The type-theory tag has no usage guidance.

**6**

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**4**answers

703 views

### What is the intuitive meaning of star and box in a pure type system?

The systems of the λ-cube have the axiom $\star:\square$.
I've listed a few meanings that the Curry-Howard isomorphism gives to $t : T$ below. What are the intuitive meanings of $\star$ and ...

**1**

vote

**2**answers

957 views

### What is a semigroup or, what do I do with that associativity proof?

Mathematically, I know what a semigroup is: It is a set S along with an associative binary operation $* : S \times S \rightarrow S$. So far, so good.
From a computational perspective, one can ...

**11**

votes

**4**answers

887 views

### Reference request for type theory

I am interested in learning the theory of types, especially in how they can provide a foundation to mathematics different to sets and how they can avoid self-referential paradoxes by stipulating that ...

**12**

votes

**2**answers

2k views

### What is the manner of inconsistency of Girard's paradox in Martin Lof type theory

I am aware that assigning the type of Type to be Type (rather than stratifying to a hierarchy of types) leads to an inconsistency. But does this inconsistency allow the construction of a well-typed ...

**7**

votes

**4**answers

954 views

### Can dependent sums be encoded as dependent products?

Please forgive any unorthodox notation or obvious errors here... I'm trying to get an intuition for dependently typed languages, so I'm starting out by seeing which analogies I can take from the ...

**3**

votes

**1**answer

485 views

### What notions of universe does predicative type theory admit?

Palmgren (1997), On universes in type theory, discusses work of several theorists that provide what we might call a family of Large Universe Axioms (LUAs) for predicative type theory, culminating in ...

**6**

votes

**2**answers

671 views

### Recursively dependent types?

Is there such a thing as "recursively dependent types"? Specifically, I would like a dependent type theory containing a type $A(x)$ which depends on a variable $x: A(z)$, where $z$ is a particular ...

**13**

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**7**answers

2k views

### What is lambda calculus related to?

So I'm not much of a math guy but I've really enjoyed programming in Lisp and have become interested in the ideas of lambda calculus which it is based.
I was wondering if anyone had a suggestion ...