If I say "A provided that B", I generally mean that if A is true, it must be that B is true. Doesn't that make sense? So that would translate to A => B. Context is always important, however.
What does "provided that" mean in terms of logic symbols? Is it a conditional implication (=>) or a biconditional implication(<=>)? I see it used primarily as a biconditional implication but sometimes (in the same book) it appears as though it is being used as a conditional implication.
Am I the only one who thinks that "A provided that B" means B => A? In my opinion, this is equivalent to "A assuming that B" and one usually assumes hypotheses, not conclusions...
This Wikibooks page has the same opinion. (Consider it as a sanity check, not necessarily as an authoritative source.)
It might be a peculiar subtlety of English.
"A provided that B" suggests to me "A -> B"
"If we are provided B then we have A" suggests to me "B -> A".
"I'll give you an apple provided that you give me a banana" would likely be taken as "I won't give you an apple unless you give me a banana", i.e. "A -> B"
"If we are provided that 'I give you give me a banana' is true then 'I give you ana apple' is true" means "If you give me a banana then I give you an apple", i.e., "B -> A".
The site says that "provided" can be swapped for if and is equivalent to p => q
and that "only provided" can be swapped for (if and only if)
I guess my problem now lies in the ambiguity that this causes.
two nonzero vectors v and w are perpendicular if the dot product of v and w is equal to zero
In this case if means if and only if
and often do i see provided being used in this way, i.e.
two nonzero vectors v and w are perpendicular provided that the dot product of v and w is equal to zero
so, you can see my problem (or my rage at people who choose to use this word) is there a definitive way to use provided?