1. ## provided that

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.

2. 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.

3. I sometimes come across: A ==> B, provided that C

4. Originally Posted by Noxide
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.
A provided B, means that A happens only if B happens.
“A only if B “ means “If A then B”.

5. delete

6. 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.)

7. It might be a peculiar subtlety of English.

"A provided that B" suggests to me "A -> B"

but

"If we are provided B then we have A" suggests to me "B -> A".

Example:

"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"

but

"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".

8. Originally Posted by emakarov
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.)
I thought the same thing, but then started to doubt my thoughts...

9. Surely “A provided B” means “B is necessary for A”.

10. 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.

for example

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?

11. Originally Posted by Noxide
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.

for example

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?
If and only if is not used here because this is a definition, not a theorem. You're saying if this happens then I'll call it this'.

12. is there a definitive way to use provided?
Yes: not to use it (PDF document).

13. Originally Posted by Swlabr
If and only if is not used here because this is a definition, not a theorem. You're saying if this happens then I'll call it this'.

That's just completely wrong. Those two statements are equivalent and by convention if is used to denote if and only if when used in a definition.

14. Originally Posted by Noxide
That's just completely wrong. Those two statements are equivalent and by convention if is used to denote if and only if when used in a definition.
And conventions differ between different people. I, for one, have never seen definitions use "if and only if".

15. For me, definitions are always if-and-only-if. Sometimes it's explicit; but if not, I've always understood that feature to be implicit.