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by exceptione 386 days ago
> But in a logic,

I am not sure if I misunderstand you. Types are for domain, real world semantics, they help to disambiguate human language, they make context explicit which humans just assume when they talk about their domain.

Logic is abstract. If you implied people should be able to express a type system in their host language, that would be interesting. I can see something like Prolog as type annotations, embedded in any programming language, it would give tons of flexibility, but then you shift quite some burden onto the programmer.

Has this idea been tried?
1 comments
practal 386 days ago
Types for real-world semantics are fine, they are pretty much like predicates if you understand them like that.

The idea to use predicates instead of types has been tried many times; the main problem (I think) is that you still need a nice way of binding variables, and types seem the only way to do so, so you will introduce types anyway, and what is the point then? The nice thing about AL is that you can have a general variable binding mechanism without having to introduce types.

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frumplestlatz 386 days ago
AL as described sounds like it reinvents parts of the meta-theoretic infrastructure of Isabelle/HOL, but repackaged as a clean break from type theory instead of what it seems to be — a notational reshuffling of well-trod ideas in type theory.

What am I missing?

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practal 386 days ago
Given that I am an Isabelle user and/or developer since about 1996, similarities with Isabelle are certainly not accidental. I think Isabelle got it basically right: its only problem (in my opinion) is that it is based on intuitionistic type theory as a metalogic and not abstraction logic (nevertheless, most type theorists pretty much ignored Isabelle!). Abstraction logic has a simple semantics; ITT does not. My bet is that this conceptual simplicity is relevant in practice. We will see if that is actually the case or not. I've written a few words about that in the abstraction logic book on page 118, also available in the sample.

> — a notational reshuffling of well-trod ideas in type theory

Always fun to encounter disparaging comments (I see that you deleted the other one in the other thread), but I wrote this answer more for the other readers than for you.

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frumplestlatz 385 days ago
> Always fun to encounter disparaging comments

I can't imagine why anyone would ever treat your bombastic proclamations about type theory with suspicion.

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practal 385 days ago
What is bombastic about them?
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