The Justification of Identity Elimination in Martin-Löf’s Type Theory

Klev, Ansten

doi:10.1007/s11245-017-9509-1

Cited by 8 publications

(8 citation statements)

References 12 publications

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“…In the homotopy type theory book (UFP, 2013), the theory is presented as an augmentation of constructive type theory with two axioms for equality, namely, the univalence axiom and higher inductive types, which populate the identity type with new terms (other than refl x ) for obtaining equality proofs. This conventional presentation of homotopy type theory is the one considered by Ladyman and Presnell (2015, 2016, 2017 in their series of papers. 2 To avoid confusion, I shall term it 'Book HoTT' and use 'homotopy type theory' in the broad sense described above.…”

Section: Homotopy Type Theorymentioning

confidence: 99%

“…From the discussion in the preceding section it can be seen that the constant concern of Ladyman and Presnell (2015, 2016, 2017 about the status of homotopy type theory as an autonomous foundation for mathematics is well founded. Homotopy type theory is just a particular form of type theory, and while type theories are usually thought of having a justification in the meaning explanations, that is not the case for Book HoTT.…”

Section: Path Inductionmentioning

confidence: 99%

“…Finally, another possibility to be considered is that the homotopy interpretation criticized by Ladyman and Presnell (2015, 2016, 2017 can be perhaps made pre-mathematical through the use of spatial intuitions in the sense of Tsementzis (forthcoming), where an intuitive semantics for homotopy type theory is given based on notions such as "shapes", "points", "symmetries" etc. But a thorough analysis of the pre-mathematical nature of the primitive assumptions the interpretation relies on and what rules it justifies (Tsementzis, forthcoming, §3.1-2) would exceed the scope of this paper and is thus seen as future work.…”

Section: Types As Concepts or Programs?mentioning

confidence: 99%

“…In a series of three long papers, Ladyman and Presnell (2015, 2016, 2017 raise an interesting challenge to a promising candidate for foundation of mathematics known as homotopy type theory: the standard justifications for the axioms and rules of inference of the framework appear to be largely based on sophisticated concepts from homotopy theory, whereas the primitive concepts of a supposedly autonomous foundation (in the sense being considered) should not be grounded on notions borrowed from other fields of mathematics. 1 In response to this observation, Ladyman and Presnell (2016) announced a program for developing an informal semantics for homotopy type theory based on an interpretation of types as mathematical concepts, which is used to claim that homotopy type theory can be an autonomous foundation for mathematics.…”

Section: Introductionmentioning

confidence: 99%

“…Except thatLadyman and Presnell (2015, 2016, 2017 have never discussed higher inductive types in their papers, so we omit the subject here for conciseness. The univalence axiom is introduced in Section 3.3.…”

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confidence: 99%

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What Types Should Not Be†

Bentzen¹

2019

Philosophia Mathematica

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In a series of papers Ladyman and Presnell raise an interesting challenge of providing a pre-mathematical justification for homotopy type theory. In response, they propose what they claim to be an informal semantics for homotopy type theory where types and terms are regarded as mathematical concepts. The aim of this paper is to raise some issues which need to be resolved for the successful development of their types-as-concepts interpretation.

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Section: Homotopy Type Theorymentioning

confidence: 99%

Section: Path Inductionmentioning

confidence: 99%

Section: Types As Concepts or Programs?mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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What Types Should Not Be†

Bentzen¹

2019

Philosophia Mathematica

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Dialogues, Reasons and Endorsement

Rahman

2019

Constructive Semantics

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The main aim of the present paper is to show that, if we follow the dialogical insight that reasoning and meaning are constituted during interaction, and we develop this insight in a dialogical framework for Martin-Löf's Constructive Type Theory, a conception of knowledge emerges that has important links with Robert Brandom's (1994 inferential pragmatism. However, there are also some significant differences that are at center of the dialogical approach to meaning. The present paper does not discuss explicitly phenomenology, however, one might see our proposal as setting the basis for a further study linking phenomenology and the dialogical conception of meaningthe development of such a link is part of several ongoing researches. I Shahid Rahman 2 2 Local reasonsRecent developments in dialogical logic show that the Constructive Type Theory approach to meaning is very natural to the game-theoretical approaches in which (standard) metalogical features are explicitly displayed at the object language-level. 3 This vindicates, albeit in quite a 2 In fact, the present paper relies on the main technical and philosophical results of Rahman/McConaugey/ Klev/Clerbout (2018). 3 Such as developed in Rahman/McConaugey/Klev/Clerbout (2018) and also in Clerbout/Rahman (2015). different fashion, Hintikka's plea for the fruitfulness of game-theoretical semantics in the context of epistemic approaches to logic, semantics, and the foundations of mathematics. 4 From the dialogical point of view, the actions-such as choices-that the particle rules associate with the use of logical constants are crucial elements of their full-fledged (local) meaning: if meaning is conceived as constituted during interaction, then all of the actions involved in the constitution of the meaning of an expression should be made explicit; that is, they should all be part of the object-language.This perspective roots itself in Wittgenstein's remark according to which one cannot position oneself outside language in order to determine the meaning of something and how it is linked to syntax; in other words, language is unavoidable: this is his Unhintergehbarkeit der Sprache. According to this perspective of Wittgensteins, language-games are supposed to accomplish the task of studying language from a perspective that acknowledges its internalized feature. This is what underlies the approach to meaning and syntax of the dialogical framework in which all the speech-acts that are relevant for rendering the meaning and the "formation" of an expression are made explicit. In this respect, the metalogical perspective which is so crucial for model-theoretic conceptions of meaning does not provide a way out. It is in such a context that Lorenz writes:Also propositions of the metalanguage require the understanding of propositions, […] and thus cannot in a sensible way have this same understanding as their proper object. The thesis that a property of a propositional sentence must always be internal, therefore amounts to articulating the insight that in propositions about ...

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“Sind die Zahlformeln beweisbar?”

Klev

2024

Logic, Epistemology, and the Unity of Science

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The Justification of Identity Elimination in Martin-Löf’s Type Theory

Cited by 8 publications

References 12 publications

What Types Should Not Be†

What Types Should Not Be†

Dialogues, Reasons and Endorsement

“Sind die Zahlformeln beweisbar?”

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