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Results tagged with lo.logic
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user 2926
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.
2
votes
In which sense "closure" is a closure?
Actually, there is a topological meaning to closure of logical formulas, if one represents formulas by string diagrams. A description of such a string diagram calculus, interpreting Peirce's existenti …
- 53.3k
2
votes
Accepted
Can the Multiplicative Fragment of Linear Logic be shown to be non-truth-functional?
I don't have a complete proof, but I'm rather skeptical of the existence of such formulas. Certainly in the unit-free fragment of MLL it's hopeless, by applying a proof net criterion. For example, usi …
- 53.3k
15
votes
Accepted
Logical complexity of algebraically closed fields
From Dirk van Dalen's Logic and Structure: the theory of algebraically closed fields is not finitely axiomatizable (see page 109 and preceding).
- 53.3k
2
votes
Injecting premises into two implicational premises connected by a tensor (multiplicative con...
Well, this is very easy, but because linear logic might be considered a little too specialized for Mathematics StackExchange, I'll answer.
Since the natural semantics of MLL (multiplicative linear l …
- 53.3k
3
votes
"Introduction to mathematical logic" book from a formalist perspective
I don't know of logic textbooks which adopt an explicitly formalist viewpoint (which is not to say I think none exist), but for what it's worth, I'd say that books on categorical logic tend not to ado …
0
votes
How do quantifiers limit scope?
You have a right idea, but it might help to refine your notation this way: write $\forall_{x: X} Q(x)$ and $\exists_{x: X} Q(x)$ whenever you mean to interpret your variable $x$ as ranging over elemen …
- 53.3k
4
votes
Accepted
name my cat: regular categories where inverse images also have right adjoint
From Freyd and Scedrov's book Categories, Allegories: a logos is a regular category in which $Sub(A)$ is a lattice for each object $A$, and in which the inverse-image operation $f^*: Sub(B) \to Sub(A) …
- 53.3k
6
votes
Accepted
Calculus of Binary Relations
As I understand it, and in more modern-day terms, the question asks whether it is possible to define the operations $0, 1, \cap, \cup, \neg$ on $P(X^2)$ (operations belonging to the "static component" …
- 53.3k
4
votes
Is there any literature about inner-replacement rule?
C.S. Peirce, in his calculus of Existential Graphs, had a rule very similar to this which he called "weakening" or "rules of insertion and erasure". The Wikipedia article is a little skimpy on this, b …
- 53.3k
17
votes
Is there a constructive proof of Cantor–Bernstein–Schroeder theorem ?
If you accept that toposes are models of constructive set theory, then another way to answer the question is to give a (non-Boolean) topos where the CBS theorem fails; that would show that this theore …
- 53.3k
9
votes
Does "compact iff projections are closed" require some form of choice?
In case anyone is interested, this is a rendition of the proof I was looking for, contained in the article pointed out by Andrej Bauer. (I'm not looking for upvotes; this is just to round out the disc …
- 53.3k
4
votes
Accepted
For the symmetric group on an infinite set, is there a generating set of strictly smaller ca...
It seems clear that the answer to the first and third questions is 'no'. Indeed, if a set of generators $X$ is of infinite cardinality $\alpha$, then the group so generated cannot have cardinality gre …
- 53.3k
10
votes
Basic results with three or more hypotheses
A compact convex subset of $\mathbb{R}^n$ with nonempty interior is homeomorphic to the $n$-dimensional ball.
44
votes
Accepted
Completion of a category
Yes, it's a general construction which is related to so-called Isbell conjugation.
Let $C$ be a small category. It is well-known that the free colimit cocompletion is given by the Yoneda embedding i …
- 53.3k
8
votes
Accepted
Adjoining an arrow to a CCC
They mean this: given a cartesian closed $\mathcal{A}$ and objects $A, B$ of $\mathcal{A}$, the inclusion $i: \mathcal{A} \to \mathcal{A}[x]$ is universal with respect to strict cartesian closed funct …
- 53.3k