considered as a case study within the programme of proof-theoretic semantics. In this programme, the question is raised whether and how proof-theoretic information can be used as a basis for semantics

Default negation as explicit negation plus update

Funding Information: Acknowledgements. This work is partially supported by the Udo Keller Foundation and by the Portuguese Science Foundation, FCT, through the project UID/MAT/00297/2020 (Centro de Matemática e Aplica¸cões). The author is grateful to an anonymous referee for helpful comments.We argue that under the stable model semantics default negation can be read as explicit negation with update. We show that dynamic logic programming which is based on default negation, even in the heads, can be interpreted in a variant of updates with explicit negation only. As corollaries, we get an easy description of default negation in generalized and normal logic programming where initially negated literals are updated. These results are discussed with respect to the understanding of negation in logic programming.publishersversionpublishe

Feferman on Foundations: Logic, Mathematics, Philosophy

Book reviewed: Gerhard Jäger and Wilfried Sieg (editors): Feferman on Foundations: Logic, Mathematics, Philosophy. Contributions to Logic, vol. 13, Springer, 2017authorsversionpublishe

Paradoxes, Intuitionism, and Proof-Theoretic Semantics

Publisher Copyright: © The Author(s) 2024.In this note, we review paradoxes like Russell’s, the Liar, and Curry’s in the context of intuitionistic logic. One may observe that one cannot blame the underlying logic for the paradoxes, but has to take into account the particular concept formations. For proof-theoretic semantics, however, this comes with the challenge to block some forms of direct axiomatizations of the Liar. A proper answer to this challenge might be given by Schroeder-Heister’s definitional freedom.publishersversionpublishe

VARIANTS of KREISEL'S CONJECTURE on A NEW NOTION of PROVABILITY

Kreisel's conjecture is the statement: if, for all n ∈ ℕ, PA ⊢ksteps φ(n), then PA ⊢ ∀x.φ(x). For a theory of arithmetic T, given a recursive function h, T ⊢≤h φ holds if there is a proof of φ in T whose code is at most h(#φ). This notion depends on the underlying coding. PhT(x) is a provability predicate for ⊢≤h in T. It is shown that there exists a sentence φ and a total recursive function h such that T ⊢≤h PrT(⌈PrT (⌈φ⌉) → φ⌉), but T ⊢/≤h φ, where PrTstands for the standard provability predicate in T. This statement is related to a conjecture by Montagna. Also variants and weakenings of Kreisel's conjecture are studied. By use of reexion principles, one can obtain a theory ThΓ that extends T such that a version of Kreisel's conjecture holds: given a recursive function h and φ(x) a Γ- formula (where Γ is an arbitrarily fixed class of formulas) such that, for all n ∈ N, T ⊢≤h φ(n), then ThΓ⊢ ∀x.φ(x). Derivability conditions are studied for a theory to statisfy the following implication: if T ⊢ ∀x.PhT(pφ(x)q), then T ⊢ ∀x.φ(x). This corresponds to an arithmetization of Kreisel's conjecture. It is shown that, for certain theories, there exists a function h such that ⊢k steps ⊆ ⊢≤h.authorsversionepub_ahead_of_prin

k-Provability in PA

We study the decidability of k-provability in PA —the relation ‘being provable in PA with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton (the list of axioms and rules that were used). The decidability of k-provability for the usual Hilbert-style formalisation of PA is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms.publishersversionpublishe

Verantwortung und Vertrauen

Diese Arbeit wurde u.a. von der Udo Keller-Stiftung und die Portugiesische Forschungsgemeinschaft FCT über das Centro de Matemática e Aplicações, UID/MAT/00297/2020, gefördert.Bei der Diskussion um die Zukunft der Künstlichen Intelligenz finden sich auch dystopische Szenarien, die soweit reichen, daß sich die Menschheit von einer künstlichen „Superintelligenz“ in ihrer Existenz bedroht fühlen müßte. In diesem Beitrag argumentieren wir dafür, daß der Mensch die Zukunft der KI in erster Linie im Hinblick auf Verantwortung und Vertrauen hin kontrollieren muß.publishersversionpublishe