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.
Boring Infinite Descent Tahko, Tuomas E.
Metaphilosophy,
April 2014, Letnik:
45, Številka:
2
Journal Article
Recenzirano
Odprti dostop
In formal ontology, infinite regresses are generally considered a bad sign. One debate where such regresses come into play is the debate about fundamentality. Arguments in favour of some type of ...fundamentalism are many, but they generally share the idea that infinite chains of ontological dependence must be ruled out. Some motivations for this view are assessed in this article, with the conclusion that such infinite chains may not always be vicious. Indeed, there may even be room for a type of fundamentalism combined with infinite descent as long as this descent is "boring," that is, the same structure repeats ad infinitum. A start is made in the article towards a systematic account of this type of infinite descent. The philosophical prospects and scientific tenability of the account are briefly evaluated using an example from physics.
I put forward precise and appealing notions of reference, self-reference, and well-foundedness for sentences of the language of first-order Peano arithmetic extended with a truth predicate. These ...notions are intended to play a central role in the study of the reference patterns that underlie expressions leading to semantic paradox and, thus, in the construction of philosophically well-motivated semantic theories of truth.
Recent advances in unfolding technique Bonet, Blai; Haslum, Patrik; Khomenko, Victor ...
Theoretical computer science,
09/2014, Letnik:
551
Journal Article
Recenzirano
Odprti dostop
We propose a new, and to date the most general, framework for Petri net unfolding, which broadens its applicability, makes it easier to use, and increases its efficiency. In particular: (i) we ...propose a user-oriented view of the unfolding technique, which simply tells which information will be preserved in the final prefix and how to declare an event a cut-off in the algorithm, while hiding the technical parameters like the adequate order; (ii) the notion of the adequate order is generalised to a well-founded relation, and the requirement that it must refine ⊂ is replaced by a weaker one; and (iii) the order in which the unfolding algorithm selects the possible extensions of the prefix is entirely disentangled from the cut-off condition. We demonstrate the usefulness of the developed theory on some case studies.
Abstract
We present an effective proof (with explicit bounds) of the Podelski and Rybalchenko Termination Theorem. The sub-recursive bounds we obtain make use of bar recursion, in the form of the ...product of selection functions, as this is used to interpret the Weak Ramsey Theorem for pairs. The construction can be seen as calculating a modulus of well-foundedness for a given program given moduli of well-foundedness for the disjunctively well-founded finite set of covering relations. When the input moduli are in system
T
, this modulus is also definable in system
T
by a result of Schwichtenberg on bar recursion.
In this paper, we study
operational termination
, a proof theoretical notion for capturing the termination behavior of computational systems. We prove that operational termination can be ...characterized at different levels by means of well-founded relations on specific formulas which can be obtained from the considered system. We show how to obtain such well-founded relations from logical models which can be automatically generated using existing tools.
Anticipation proof obligations for stated variants need to be proved in Event-B even if the variant has no variable in common with an anticipated event. This often leads to models that are ...complicated by additional auxiliary variables and variants that need to take into account these variables. Because of such “encodings” of control flow information in the variants the corresponding proof obligations can usually not be discharged automatically.
We present a new proof obligation for anticipated events that does not have this defect and prove it correct. The proof is fairly intricate due to the nondeterminism of the simulations that link refinements. An informal soundness argument suggests using a lexicographic product in the soundness proof. However, it turns out that a weaker order is required which we call quasi-lexicographic product.
•An improvement of the Event-B proof obligations for anticipation and convergence.•Correctness proof for inductive convergence in Event-B using anticipation and convergence.•Introduction of new required mathematical notions, in particular, quasi-lexicographic products.
Reference and Truth Picollo, Lavinia
Journal of philosophical logic,
06/2020, Letnik:
49, Številka:
3
Journal Article
Recenzirano
Odprti dostop
I apply the notions of alethic reference introduced in previous work in the construction of several classical semantic truth theories. Furthermore, I provide proof-theoretic versions of those notions ...and use them to formulate axiomatic disquotational truth systems over classical logic. Some of these systems are shown to be sound, proof-theoretically strong, and compare well to the most renowned systems in the literature.
Russell's paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order ...logic—the first-order logical form of this proposition is inconsistent. This explains why Russell's paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets. Burali-Forti's paradox, like Russell's paradox, is portable. I offer the following explanation for this fact: Burali-Forti's paradox, like Russell's, is purely logical. Concretely, I show that if we enrich the language ℒ of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal. Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti's. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti's paradox is explained in the same way as the portability of Russell's: both paradoxes involve an inconsistent logical form—Russell's involves an inconsistent form expressible in ℒ and Burali-Forti's involves an inconsistent form expressible in ℒ + W.
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