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I already know that a false statement implies anything. Because I ask only for intuition, please do NOT prove this or use truth tables (which I already understand).

Source: p 333, A Concise Introduction to Logic (12 Ed, 2014), by Patrick J. Hurley

The truth table shows that the biconditional is true when its two components have the same truth value and that otherwise it is false. These results are required by the fact that $P ≡ Q$ is simply a shorter way of writing $(P ⊃ Q) \wedge (Q ⊃ P)$. If P and Q are either both true or both false, then $P ⊃ Q$ and $Q ⊃ P$ are both true, making their conjunction true. ...

I already understand the above, but am seeking an even more intuitive explanation.

Community
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    It may help to further describe what kind of intuition you are after here; for example, are you looking for something linguistic in nature that substantiates the raw logic of biconditionals? Or something else? – Daniel W. Farlow May 03 '15 at 03:40
  • @Law The link you provided, gives a very good example. It does not work for you? – Mehdi Jafarnia Jahromi May 03 '15 at 03:43
  • @MehdiJafarniaJahromi It concerns just a (single) conditional, though. I don't know the intuition for a biconditional. –  May 03 '15 at 03:44
  • @MagicMan Perhaps a linguistic or real-life example will help, such as that in the link? Does this help? I admit that I grapple with pinpointing what KIND of intuition I seek, because I haven't (yet) been exposed to the desired intuition. –  May 03 '15 at 03:46
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    What's more intuitive than "$P$ and $Q$ have the same truth value"? They must be both true or both false. –  May 03 '15 at 03:59
  • Anyway... Just change the example you like to "If and only if pigs fly, I will give you $1,000,000". –  May 03 '15 at 04:01

5 Answers5

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Consider the sentence:

I will jump off the cliff if and only if you do it as well

You did not jump off the cliff - so why should I?

The moral of the story is that a biconditional statements only states that $\alpha$ holds whenever $\beta$ is the case, they are, say, 'logically attached'. In the cliff analogy, they either jump 'together' or not.

Bruno Bentzen
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In law, it is sometimes the case that a contract is binding if, and only if, both parties signed it. So, if $P$ stands for "the contract is binding" and $Q$ stands for "Both parties signed the contract", then $P\iff Q$ stands for "the contract is binding if, and only if, both parties signed it". Now, when would the claim $P\iff Q$ be true in a particular situation? If both parties signed the contract and the contract is binding, then $P\iff Q$ is true. If not both parties signed it, nor is the contract binding, then $P\iff Q$ is still true. In any other case it is false.

Ittay Weiss
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To adapt the example given in the link, consider "I will give you $1000000 if and only if pigs fly".

It means that I am obligated to give you \$1000000 if pigs fly, and that pigs must exist if I am to give you \$1000000. Just actively see it (at least at first) as two conditionals; work out a concrete causal picture and then find a way to invert the direction so that it makes sense.

Nigel
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  • Sorry, but would you please clarify how this answers my question? I don't understand what you mean by 'actively see it' and work out a concrete causal picture and then find a way to invert the direction so that it makes sense. –  May 03 '15 at 04:02
  • Actively interpret it as two conditionals. In other words, apply your intuition for the conditional twice. – Nigel May 03 '15 at 04:05
  • But that's my problem. I don't know how to apply your intuition for the conditional twice. Can you please explain this explicitly? –  May 03 '15 at 04:06
  • You mentioned that you already have an intuitive feel for the conditional. The biconditional is just two of them - one in each direction. Interpret them one by one. There isn't really a better way as far as I can see. – Nigel May 03 '15 at 04:08
  • Sorry, but your latest comment just sounds like the quote, which I already understand. I wish to intuit the biconditional directly. So how does this answer my question? –  May 03 '15 at 04:12
  • I am inclined to suggest that your question, in those terms, has no answer. In math, for instance, proofs of logical equivalence between two conditions are often impossible to directly prove with a series of biconditionals; one instead has to prove one conditional and then its converse (because the flow of reasoning underlying each conditional is very different). – Nigel May 03 '15 at 04:35
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"$\Leftrightarrow$" and "$\leftrightarrow$" have completely different meanings in mathematics. The first one means what is on the left is equivalent to what is on the right. However, the second one is what you are looking for as bidirectional and says nothing about the equivalency.

So if I say that "pigs can fly if and only if the moon is made up of cheese", what is your reason for proving me wrong. You may just prove under one of the following conditions:

condition 1: pigs can fly but the moon is not made up of cheese.

condition 2: The moon is made up of cheese but pigs cannot fly.

Mehdi Jafarnia Jahromi
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Suppose before you roll a 6-sided die, I say "if the die roll is an even number, then it will be a 4." (maybe I'm just guessing). Next, suppose it comes up a 5. In that case, surely my predicted claim was not false (since i merely said it would be 4 IF it was even). However, either my prediction was true or it was false (law of the excluded middle), so since we've shown it wasn't false, it must have been true. In other words, when both the antecedent and the consequent are false, the one-way implication is true. Finally, a bidirectional implication is true if both of the underlying one-way implications are true, and based on the above, this will hold in cases where both sides are false.

Andy
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