About the characterization of a fine line that separates generalizations and boundary-case exceptions for the Second Incompleteness Theorem under semantic tableau deduction

2020 ◽  
Author(s):  
Dan E Willard
Keyword(s):  
Incompleteness Theorem ◽  
Fine Line ◽  
Total Function ◽  
Boundary Case ◽  
Semantic Tableau ◽  
The Law ◽  
Excluded Middle

Abstract Our previous research showed that the semantic tableau deductive methodology of Fitting and Smullyan permits boundary-case exceptions to the second incompleteness theorem, if multiplication is viewed as a 3-way relation (rather than as a total function). It is known that tableau methodologies prove a schema of theorems verifying all instances of the law of the excluded middle. But if one promotes this schema of theorems into formalized logical axioms, then the meaning of the pronoun of ‘I’, used by our self-referencing engine, changes quite sharply. Our partial evasions of the second incompleteness theorem shall then come to a complete halt.

scholarly journals An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency

2005 ◽  
Vol 70 (4) ◽  
pp. 1171-1209 ◽  
Author(s):  
Dan E. Willard
Keyword(s):  
Modus Ponens ◽  
Axiom System ◽  
Incompleteness Theorem ◽  
Deduction System ◽  
Total Function ◽  
Boundary Case ◽  
Axiom Systems

AbstractThis article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: (1) α treats multiplication as a 3-way relation (rather than as a total function), and that (2) D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 and Σ1 formulae.Part of what will make this boundary-case exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundary-case exceptions in any of several further directions.

Reconsidering Kant’s Rejection of Indirect Arguments in Transcendental Philosophy

2021 ◽  
pp. 1-19
Author(s):  
Marcel Buß
Keyword(s):  
Immanuel Kant ◽  
Explanatory Power ◽  
Transcendental Philosophy ◽  
Point Of View ◽  
Transcendental Arguments ◽  
Mathematical Proofs ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Excluded Middle ◽  
Insight Into

Abstract Immanuel Kant states that indirect arguments are not suitable for the purposes of transcendental philosophy. If he is correct, this affects contemporary versions of transcendental arguments which are often used as an indirect refutation of scepticism. I discuss two reasons for Kant’s rejection of indirect arguments. Firstly, Kant argues that we are prone to misapply the law of excluded middle in philosophical contexts. Secondly, Kant points out that indirect arguments lack some explanatory power. They can show that something is true but they do not provide insight into why something is true. Using mathematical proofs as examples, I show that this is because indirect arguments are non-constructive. From a Kantian point of view, transcendental arguments need to be constructive in some way. In the last part of the paper, I briefly examine a comment made by P. F. Strawson. In my view, this comment also points toward a connection between transcendental and constructive reasoning.

The Law of Excluded Middle

1998 ◽  
pp. 92-112
Author(s):  
Quentin Gibson
Keyword(s):  
Law Of Excluded Middle ◽  
The Law ◽  
Excluded Middle

La zone économique exclusive dans la pratique des États

1998 ◽  
Vol 35 ◽  
pp. 3-55 ◽  
Author(s):  
Francis Rigaldies
Keyword(s):  
United Nations ◽  
Scientific Research ◽  
Law Of The Sea ◽  
Basic Principles ◽  
Judicial Decisions ◽  
State Practice ◽  
United Nations Convention ◽  
The United Nations ◽  
The Law

SummaryThe use of the concept of an exclusive Economie zone has increased since the adoption of the United Nations Convention on the Law of the Sea. However, the characterization of this zone varies greatly between States. This article presents an exhaustive survey of the concept of an exclusive Economie zone. The author discusses the types of jurisdiction exercised by States in their uses of an exclusive Economie zone. Disparity exists between the provisions of the Convention and State practice in some specific areas: for example, the provisions on the environment and on scientific research. Despite these exceptions, the author maintains that the basic tenets of the Convention are respected in State practice. State declarations as well as arbitral and judicial decisions show that the Convention and State practice are together evolving to reinforce the basic principles of the concept of an exclusive Economie zone.

scholarly journals Methodological Frames: Paul Bernays, Mathematical Structuralism, and Proof Theory

2020 ◽  
pp. 352-382
Author(s):  
Wilfried Sieg
Keyword(s):  
Proof Theory ◽  
Incompleteness Theorem ◽  
Mathematical Structures ◽  
General Characterization ◽  
Mathematical Structuralism ◽  
Mathematical Experience

Mathematical structuralism is deeply connected with Hilbert and Bernays’s proof theory and its programmatic aim to ensure the consistency of all of mathematics. That aim was to be reached on the basis of finitist mathematics. Gödel’s second incompleteness theorem forced a step from absolute finitist to relative constructivist proof-theoretic reductions. This mathematical step was accompanied by philosophical arguments for the special nature of the grounding constructivist frameworks. Against that background, this chapter examines Bernays’s reflections on proof-theoretic reductions of mathematical structures to methodological frames via projections. However, these reflections are focused on narrowly arithmetic features of frames. Drawing on broadened meta-mathematical experience, this chapter proposes a more general characterization of frames that has ontological and epistemological significance. The characterization is given in terms of accessibility: domains of objects are accessible if their elements are inductively generated, and principles for such domains are accessible if they are grounded in our understanding of the generating processes.

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