Logic structure versus grammar rules

Question

When it comes to the some words in a logic or math context, say implizieren and enthalten for sake of concreteness, the grammar rules and the logic structure do not always seem to peacefully coexist. When writing math in German which one should be given preference?

Enthalten

If you want to mention a set inclusion. If A and B are sets. If I mean A ⊂ B, I am always unsure about the syntax. Both seem be objectionable:

Option a.

(1.a) Daher enthält B A

Here since B and A are not given cases, I don't know whether it's clear which contains which.

or

Option b.

(1.b) Daher B enthält A.

Ja, aber... then the verb is not at the second place, which is a grammar mistake.

Implizieren

The same happens while facing subordinate clauses. If p and q are two logical statements,

Option a. One violates the Verb-am-Ende-rule, but it's clear which implies which:

(2.a) (…), woraus folgt, dass p impliziert q

or

Option b. Good grammar but, which implies which? If p and q cannot be given a case, then it's hard to decide:

(2.b) (…), woraus folgt, dass p q impliziert.

Existieren

I've usually seen (e.g. here in Aufgabe 47 (c) ) something which, simplified, would read:

(…), dass genau ein x ∈ X existiert mit x>0.

rather than the expected

(…), dass genau ein x ∈ X mit x>0 existiert.

Of course, usually you don't only get x>0, but a Wurst of conditions, which makes leaving the verb at the end unreadable.

As far as I know (and from my quite short experience by reading mathematical texts in German since all math is in English) preference is given to the logical structure over the grammar. I'd like this to be confirmed here. I'm not wishing to reorder. I would like to know what to do, when there is no other option.

Answer 1

(1.b) und (2.a) "gehen gar nicht". (1.a) und (2.b) sind eindeutig, können mündlich problemlos benutzt werden, und (2.b) hört sich auch flüssig an. Allerdings sollten in einem gesetzten Text nicht zwei Formeln direkt aufeinander folgen, so dass Du etwas dazwischen einfügen musst, zum Beispiel

Daher enthält $B$ die Menge $A$.

Und wenn wir schon Wörter einfügen, dann vielleicht gleich

Daher ist $A$ eine Teilmenge von $B$.

Das löst auch das Problem, dass "B enthält A" sowohl "A ist Teilmenge von B" als auch "A ist Element von B" bedeuten kann.

Das Beispiel mit der Aufgabe ist nicht ganz vergleichbar, da die erste Formulierung zwar sprachlich etwas holprig, aber nicht grammatisch falsch ist.

Answer 2

I quite disagree. 1.b and 2.a read horribly. 1.a and 2.b are much preferable even though they juxtapose subject and object with no case markers.

I agree that a way should be sought to make those sentences flow better, but not by reordering. You might say "Daher enthält A das Element B", or "woraus folgt: P impliziert Q." or even "Daraus folgt: P impliziert Q."

Answer 3

I have come across the kind of expression that you mean once in a while, but never ever outside mathematical (or strictly logical) context. It is a very specific form sometimes used in the way of seeing the logical expression (p impliziert q) in its role as the expression of a thought, and not in its role of being a grammatical part of a sentence. Like a semantic wildcard.

Woraus folgt, dass {insert thought here}.

I used to just accept it, because a certain neglect of good language seemed to be common or even intrinsic to mathematical papers. However, I'd say that 1.b and 2.a are definitely wrong.

On second thought, there is also a possibility someone wanted to arrange a sentence to look like this:

Woraus folgt, dass:
{Absatz, Kästchen, farbige Hinterlegung} p impliziert q

In that case, the phrasing would not follow logic, but the text layout, assuming that someone who wants to follow the important facts rather than the prose would pay more attention to the highlighted areas. "Dass" would still look out of place, but the colon explicitly separates the two, so it would be grammatically correct.