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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.

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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It can be used as a bicondition like so: "I'll buy you a new wallet if you need one" –  Matt Эллен Jul 19 '12 at 8:01
    
@MattЭллен Thanks for the "original" example :P –  Ice Boy Jul 19 '12 at 8:05
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Wouldn't that be "if and only if" (sometimes written as "iff")? –  Joachim Sauer Jul 19 '12 at 8:06
    
My point, if you've missed it, is that you don't explain what you don't understand. The example you give makes sense, given the definition of biconditional if in your questions. –  Matt Эллен Jul 19 '12 at 8:07
    
@MattЭллен OP seems to ask: how to make it clear that the "if" is indeed bi-conditional in a sentence, as in "I'll buy you a new wallet if you need one", and not as in "it is cloudy if it is raining". So that the reader may unambiguously understand such an implication. –  Kris Jul 19 '12 at 8:16

2 Answers 2

up vote 15 down vote accepted

This is the Cooperative Principle in action: enter image description here (link to comic)

If you have no particular reason to think I will buy you a wallet and I say:

I will buy you a new wallet if you need one.

then, under the Cooperative Principle, you can safely assume that this is all you want / need to know. But if there were other conditions in which I'd buy you a wallet, you'd want to know about them. Therefore, this is the only condition under which I will buy you a wallet, so it is bi-conditional.

But this is quite a tenuous line of reasoning. For instance, I might be deliberately violating the Cooperative Principle in order to hide the fact that I intend to buy you one for your birthday. So, to be extra clear, I could add an "only":

I will buy you a new wallet only if you need one.

In this context, this is now an unequivocal biconditional.

(T-Rex might argue that "only if" should only mean the reverse conditional, not the bi-conditional. I would tell him not to confuse mathematical jargon with Plain English, and that if he doesn't buy the wallet he's promised me I won't hesitate to brand him a disingenuous and unhelpful jerk. :)

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"...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? –  Ice Boy Jul 19 '12 at 16:31
    
"I will buy you a wallet if you need it. I make no promises about what I will do if you don't need it." = conditional "I will buy you a wallet you need it. If you don't need it, I won't buy you one." = bi-conditional –  Pitarou Jul 19 '12 at 21:45

Firstly, written expressions suffer from lack of the intonational advantage of speech.

Furthermore, there are such ambiguities galore in the English language, probably far more than in most of the other 'conservative' languages, i.e., those that well-defined and relatively static.

In writing, I might use an expression like:

"I'll buy you a new wallet , only if you need one."

to avoid ambiguity. Note the comma use as well.

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In many real-world contexts the speaker would say "but only if..." Another common way to explicitly stress the "exclusive if" meaning is to say "A, if B, but not otherwise" (or context-dependent variants such as "I'll buy you a new wallet if you need one, but not if you don't".) –  FumbleFingers Jul 19 '12 at 11:34
    
@FumbleFingers "...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? –  Ice Boy Jul 19 '12 at 16:27
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@Former_Math_Addict: I don't want to get bogged down in whether "exclusive if" is valid terminology. Let's just say "I [will] buy you a wallet" is condition A, and "You need a new wallet" is condition B. In normal English, "A if B" is potentially ambiguous - it often implies "B if A" (i.e. - it's often "biconditional"). But it doesn't have to be - it might just be "conditional" (A is definitely true if B is true, but A might still be true even if B isn't true). –  FumbleFingers Jul 19 '12 at 19:49

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