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Results tagged with lo.logic
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user 1610
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
41
votes
Accepted
Presburger Arithmetic
Presburger arithmetic does NOT prove its own consistency. Its only function symbols are addition and successor, which are not sufficient to represent Godel encodings of propositions.
However, consiste …
- 35.5k
5
votes
Does type theory help us avoid the "defining postulate"?
Indeed, type theory does not need to add new axioms to represent definitions. The basic idea is that the formal language of type theory contains a binding form for terms -- something like:
$\mathsf{le …
- 9,201
21
votes
Proof assistants for mathematics
Honestly, part of the reason that proof assistants are focused on proving programs is precisely because of our very limited understanding of how to actually represent mathematics in formal logical sys …
- 1,446
65
votes
Accepted
Is there any formal foundation to ultrafinitism?
Wikipedia also says that Troelstra said in 1988 that there were no satisfactory foundations for ultrafinitism. Is this still true? Even if so, are there any aspects of ultrafinitism that you can get …
- 33.8k
7
votes
Are there any good nonconstructive "existential metatheorems"?
A nice source of examples of this kind are decidable theories whose decision procedures have very high computational complexities.
For example, consider Presburger arithmetic, the first-order theory …
- 47.7k
2
votes
Is there any proof assistant based on first-order logic?
Try Richard Bornat's Jape system. It's a teaching tool with a module for natural deduction, so there's a GUI. Proof automation is very limited, since the point is to teach people how to do formal proo …
- 103
18
votes
Can we disallow finite choice?
This is possible in constructive mathematics, because it distinguishes between finite sets and sets with a counted number of elements. (I'm not quite sure what the standard terminology is, though.)
A …
CommunityBot
- 1
4
votes
What are other theories of causality besides graphical models and Bayesian networks?
The standard account of causality is Lewis's theory of counterfactuals. He wrote a small, very readable book called Counterfactuals, which the SEP summarizes here. The idea is to take the viewpoint of …
- 9,201
17
votes
Is there any relationship between Bourbaki's Epsilon Calculus and Lambda Calculus? Is $\lamb...
Bourbaki's tau-box notation is somewhat insane (e.g., see Adrian Mathias's A Term of Length 4,523,659,424,929), so I'll eventually answer in terms of Hilbert's epsilon-calculus.
But first, the laws o …
- 9,201
5
votes
Accepted
Formal verification of simple equational proofs (as in Universal Algebra...)?
SMT (Satisfaction Modulo Theories) solving is pretty much the go-to technology for this these days, and works shockingly well in practice, often even on undecidable theories. Here are links to a few s …
- 9,201
20
votes
Proof strength of Calculus of (Inductive) Constructions
IIRC, the calculus of inductive constructions is equi-interpretable with ZFC plus countably many inaccessibles -- see Benjamin Werner's "Sets in Types, Types in Sets". (This is because of the presence …
- 9,201
16
votes
Logic in mathematics and philosophy
I agree with Timothy and Andrej's answers, and will complement them by suggesting a few books by philosophers and philosophically-inclined logicians which I have found very interesting. I am sure the …
- 9,201
9
votes
Accepted
Can a typing judgment admit essentially different derivations?
This property is called "coherence", and no, it doesn't always hold.
Establishing this property holds for a given semantics of proofs is a proof obligation. An example of when it doesn't arises with …
- 9,201
12
votes
The use of the word "model" in Mathematical Logic vs the same word in Natural Sciences
No, it's not strange. Understand why it's not strange, and you understand the essence of one of Frege's great innovations in logic: the so-called "linguistic turn", in which he taught us to shift from …
- 9,201
3
votes
What is the proper name for "compact closed" multiplicative intuitionistic linear logic?
Compact closed categories are models of classical linear logic when tensor and par collapse.
As an aside, I'm not sure that the particular resource interpretation you're suggesting genuinely works, s …
- 9,201