# General hypothesis vs particular hypothesis

everyone. It is my first time here.

I have always had a doubt in deciding whether a proposition can be true, false or neither, depending on the general hypothesis of the definitions in the context. It will be clearer with some examples:

1) Supose we are working in $$\mathbb{Z}$$ and define for $$a, b \in \mathbb{Z}, a \neq 0$$, $$a|b$$ if there exists $$k \in \mathbb{Z}$$ such that $$b = k \cdot a$$.

What can I say about the following sentence: $$0|1$$?

One could say that it is false because there not exists an integer $$k$$ such that $$1 = k \cdot 0$$, but I think it wouldn´t be correcte because $$a=0$$ was excluded in the definition of divisibility. Does this mean that the sentence $$0|1$$ has no sense? It can be neither true nor false? Is it even a proposition?

2) In real analysis, we know that, in the definition of limit of a real function of real variable, in the expression $$\lim_{x \rightarrow a} f(x)$$, we assume that $$a$$ is a limit point (or accumulation point) of $$Dom(f)$$. Then, if we consider a function $$f \colon D \rightarrow \mathbb{R}$$, with $$Dom(f)=\{2\} \cup (3, +\infty)$$, what can I say about the sentence: $$\lim_{x \rightarrow 2} f(x)=f(2)$$? Is it true, false, or none? Has it any sens? Is it a proposition?

I hope my question to be clear.

Thank you very much and sorry for my English.

• As you've suggested, both formulae are ill-defined; thus, neither is a proposition or even an open formula that has a varying truth value. May 14 at 8:13
• Thanks for your answer, ryang. I am new here and I tried to vote your comment. Accidentally I have undone my vote and now I cannot vote again. But I appreciate a lot your comment.
– Luks
May 14 at 15:59