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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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closed as not a real question by Pedro Tamaroff, Amzoti, Martin, O.L., tomasz May 31 '13 at 18:36

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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I'm not sure there is a mathematical question here. It may belong better on English.SE? –  Henning Makholm Jul 13 '12 at 10:45
    
@HenningMakholm As the wikipedia article states: "One unambiguous way of stating a biconditional in plain English is .." Is not stating conditions unambiguously in plain English of interest to students of math? –  skullpatrol Jul 13 '12 at 10:53
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The interest of mathematics students is to express mathematical statements unambiguously. Luckily, real world objects such as wallets, money, and the acts of buying are not mathematical statements. –  Asaf Karagila Jul 13 '12 at 11:43

3 Answers 3

up vote 1 down vote accepted

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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Thank you for the answer. Are you saying that " buying a wallet for someone who doesn't need one is quite silly" is the same as "the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed"? If so, the parenthetical statement in the article should be changed from "(as in a conditional)" to (as in a biconditional). –  skullpatrol Jul 13 '12 at 10:43
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@Former_Math_Addict: the cited phrase is not beautiful, but it means that the dismissed scenario (namely buying whether or not) would be valid if (only) the conditional is claimed, but is not valid if the biconditional is claimed. So the parenthetical statement says what it intends to. –  Marc van Leeuwen Jul 13 '12 at 11:38
    
@MarcvanLeeuwen Thank you for clearing that up. –  skullpatrol Jul 13 '12 at 11:53

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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Thank you for your answer. Should the parenthetical statement in the article be changed from "(as in a conditional)" to (as in a biconditional)? –  skullpatrol Jul 13 '12 at 11:02
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The original "(as in a conditional)" seems to be correct. –  yoyostein Jul 15 '12 at 15:42
    
"...buying the wallet whether or not the wallet is needed" cannot be the "exclusive-if," but then the wiki example has the parenthetical statement: "(as in a conditional)" this is where I get confused. What is the parenthetical "conditional" referring to? –  skullpatrol Jul 19 '12 at 16:34
    
@skullpatrol: The parenthetical (as in a conditional) refers to buying the wallet whether or not the wallet is needed . –  yoyostein Jul 22 '12 at 14:47

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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