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Tytuł:
Remarks on the Gödelian Anti-Mechanist Arguments
Autorzy:
Raatikainen, Panu
Powiązania:
https://bibliotekanauki.pl/articles/1796974.pdf
Data publikacji:
2020
Wydawca:
Polskie Towarzystwo Semiotyczne
Tematy:
Gödel
incompleteness
mechanism
Opis:
Certain selected issues around the Gödelian anti-mechanist arguments which have received less attention are discussed.
Źródło:
Studia Semiotyczne; 2020, 34, 1; 267-278
0137-6608
Pojawia się w:
Studia Semiotyczne
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Gödel’s Incompleteness Theorem and the Anti-Mechanist Argument: Revisited
Autorzy:
Cheng, Yong
Powiązania:
https://bibliotekanauki.pl/articles/1796969.pdf
Data publikacji:
2020
Wydawca:
Polskie Towarzystwo Semiotyczne
Tematy:
Gödel’s incompleteness theorem
the Anti-Mechanist Argument
Gödel’s Disjunctive Thesis
intensionality
Opis:
This is a paper for a special issue of Semiotic Studies devoted to Stanislaw Krajewski’s paper (2020). This paper gives some supplementary notes to Krajewski’s (2020) on the Anti-Mechanist Arguments based on Gödel’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4–6 in Krajewski’s (2020) and classify some misunderstandings of Gödel’s incompleteness theorem related to AntiMechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel’s Disjunctive Thesis, Gödel’s Undemonstrability of Consistency Thesis and the definability of natural numbers as in Section 7–8 in Krajewski’s (2020), describing how recent advances bear on these issues.
Źródło:
Studia Semiotyczne; 2020, 34, 1; 159-182
0137-6608
Pojawia się w:
Studia Semiotyczne
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Understanding, Expression and Unwelcome Logic
Autorzy:
Holub, Štěpán
Powiązania:
https://bibliotekanauki.pl/articles/1796970.pdf
Data publikacji:
2020
Wydawca:
Polskie Towarzystwo Semiotyczne
Tematy:
mechanism
Gödel’s theorem
Turing machine
hermeneutics
Opis:
In this paper I will attempt to explain why the controversy surrounding the alleged refutation of Mechanism by Gödel’s theorem is continuing even after its unanimous refutation by logicians. I will argue that the philosophical point its proponents want to establish is a necessary gap between the intended meaning and its formulation. Such a gap is the main tenet of philosophical hermeneutics. While Gödel’s theorem does not disprove Mechanism, it is nevertheless an important illustration of the hermeneutic principle. The ongoing misunderstanding is therefore based in a distinction between a metalogical illustration of a crucial feature of human understanding, and a logically precise, but wrong claim. The main reason for the confusion is the fact that in order to make the claim logically precise, it must be transformed in a way which destroys its informal value. Part of this transformation is a clear distinction between the Turing Machine as a mathematical object and a machine as a physical device.
Źródło:
Studia Semiotyczne; 2020, 34, 1; 183-202
0137-6608
Pojawia się w:
Studia Semiotyczne
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On Martin-Löf’s Constructive Optimism
Autorzy:
Peluce, V. Alexis
Powiązania:
https://bibliotekanauki.pl/articles/1796975.pdf
Data publikacji:
2020
Wydawca:
Polskie Towarzystwo Semiotyczne
Tematy:
optimism
pessimism
Martin-Löf
Gödel’s disjunction
Opis:
In his 1951 Gibbs Memorial Lecture, Kurt Gödel put forth his famous disjunction that either the power of the mind outstrips that of any machine or there are absolutely unsolvable problems. The view that there are no absolutely unsolvable problems is optimism, the view that there are such problems is pessimism. In his 1995—and, revised in 2013—Verificationism Then and Now, Per Martin-Löf presents an illustrative argument for a constructivist form of optimism. In response to that argument, Solomon Feferman points out that Martin-Löf’s reasoning relies upon constructive understandings of key philosophical notions. In the vein of Feferman’s analysis, one might be object to Martin-Löf’s argument for either its reliance upon constructivist (as opposed to classical) considerations, or for its appeal to non-unproblematically mathematical premises. We argue that both of these responses fall short. On one hand, to be critical of Martin-Löf’s reasoning for its constructiveness is to reject what would otherwise be a scientific advance on the basis of the assumption of constructivism’s falsehood or implausibility, which is of course uncharitable at best. On the other hand, to object to the argument for its use of non-unproblematically mathematical premises is to assume that there is some philosophically neutral mathematics, which is implausible. Martin-Löf’s argument relies upon his third law, the claim that from the impossibility of a proof of a proposition we can construct a proof of its negation. We close with a discussion of some ways in which this claim can be criticized from the constructive point of view. Specifically, we contend that Martin-Löf’s third law is incompatible with what has been called “Poincaré’s Principle of Epistemic Conservation”, the thesis that genuine increase in mathematical knowledge requires subject-specific insight.
Źródło:
Studia Semiotyczne; 2020, 34, 1; 233-242
0137-6608
Pojawia się w:
Studia Semiotyczne
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A Note on Gödel-Dummet Logic LC
Autorzy:
Robles, Gemma
Méndez, José M.
Powiązania:
https://bibliotekanauki.pl/articles/2033854.pdf
Data publikacji:
2021-07-01
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
Intermediate logics
Gödel-Dummet logic LC
Opis:
Let \(A_{0},A_{1},...,A_{n}\) be (possibly) distintict wffs, \(n\) being an odd number equal to or greater than 1. Intuitionistic Propositional Logic IPC plus the axiom \((A_{0}\rightarrow A_{1})\vee ...\vee (A_{n-1}\rightarrow A_{n})\vee (A_{n}\rightarrow A_{0})\) is equivalent to Gödel-Dummett logic LC. However, if \(n\) is an even number equal to or greater than 2, IPC plus the said axiom is a sublogic of LC.
Źródło:
Bulletin of the Section of Logic; 2021, 50, 3; 325-335
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The Problematic Nature of Gödel’s Disjunctions and Lucas-Penrose’s Theses
Autorzy:
Avron, Arnon
Powiązania:
https://bibliotekanauki.pl/articles/1796961.pdf
Data publikacji:
2020
Wydawca:
Polskie Towarzystwo Semiotyczne
Tematy:
Gödel disjunction
Lucas-Penrose argument
mechanism
mind
computationalism
Opis:
We show that the name “Lucas-Penrose thesis” encompasses several different theses. All these theses refer to extremely vague concepts, and so are either practically meaningless, or obviously false. The arguments for the various theses, in turn, are based on confusions with regard to the meaning(s) of these vague notions, and on unjustified hidden assumptions concerning them. All these observations are true also for all interesting versions of the much weaker (and by far more widely accepted) thesis known as “Gö- del disjunction”. Our main conclusions are that pure mathematical theorems cannot decide alone any question which is not purely mathematical, and that an argument that cannot be fully formalized cannot be taken as a mathematical proof.
Źródło:
Studia Semiotyczne; 2020, 34, 1; 83-108
0137-6608
Pojawia się w:
Studia Semiotyczne
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Relation between (fuzzy) Gödel ideals and (fuzzy) Boolean ideals in BL-algebras
Autorzy:
Paad, Akbar
Powiązania:
https://bibliotekanauki.pl/articles/728828.pdf
Data publikacji:
2016
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
BL-algebra
(fuzzy) filter
(fuzzy) Boolean ideal
(fuzzy) Gödel ideal
Opis:
In this paper, we study relationships between among (fuzzy) Boolean ideals, (fuzzy) Gödel ideals, (fuzzy) implicative filters and (fuzzy) Boolean filters in BL-algebras. In [9], there is an example which shows that a Gödel ideal may not be a Boolean ideal, we show this example is not true and in the following we prove that the notions of (fuzzy) Gödel ideals and (fuzzy) Boolean ideals in BL-algebras coincide.
Źródło:
Discussiones Mathematicae - General Algebra and Applications; 2016, 36, 1; 45-58
1509-9415
Pojawia się w:
Discussiones Mathematicae - General Algebra and Applications
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Gödel, Wittgenstein and the Sensibility of Platonism
Autorzy:
Poręba, Marcin
Powiązania:
https://bibliotekanauki.pl/articles/1357900.pdf
Data publikacji:
2021-06-30
Wydawca:
Uniwersytet Warszawski. Wydział Filozofii
Tematy:
concepts
Gödel
intuition
mathematics
Platonism
realism
rule-following
Wittgenstein
Opis:
The paper presents an interpretation of Platonism, the seeds of which can be found in the writings of Gödel and Wittgenstein. Although it is widely accepted that Wittgenstein is an anti-Platonist the author points to some striking affinities between Gödel’s and Wittgenstein’s accounts of mathematical concepts and the role of feeling and intuition in mathematics. A version of Platonism emerging from these considerations combines realism with respect to concepts with a view of concepts as accessible to feeling and able to guide our behavior through feeling.
Źródło:
Eidos. A Journal for Philosophy of Culture; 2021, 5, 1; 108-125
2544-302X
Pojawia się w:
Eidos. A Journal for Philosophy of Culture
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Ontologiczny dowód Gödla z ograniczoną redukcją modalności
Autorzy:
Świętorzecka, Kordula
Powiązania:
https://bibliotekanauki.pl/articles/705971.pdf
Data publikacji:
2012-09-01
Wydawca:
Polska Akademia Nauk. Czytelnia Czasopism PAN
Tematy:
dowód ontologiczny
K. Gödel
dowód na istnienie Boga
teodycea,formalizacja
Opis:
Prezentowane rozważania są efektem poszukiwania możliwie słabej podstawy formalnej dla modalnej wersji ontologicznego argumentu na konieczne istnienie Boga, naszkicowanego przez K. Gödla. Dotychczasowe modalne rekonstrukcje notatki Gödla Ontologischer Beweis (1970) najczęściej opierają argumentację Gödla na różnych kwantyfikatorowych rozszerzeniach logiki modalnej S5 lub B. System S5, jako podstawa formalna zamierzona przez samego autora, umożliwia określoną konstrukcję argumentu ontologicznego, jednak z drugiej strony ten sposób rozumienia modalności może być uważany także za źródło słabości opartej na nim teorii Absolutu - redukcja modalności S5 (i B) może dawać okazję do formułowania krytyki w stylu Gaunilona. Standardowe rozszerzenie S5 lub B do logiki kwantyfikatorowej jest uwikłane w dalsze komplikacje: w odpowiednio rozbudowanej standardowej semantyce światów możliwych rozstrzyga się, że modele tych logik mają stałe uniwersum indywiduów. Tymczasem to rozstrzygnięcie nie ma związku z zasadniczym problemem rozważanym w formalizmie Gödla. W proponowanej wersji argumentu Gödla ograniczam redukcję modalności S5 do wybranego specyficznego kontekstu dotyczącego istnienia Absolutu. Logiką, która pozwala zachować konstrukcję argumentacji Gödla, okazuje się system S4. Otrzymaną teorię wiążę z semantyką światów możliwych z możliwie zmiennymi uniwersami. Istnienie indywiduów wyrażam za pomocą kwantyfikatora Ǝ interpretowanego aktualistycznie, bez użycia pierwotnego predykatu istnienia.
Źródło:
Przegląd Filozoficzny. Nowa Seria; 2012, 3; 21-34
1230-1493
Pojawia się w:
Przegląd Filozoficzny. Nowa Seria
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Syntax-Semantics Interaction in Mathematics
Autorzy:
Heller, Michael
Powiązania:
https://bibliotekanauki.pl/articles/561342.pdf
Data publikacji:
2018
Wydawca:
Polskie Towarzystwo Semiotyczne
Tematy:
philosophy of mathematics
categorical logic
syntax-semantic interaction
Bell’s program
Gödel-like limitations
Opis:
Mathematical tools of category theory are employed to study the syntaxsemantics problem in the philosophy of mathematics. Every category has its internal logic, and if this logic is sufficiently rich, a given category provides semantics for a certain formal theory and, vice versa, for each (suitably defined) formal theory one can construct a category, providing a semantics for it. There exists a pair of adjoint functors, Lang and Syn, between a category (belonging to a certain class of categories) and a category of theories. These functors describe, in a formal way, mutual dependencies between the syntactical structure of a formal theory and the internal logic of its semantics. Bell’s program to regard the world of topoi as the univers de discours of mathematics and as a tool of its local interpretation, is extended to a collection of categories and all functors between them, called “categorical field”. This informal idea serves to study the interaction between syntax and semantics of mathematical theories, in an analogy to functors Lang and Syn. With the help of these concepts, the role of Gödel-like limitations in the categorical field is briefly discussed. Some suggestions are made concerning the syntax-semantics interaction as far as physical theories are concerned.
Źródło:
Studia Semiotyczne; 2018, 32, 2; 87-105
0137-6608
Pojawia się w:
Studia Semiotyczne
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Can a Robot Be Grateful? Beyond Logic, Towards Religion
Autorzy:
Krajewski, Stanisław
Powiązania:
https://bibliotekanauki.pl/articles/451269.pdf
Data publikacji:
2018-12-28
Wydawca:
Uniwersytet Warszawski. Wydział Filozofii
Tematy:
computer science
robot
Gödel’s theorem
digitalization
Pythagoreanism
context
Church’s Thesis
philosophy of dialogue
gratitude
prayer
Opis:
Philosophy should seriously take into account the presence of computers. Computer enthusiasts point towards a new Pythagoreanism, a far reaching generalization of logical or mathematical views of the world. Most of us try to retain a belief in the permanence of human superiority over robots. To justify this superiority, Gödel’s theorem has been invoked, but it can be demonstrated that this is not sufficient. Other attempts are based on the scope and fullness of our perception and feelings. Yet the fact is that more and more can be computer simulated. In order to secure human superiority over robots, reference to the realm of human relations and attitudes seems more promising. Insights provided by philosophy of dialogue can help. They suggest an ultimate extension of the Turing test. In addition, it seems that in order to justify the belief in human superiority one must rely on the individual experiences that indicate a realm that is not merely subjective. It makes sense to call it religious.
Źródło:
Eidos. A Journal for Philosophy of Culture; 2018, 2, 4(6); 4-13
2544-302X
Pojawia się w:
Eidos. A Journal for Philosophy of Culture
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On the Anti-Mechanist Arguments Based on Gödel’s Theorem
Autorzy:
Krajewski, Stanisław
Powiązania:
https://bibliotekanauki.pl/articles/1796977.pdf
Data publikacji:
2020
Wydawca:
Polskie Towarzystwo Semiotyczne
Tematy:
Gödel’s theorem
mechanism
Lucas’s argument
Penrose’s argument
computationalism
mind
consistency
algorithm
artificial intelligence
natural number
Opis:
The alleged proof of the non-mechanical, or non-computational, character of the human mind based on Gödel’s incompleteness theorem is revisited. Its history is reviewed. The proof, also known as the Lucas argument and the Penrose argument, is refuted. It is claimed, following Gödel himself and other leading logicians, that antimechanism is not implied by Gödel’s theorems alone. The present paper sets out this refutation in its strongest form, demonstrating general theorems implying the inconsistency of Lucas’s arithmetic and the semantic inadequacy of Penrose’s arithmetic. On the other hand, the limitations to our capacity for mechanizing or programming the mind are also indicated, together with two other corollaries of Gödel’s theorems: that we cannot prove that we are consistent (Gödel’s Unknowability Thesis), and that we cannot fully describe our notion of a natural number.
Źródło:
Studia Semiotyczne; 2020, 34, 1; 9-56
0137-6608
Pojawia się w:
Studia Semiotyczne
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Diagonal Anti-Mechanist Arguments
Autorzy:
Kashtan, David
Powiązania:
https://bibliotekanauki.pl/articles/1796972.pdf
Data publikacji:
2020
Wydawca:
Polskie Towarzystwo Semiotyczne
Tematy:
mechanism
mind
computability
incompleteness theorems
computation-al theory of mind
the cogito
diagonal arguments
Gödel
Descartes
Tarski
Turing
Chomsky
Opis:
Gödel’s first incompleteness theorem is sometimes said to refute mechanism about the mind. §1 contains a discussion of mechanism. We look into its origins, motivations and commitments, both in general and with regard to the human mind, and ask about the place of modern computers and modern cognitive science within the general mechanistic paradigm. In §2 we give a sharp formulation of a mechanistic thesis about the mind in terms of the mathematical notion of computability. We present the argument from Gödel’s theorem against mechanism in terms of this formulation and raise two objections, one of which is known but is here given a more precise formulation, and the other is new and based on the discussion in §1.
Źródło:
Studia Semiotyczne; 2020, 34, 1; 203-232
0137-6608
Pojawia się w:
Studia Semiotyczne
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
What are the limits of mathematical explanation? Interview with Charles McCarty by Piotr Urbańczyk
Autorzy:
McCarty, David Charles
Urbańczyk, Piotr
Powiązania:
https://bibliotekanauki.pl/articles/691211.pdf
Data publikacji:
2016
Wydawca:
Copernicus Center Press
Tematy:
mathematics
logic
mathematical explanation
limits of explanation
mathematical proof
proof-core
intuitionism
constructivsim
Gödel’s Incompleteness Theorems
intuitionistics mathematics
classical mathematics
Axiom of Choice
Opis:
An interview with Charles McCarty by Piotr Urbańczyk concerning  mathematical explanation.
Źródło:
Zagadnienia Filozoficzne w Nauce; 2016, 60; 119-137
0867-8286
2451-0602
Pojawia się w:
Zagadnienia Filozoficzne w Nauce
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A Logic for Dually Hemimorphic Semi-Heyting Algebras and its Axiomatic Extensions
Autorzy:
Cornejo, Juan Manuel
Sankappanavar, Hanamantagouda P.
Powiązania:
https://bibliotekanauki.pl/articles/43189647.pdf
Data publikacji:
2022
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
semi-intuitionistic logic
dually hemimorphic semi-Heyting logic
dually quasi-De Morgan semi-Heyting logic
De Morgan semi-Heyting logic
dually pseudocomplemented semi-Heyting logic
regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1
implicative logic
equivalent algebraic semantics
algebraizable logic
De Morgan Gödel logic
dually pseudocomplemented Gödel logic
Moisil's logic
3-valued Łukasiewicz logic
Opis:
The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.
Źródło:
Bulletin of the Section of Logic; 2022, 51, 4; 555-645
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
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