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I am having a hard time understanding this section in Wikipedia's article on Logical biconditionals:

Colloquial usage

One unambiguous way of stating a biconditional in plain English is of the form "b if a and a if b". Another is "a if and only if b". Slightly more formally, one could say "b implies a and a implies b". The plain English "if'" may sometimes be used as a biconditional. One must weigh context heavily.

For example, "I'll buy you a new wallet if you need one" may be meant as a biconditional, since the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional). However, "it is cloudy if it is raining" is not meant as a biconditional, since it can be cloudy while not raining.

Should the example read:

"I'll buy you a new wallet if you need one" may be meant as a biconditional, since the speaker does intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a biconditional).

My question is how can the plain English "if'" sometimes be used as a biconditional? I'm OK with the word "biconditional." I don't understand how the reader is to know the "speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional)" especially how this amounts to "(as in a conditional)".

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    English is not math, and the word 'if' in English should not be taken to be equivalent to mathematical implication. For example, there's the biscuit conditional: "There's a biscuit on the counter, _if_ you're hungry." There's no conditional implied: it's there whether or not you're actually hungry.2016-05-16

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I think the biconditional means:

Buy wallet $\iff$ you need the wallet

So, by taking the contrapositive,

Don't buy wallet $\iff$ You don't need the wallet.

Hence, Wikipedia means that the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed , since if the wallet is not needed, the outcome is "Don't buy wallet".

If it were a conditional, i.e. "You need wallet $\Rightarrow$ Buy wallet", note that if "You need wallet" is false, "Buy wallet" could still be true and satisfy the truth table of the conditional.

Hope it helps.

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    @skullpatrol: The parenthetical (as in a conditional) refers to buying the wallet whether or not the wallet is needed .2012-07-22
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It is lucky that expressing a biconditional in a mathematical statement is easy; there is an unambiguous mathematical meaning for "if and only if". Perhaps the philosophy.SE would be more inclined to give you a more in-depth answer. Implication is a subject of interest there since it isn't always merely the material conditional, $P\rightarrow Q$ meaning $Q\vee \neg P$, but can also have modal quantifiers. That would be something like "Necessarily, P implies Q" vs. "P implies (necessarily Q)".

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No. Let $Q$ be the statement "I will buy you a wallet" and $P$ the statement "You need a wallet. "I'll buy you a wallet if you need one" could conceivably be translated as $P \Rightarrow Q.$ However, I hope you would agree that buying a wallet for someone who doesn't need one is quite silly - thus buying someone a wallet implies that they had needed one, ie) $Q \Rightarrow P.$ Hence $(Q\Rightarrow P ) \wedge (P \Rightarrow Q),$ or $P \iff Q$

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    @MarcvanLeeuwen Thank you for clearing that up.2012-07-13