Is the following a valid logical proposition?(adsbygoogle = window.adsbygoogle || []).push({});

Proposition:If##\mathcal{S}## is a collection of subsets of ##X## that covers ##X##thenthe collection ##\mathcal{B}## of all finite intersections of elements of ##\mathcal{S}## is a basis for a topology on ##X##.

Firstly, when I state a proposition, is it implied that what I mean is "proposition: it is true thatif...then..."? if not, what does it mean to propose an implication?

Anyway, I ask if the above is a valid logical proposition because the second statement in that implication is not defined on its own, it needs the first statement since ##\mathcal{S}## is defined as a collection of subsets of ##X##onlyon the first statement. Would ##\mathcal{S}## qualify as a free variable? In which case the proposition is not a logical proposition in classical logic?

I guess my question could be generalized to: in classical logic, if I propose that ##x\rightarrow y##, is it a convention that what I am proposing is that ##x\rightarrow y=1##, and also, could I make that proposition if ##y## only makes sense if ##x## is true?

NOTE: The topology example is only anexampleto help me illustrate my question, it is not the actual question.

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# Defining things in logical statements

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