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2019 | OriginalPaper | Chapter

4. Logical Preliminaries

Authors : Patrick Schultz, David I. Spivak

Published in: Temporal Type Theory

Publisher: Springer International Publishing

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Abstract

In this chapter, we transition from an external point of view to an internal one. In Chap. 2 we defined the interval domain \(\mathbb {I\hspace {1.1pt} R}\), and in Chap. 3 we defined a quotient \(\mathcal {B}\cong \mathsf {Shv}(S_{\mathbb {I\hspace {1.1pt} R}/\rhd })\) of its topos of sheaves. A main goal of this book is to define a temporal type theory—including one atomic predicate and ten axioms—that has semantics in \(\mathcal {B}\); we do this in Chap. 5. In the present chapter, we attempt to provide the reader with a self-contained account of the sort of type theory and logic we will be using, as well as some important concepts definable therein.

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previous chapter Translation Invariance
next chapter Axiomatics
Footnotes
1
The dependent type theory in [Mai05] is sound and complete for 1-toposes, relying heavily on extensivity of types. This is fine for 1-toposes, but it makes computability impossible and hence is not useful for a proof assistant like Coq. On the other hand, the Calculus of Constructions [CH88] and related formalisms used in Coq and Lean are formulated using a hierarchy of universes. We believe this aspect of the Calculus of Constructions is sound for toposes, and we believe this follows from [Str05], but could not find an explicit reference.
 
2
Note that what we call sum types are called “coproduct types” in [Jac99, Section 2.​3].
 
3
The propositions ¬P and P ⇔ Q are just shorthands for P ⇒⊥ and (P ⇒ Q) ∧ (Q ⇒ P).
 
4
It is natural to want to extend this idea to contexts with more than one variable, but that takes us into dependent types; see Sect. 4.1.4.
 
5
As with subtypes, it is natural to want to extend the above idea to contexts with more than two variables of the same type, but doing so takes us into dependent types; see Sect. 4.1.4.
 
6
It will turn out that the type https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-00704-1_4/457670_1_En_4_IEq108_HTML.gif of real numbers is constant in our main topos of interest, \(\mathcal {B}\), but not in many of the subtoposes we consider. For example, the type https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-00704-1_4/457670_1_En_4_IEq110_HTML.gif of real numbers in \(\mathcal {B}_\pi \) does not have decidable equality, roughly because continuous functions that are unequal globally may become equal locally.
 
7
Throughout this book, we use δ (delta) and d for “down” and υ (upsilon) and u for “up.” Thus δ classifies a “down-set” of https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-00704-1_4/457670_1_En_4_IEq119_HTML.gif , and υ classifies an “up-set.”
 
8
Throughout this section, we will use Remark 4.9 without mentioning it.
 
9
In Definition 4.21 we gave the warning that one should be a bit careful with the < relation on other numeric types. As an example of what can violate standard intuition, let https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-00704-1_4/457670_1_En_4_IEq273_HTML.gif be given by δq ⇔ q < 1 and υq ⇔−1 < q. Then x < x holds.
For the one-sided numeric types, e.g. https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-00704-1_4/457670_1_En_4_IEq274_HTML.gif , there is a good notion of ≤, namely https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-00704-1_4/457670_1_En_4_IEq275_HTML.gif . To obtain < one might try (δ ≤ δ ′) ∧ (δ ≠ δ ′), but this is semantically too strong in a general topos.
 
Literature
[BT09]
[CH88]
[Ded72]
Dedekind, R.: Stetigkeit und irrationale Zahlen. F. Vieweg und sohn (1872)
[Gol11]
Goldsztejn, A.: Modal intervals revisited, part 1: a generalized interval natural extension. Reliab. Comput. 16, 130–183 (2011). ISSN:1573–1340
[Jac99]
Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141, pp. xviii+ 760. North-Holland, Amsterdam (1999). ISBN:0-444-50170-3
[Joh02]
Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium. Oxford Logic Guides, vol. 43, pp. xxii+ 468+ 71. New York: The Clarendon Press/Oxford University Press (2002). ISBN:0-19-853425-6
[Kau80]
Kaucher, E.: Interval analysis in the extended interval space IR. In: Fundamentals of Numerical Computation (Computer-Oriented Numerical Analysis), pp. 33–49. Springer, Berlin (1980)
[LS88]
Lambek, J., Scott, P.J.: Introduction to Higher Order Categorical Logic. Cambridge Studies in Advanced Mathematics, vol. 7, pp. x+ 293. Reprint of the 1986 original. Cambridge University Press, Cambridge (1988). ISBN:0-521-35653-9
[Mai05]
[MM92]
MacLane, S., Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer, New York (1992). ISBN:0387977104
[Sai+14]
[Str05]
[Voe+13]
Voevodsky, V., et al.: Homotopy type theory: Univalent foundations of mathematics. In: Institute for Advanced Study (Princeton). The Univalent Foundations Program (2013)
Metadata
Title
Logical Preliminaries
Authors
Patrick Schultz
David I. Spivak
Copyright Year
2019
Book
Temporal Type Theory

Print ISBN: 978-3-030-00703-4

Electronic ISBN: 978-3-030-00704-1

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-00704-1_4

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