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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.
7
votes
Accepted
How much of the current logic is about syntax?
I think your observation is a very good one, but this phenomenon is limited to classical logic and does not continue to hold when we move to intuitionistic or substructural logics.
One way of underst …
- 9,201
10
votes
Impredicativity
Andrej's answer is great, but I want to elaborate on one point: often, there are multiple, incompatible, ways to strengthen a formalism. Developing a piece of mathematics using relatively weak techniq …
- 9,201
11
votes
Why are universal introduction and existential elimination valid inference rules?
You are getting tripped up by some very traditional, yet very bad, notation.
The $k$ in these formulas are not true constants of the domain of individuals, but rather are Skolem constant. The idea i …
- 9,201
2
votes
Are all mathematical theorems necessarily true?
Charles wrote:
Robert Hanna (Kant's Theory of Judgment, SEP 2009, sect. 2.2.2) interprets Kant as saying that "logically possible worlds are nothing but maximal logically consistent sets of concep …
- 9,201
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 …
- 9,201
4
votes
What does the disjunction elimination rule say?
Yes, the first version of your rule is incomplete. In natural deduction style, the elimination rule for disjunctions is:
$$\frac{\Gamma \vdash A \vee B \qquad \Gamma, A\vdash C \qquad \Gamma, B\vdas …
- 9,201
6
votes
A book explaining power and limitations of Peano Axioms?
Several people have given excellent answers to your second question, so let me address the first.
2) As far as I remember, PA do not have a "built-in" scheme for inductive definitions. So I assum …
- 9,201
7
votes
Can we prove set theory is consistent?
Is there any reason to believe that Set1 cannot prove the consistence of Set2? Or I'm just confused and what I said does not make sense?
What you're asking does make sense, but there are good inf …
- 9,201
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
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
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
23
votes
Accepted
How do proof verifiers work?
What exactly is the role of type theory in creating higher-order logics? Same goes with category theory/model theory, which I believe is an alternative.
Don't think of type theory, categori …
- 9,201
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
4
votes
When can we prove constructively that a ring with unity has a maximal ideal?
You should take a look at Coquand and Lombardi's "A Logical Approach to Abstract Algebra".
They observe that commutative rings have a purely equational description, and so there are very strong meta …
- 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