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I have an exercise book from my university which doesn't specify a quantifier.

It uses expressions like "here $A$,$B$,$C$ are sets", or "if $x \notin A$ then ..." (it uses $x$ before it is even defined, just out of nowhere).

I'm going to need an answer from my university, of course, but I want to ask in general context: Is there any implicit quantifier when one is not specified, or is it always an error? If there is an implicit quantifier, which is it?

MasterMastic
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  • Next time you ask a question of this sort don't accept the first answer you get within a few minutes. – Christian Blatter Nov 10 '14 at 15:52
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    @ChristianBlatter Why? what do you mean "this sort"? And by the way, I can still choose a different answer, so if you have a better answer I'd love to hear and accept it. – MasterMastic Nov 10 '14 at 16:01

2 Answers2

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Generally speaking it is very, very common in mathematics to write a natural language statement like "if $x \in X$ then BLAH BLAH" which then translates into the more explicitly quantified statement "for all $x \in X$ BLAH BLAH". There are many, many other dictionary entries which translate between natural language mathematical statements and explicitly quantified statements. You can probably encounter many such dictionary entries in any entry level calculus textbook, where formal quantification is usually treated on an intuitive level.

Lee Mosher
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I, personally, consider it very bad practice to not quantify every variable. Mathematicians are divided in this respect and one can even note that mathematicians from certain fields do this kind of thing more often than mathematicians in other fields.

Basically it's a matter of opinion whether it is incorrect or not.

Usually, if a quantifier is missing, the variable is assume to be quantified universally.

Git Gud
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  • I really can see why different fields would train people differently. When I saw anything used before it has even been declared I was shocked that this was academic / high-education studies, and as a programmer, especially so. I agree that it must be a very bad practice, and I wholeheartedly wish it was an absolute error (or a crime :P). Thank you! – MasterMastic Nov 10 '14 at 15:31