Note that $24\mathbb Z$ is subgroup of abelian group $(\Bbb R,+)$ and thus abelian group $\Bbb R/24\Bbb Z$ can be formed. This will do exactly what you want: $x\equiv y\pmod {24}$ if and only if $x-y\in 24\Bbb Z$, i.e. $x$ and $y$ differ by a multiple of $24$.
Also note that $\Bbb R/24\Bbb Z$ is not a ring, since $24\Bbb Z$ is not ideal in $\Bbb R$ because $\Bbb R$ is a field.
Considering Euclidean division, $\Bbb R$ is a field, and thus trivially Euclidean domain. What I mean is that for $a,b\in\Bbb R$, $b\neq 0$, one can always find $q\in \Bbb R$ such that $a = qb$: namely, $q=\frac a b$.
I'm not sure what exactly you meant by extending Euclidean division, perhaps you wanted $q\in\Bbb Z$? This will yield the same thing as I describe above.
Perhaps it would be beneficial to elaborate what I mean by $\Bbb R/24\Bbb Z$ is abelian group, but not a ring. It boils down to saying that addition is well defined, but multiplication is not. For example take reals $24.1$ and $25.2$. Then we have $$24.1\equiv 0.1\pmod{24}\\ 25.2\equiv 1.2\pmod{24}$$ and if we add the values we have $$24.1+25.2\equiv 0.1+1.2\pmod{24}$$ but $$24.1\cdot 25.2 \not\equiv 0.1\cdot 1.2\pmod{24}$$
I think the pedagogical ambiguity here is best resolved exactly by not introducing a new term: instead use the totally unambiguous phrases "syntactic entailment" and "semantic entailment" (until fluency is achieved of course).
This is especially true since logic already suffers from an abundance of terminology (signature/vocabulary/type/alphabet; countable/denumerable/enumerable; recursively enumerable/computably enumerable/recognizable/semidecidable/; etc.).
That said, suppose one is absolutely dead-set on introducing a new notation; what's a least-bad choice?
"Allows" is in my opinion a poor choice here: there's no sense in which "$\vdash$" is less compulsory than "$\models$," so I don't see what difference is being emphasized. It's also highly misleading in that it would lead to the conclusion that more restrictive axiom sets allow more things. Adding axioms in the hope of removing inconsistencies is a common mistake students make ("Let's get rid of Russell's paradox by forbidding self-containing sets"), and this wouldn't help.
"Deduces" has the advantage of being fairly unambiguous and connecting with existing terminology ("natural deduction"). However, the grammar is horrible: "$\Gamma$ deduces $\varphi$" isn't right at all. What we should say is "From $\Gamma$ we can deduce $\varphi$" or similar, but that's a mouthful. I personally do think grammar matters in this case: using a strange grammar makes the terminology feel more alien.
The best one I can think of is "justifies." The idea here is that we think of statements involving $\vdash$ as taking place in some dialogical process, with our goal being to construct an argument.
But again, I really think that no new terminology should be introduced; rather, the existing terms "syntactic entailment" and "semantic entailment" should be used until comfort is achieved. Besides the reasons mentioned above, this terminology has one very useful advantage: it emphasizes the similarity between $\vdash$ and $\models$, which is really the surprising feature (ultimately justified by the completeness theorem).
- Of course, in more advanced topics in logic we'll sometimes want to go the other way and emphasize the difference between the two notions, or work in a context where one or the other doesn't even exist, but those situations won't arise until well after we've achieved a basic level of competence - at which point this won't be an issue anymore.
Best Answer
I am partial to what Omnomnomnom mentioned in a comment: it is a proportion, and the authoritative Oxford English Dictionary supports this terminology (snippet produced below):