 Sep 2023

en.wikipedia.org en.wikipedia.org

en.wikipedia.org en.wikipedia.org

In type theory, product types (with no field names) are generally preferred due to their simplicity

 Dec 2022

math.stackexchange.com math.stackexchange.com

My freely downloadable Beginning Mathematical Logic is a Study Guide, suggesting introductory readings beginning at subMasters level. Take a look at the main introductory suggestions on FirstOrder Logic, Computability, Set Theory as useful preparation. Tackling midlevel books will help develop your appreciation of mathematical approaches to logic.
This is a reference to a great book "Beginning Mathematical Logic: A Study Guide [18 Feb 2022]" by Peter Smith on "Teach Yourself Logic A Study Guide (and other Book Notes)". The document itself is called "LogicStudyGuide.pdf".
It focuses on mathematical logic and can be a gateway into understanding Gödel's incompleteness theorems.
I found this some time ago when looking for a way to grasp the difference between firstorder and secondorder logics. I recall enjoying his style of writing and his commentary on the books he refers to. Both recollections still remain true after rereading some of it.
It both serves as an intro to and recommended reading list for the following:  classical logics  first & secondorder  modal logics  model theory<br />  nonclassical logics  intuitionistic  relevant  free  plural  arithmetic, computability, and incompleteness  set theory (naïve and less naïve)  proof theory  algebras for logic  Boolean  Heyting/pseudoBoolean  higherorder logics  type theory  homotopy type theory

 Nov 2021

www.typescriptlang.org www.typescriptlang.org

the name union comes from type theory. The union number  string is composed by taking the union of the values from each type. Notice that given two sets with corresponding facts about each set, only the intersection of those facts applies to the union of the sets themselves.

For example, if we had a room of tall people wearing hats, and another room of Spanish speakers wearing hats, after combining those rooms, the only thing we know about every person is that they must be wearing a hat.

 Mar 2021

en.wikipedia.org en.wikipedia.org

How is it that https://en.wikipedia.org/wiki/Type_theory links to https://en.wikipedia.org/wiki/Type_(model_theory) but the latter does not have any link to or mention of https://en.wikipedia.org/wiki/Type_theory
Neither mentions the relationship between them, but both of them should, since I expect that is a common question.


en.wikipedia.org en.wikipedia.org

In fact categories can themselves be viewed as type theories of a certain kind


en.wikipedia.org en.wikipedia.org
 Jul 2020

osf.io osf.io

Halford, E., Dixon, A., Farrell, G., Malleson, N., & Tilley, N. (2020). Crime and coronavirus: Social distancing, lockdown and the mobility elasticity of crime [Preprint]. SocArXiv. https://doi.org/10.31235/osf.io/4qzca

 Dec 2015

math.mit.edu math.mit.eduCT4S.pdf4

Since ducks can both swim and fly, each duck is found twice inC, once labeled as aflyer and once labeled as a swimmer. The typesAandBare kept disjoint inC, whichjustifies the name “disjoint union.”
The disjoint union reminds me of algebraic datatypes in functional programming languages, whereas a settheoretic union is more like a union in CS: the union has no label associated with it, so additional computation (or errors) may arise due to a lack of ready information about elements in the union.

An aspect of a thingxis a way of viewing it, a particular way in whichxcan be regardedor measured. For example, a woman can be regarded as a person; hence “being a person”is an aspect of a woman. A molecule has a molecular mass (say in daltons), so “havinga molecular mass” is an aspect of a molecule. In other words, byaspectwe simply meana function. The domainAof the functionf:A—Bis the thing we are measuring, andthe codomain is the set of possible “answers” or results of the measurement.
Naïvely (since my understanding of type theory is naïve), this seems to mesh with the concepts of inheritance for the "is" relationships, and also with typetheory more generally for "has" relationships, since I believe we can view any object or "compound type", as defined here, as being a subtype of another type 'o' if one of its elements is of type 'o'. Though we have to be careful for functional mapping when thinking of aspects: we can't just say Int is an aspect of Pair(Int, Int), since this is ambiguous (there are two ints)  we must denote which Int we mean.

We represent eachtype as a box containing asingular indefinite noun phrase.

In 1980 Joachim Lambek showed that the types and programs used in computerscience form a specific kind of category. This provided a new semantics for talking aboutprograms, allowing people to investigate how programs combine and compose to createother programs, without caring about the specifics of implementation. Eugenio Moggibrought the category theoretic notion of monads into computer science to encapsulateideas that up to that point were considered outside the realm of such theory.
