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Simple question as above. Given "A provided that B",

the logical equivalence seems to be "A only if B".

Very startling that my lecturer enjoy using such 'informal' speak in his lecture.

Confirmation please.

Mathematicing
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  • Was it a lecture in math or in logic? If the former, then it is important to realize that mathematics is not logic, and is essentially a unique language. – Forgottenscience Oct 22 '15 at 07:14
  • You don't need the "only" part. A simple "if" will do. – mhp Oct 22 '15 at 07:15
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    @mhp "If" is not short for "only if", it's the converse. "A provided that B" = "A if B" = "$B\implies A$, while "A only if B" = "$A\implies B$" . – bof Oct 22 '15 at 07:21
  • @bof Agreed. you make a good point. +1 – mhp Oct 22 '15 at 07:24

2 Answers2

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It's not informal: just another way of saying "as long as", "when", or "if" in a narrative.

"$n$ is not prime provided that it is even and sufficiently large", for example, is equivalent to "if $n$ is even and sufficiently large, then it is not prime".

If you find "provided that" informal, why do you accept "if"? They're all unambiguous. You could ask your professor to be more limited in their vocabulary, but it's the kind of thing it's very hard to notice yourself doing. It is good practice, though, when stating theorems formally, to use "if… then…", just because it's easiest to understand at a glance.

Patrick Stevens
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  • But hardly is "as long as", "when", et cetera used in most mathematics text book. Natural language is very ambiguous. If you are referring to the conflation between "if" and "only if" then yes I do make a distinction even in everyday speak. – Mathematicing Oct 22 '15 at 07:22
  • You must never conflate "if" with "only if". No mathematician will do that intentionally. If a mathematician says "if" or "only if", then they intend "if" or "only if" in the technical sense, with the possible exception of when they're speaking in a non-mathematical context. "When" is used in mathematical textbooks, I believe - certainly "just when" is used as a synonym for "iff". "As long as" may be less common, but it's still totally unambiguous. I'm not sure how you managed to misinterpret it if you're a native English speaker. – Patrick Stevens Oct 22 '15 at 07:30
  • Precisely my point. Most human beings conflate "if" with "only if" which is a source of trouble. After a while, especially with many years of formal mathematics education, one does restrict the use of conditions, even in everyday speak, to "if" and "only if". The computer science people have it more severe. – Mathematicing Oct 22 '15 at 07:35
  • @Mathematicing The most venerable maths lecturers I ever had were very free with what they used - "if", "when", "as long as", etc. – Patrick Stevens Oct 22 '15 at 07:39
  • @ryang "n^2 is even as long as n is even" means "if n is even, then n^2 is even". E.g. "as long as" is sort of a synonym in ordinary English for "while", and "while" is nearly a synonym for "if". But this is a fuzzy enough case that I'd generally avoid "as long as" :P – Patrick Stevens Dec 29 '21 at 09:15
  • Okay, so 'provided that' and 'as long as' both mean 'if'. I've always informally used 'when' synonymously with 'if', but now I'm wondering whether 'when/while' is closer in meaning to 'iff'. (I'm careful about using 'where'.) – ryang Dec 30 '21 at 04:43
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It is equivalent to "if." By the way, one could claim that your use of "only if" is just as 'informal' as your professors use of 'provided that.' "Only if" could potentially be confused with "if and only if," which DOES have a different meaning.

I hope this doesn't come off as condescending at all, but I think there is a phenomenon sometimes of people first learning how to "speak math" and then using it very strictly, only to learn that, as a practical manner, we often use language which you might call 'informal.' As someone else pointed out, 'if' itself doesn't really have a definition. At some point we must use these words and move on.

Lastly, if you happen to use english as a second language, then that could be an alternative source of confusion, which is a different discussion.

pancini
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    English happens to be my first language but as have been posted earlier, natural language is a source of confusion. Case in point, after a while it becomes hard to recognise what "provided A then B" means. Most computer languages and mathematics literature stick to "if" and "only if". Edited – Mathematicing Oct 22 '15 at 07:25
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    No disrespect, there are just many non-native speakers on this site. I agree; math language will always have some ambiguity, unfortunately. I guess all we can do is aim for consistency at least. – pancini Oct 22 '15 at 07:27
  • I cannot understand how you are being disrespectful. I see your point. – Mathematicing Oct 22 '15 at 07:47
  • Ironically, many non-technical sources (e.g., here and here) claim that 'provided that' means 'only if'. (Not saying that I agree.) Amen to the point about at least aiming for consistency. – ryang Dec 30 '21 at 05:49