In everyday conversation (in the US at least, but may be in other English-speaking countries too) one sometimes hears expressions like:

If Bob is a great dancer, then pigs can fly.

...meaning that Bob is a notoriously atrocious dancer.

This is an example of an everyday construct that relies on the fact that an implication whose premise is false is always true, irrespective of the truth of its consequent. Therefore, any sentence having the form "If X, then Y", where Y is a patently false statement (like "pigs can fly") is a roundabout way of asserting the falsity of X, since only in this case can the whole implication be true.

Is it possible to translate this general idea to Japanese?

The context where this query originated is the larger question of how traditional Western propositional logic maps to a non-Western language like Japanese. It seems clear to me that traditional logic is strongly influenced by the syntax of European languages. Is it difficult to translate these ideas to languages far removed from European ones?

Thanks!

There's also a similar time-based phrase:

"X will happen when hell freezes over".

I'm interested to hear about this one too!

I can find exactly what you want, however "when pigs fly" is equivilent to "ひゃくねんかせいをまつ（百年河清を俟つ)". 河清を俟つ is also seems to be okay. 例文： 百年河清を俟つとも犯罪のたねは尽きない。

For "When hell freezes over" maybe what you are looking for is
そうなるはずがないよ
Which is translated as "It will never happen" or "That will be the day."

I will look for more example sentences.

The material conditional and the if-then construction in natural language are different. Granted that P → Q is conventionally pronounced "If P, then Q" in English, this doesn't mean it's the same as the conditional statement in English as a natural language. It's the same as the fact that the validity of argument X in logic doesn't always coincide with that of X taken as a natural language.

If you're not familiar with those terms, think of an "acute angle" in elementary math. The term "acute" isn't the same as "acute" in "He has an acute sense of humor." The "angle" isn't the same as that in "You should view this problem from a different angle." And math, logic, etc. also have sentence structures and phrases which have different meanings than in natural language. "If P, then Q" is an example of this. So your example doesn't particularly rely on the material conditional in the first place.

You can translate any purely mathematical or purely logic statement into another language as long as the target language has all the jargon because it's just relabeling. A sentence in a natural language can be impossible to translate because often the time you don't have exactly the same concept in another language.

Quote:So your example doesn't particularly rely on the material conditional in the first place.

I think it does, and took pains to explain why. Please point out where you think my explanation is incorrect.

gfb345 Wrote:It seems clear to me that traditional logic is strongly influenced by the syntax of European languages. Is it difficult to translate these ideas to languages far removed from European ones?

This idea is rather trivial to translate. From a quick google search:

「昭一が愛国者だったら私は神様だ。 ありえない話です」

But maybe when you get into first-order logic, things get more complicated.

gfb345 Wrote:

Quote:So your example doesn't particularly rely on the material conditional in the first place.

I think it does, and took pains to explain why. Please point out where you think my explanation is incorrect.

Read the very first line of my previous post:

magamo Wrote:The material conditional and the if-then construction in natural language are different.

The following line is more specific about it:

magamo Wrote:Granted that P → Q is conventionally pronounced "If P, then Q" in English, this doesn't mean it's the same as the conditional statement in English as a natural language.

If you want a more specific explanation, here is one of the errors in your post:

gfb345 Wrote:an implication whose premise is false is always true, irrespective of the truth of its consequent

This holds if the "If P, then Q" structure in your example "If Bob is a great dancer, then pigs can fly" were a material conditional. But the "if-then" statement in English as a natural language is not. If you don't know what a material conditional is, here is the Wikipedia article. It also discusses a similar kind of error too:

Wikipedia Wrote:The meaning of the material conditional can sometimes be used in the natural language English "if condition then consequence" construction (a kind of conditional sentence), where condition and consequence are to be filled with English sentences. However, this construction also implies a "reasonable" connection between the condition (protasis) and consequence (apodosis) (see Connexive logic).

So, although a material conditional from a contradiction is always true, in natural language, "If there are three hydrogen atoms in H2O then the government will lose the next election" is interpreted as false by most speakers, since assertions from chemistry are considered irrelevant conditions for proposing political consequences. "If P then Q", in natural language, appears to mean "P and Q are connected and P→Q". Just what kind of connection is meant by the natural language is not clearly defined.

It also discusses what "If P, then Q" in the natural language English can be considered in logic. It's not comprehensive, but I think it's more than enough to understand your error.

Quote:So, although a material conditional from a contradiction is always true, in natural language, "If there are three hydrogen atoms in H2O then the government will lose the next election" is interpreted as false by most speakers, since assertions from chemistry are considered irrelevant conditions for proposing political consequences. "If P then Q", in natural language, appears to mean "P and Q are connected and P→Q". Just what kind of connection is meant by the natural language is not clearly defined.

It seems perfectly clear to me that this analysis does not hold for the particular expression I gave as my original example. You disagree with this. So I guess we will have to agree to disagree.

gfb345 Wrote:

Quote:So, although a material conditional from a contradiction is always true, in natural language, "If there are three hydrogen atoms in H2O then the government will lose the next election" is interpreted as false by most speakers, since assertions from chemistry are considered irrelevant conditions for proposing political consequences. "If P then Q", in natural language, appears to mean "P and Q are connected and P→Q". Just what kind of connection is meant by the natural language is not clearly defined.

It seems perfectly clear to me that this analysis does not hold for the particular expression I gave as my original example. You disagree with this. So I guess we will have to agree to disagree.

As I said, that example you just quoted from Wikipedia is a similar kind, not the same. If you don't understand, here is what's happening in the particular example you wrote in your OP:

The "If P, then Q" statement in English as a natural language assumes, unlike in logic, a certain reasonable connection between P and Q. In your example, P is "Bob is a great dancer," and Q is "pigs can fly." Because this is supposed to be taken from everyday conversation, this statement assumes a connection between P and Q.

Now Q is so obviously false, by the assumed connection between P and Q, people conclude that P is as obviously false as Q is. This is the reason why people think Bob is a bad dancer. Besides, since this isn't a statement in logic, just because P is false doesn't mean the whole statement is true. Hence, your argument is erroneous.

If you want to see this in action, try to make up a similar phrase with different Q, i.e., fill Q in "If Bob is a great dancer, then Q" with your own words so that the resulting sentence has the same kind of effect. You will notice that the more absurd Q is, the stronger the sense of P's falsity gets. And if Q is mocking something, people would take that you're mocking Bob. This shouldn't happen if you do not assume a connection between P and Q.

You said Bob is a "notoriously atrocious" dancer. Don't you think this is too strong a wording for a simple negation of P, i.e., "Bob is not a great dancer"? If your argument were correct, "Bob is not a great dancer" would be the conclusion. However, usually people assume worse than "not great." Why? Because P is supposed to be as absurd as saying pigs can fly.

So here are the fatal errors you made in your OP post. You:

1. wrongly assumed that the material conditional in logic and the if-then construction in natural language were the same,
2. did not notice that the sense of P being false comes from its connection to the falsity of Q,
3. wrongly concluded that the sense of P's falsity stemmed from the truth table of a Boolean function.

I hope this clears things up.

I'm going to have to agree with magamo. It seems like you're (gfb345) confusing the material conditional (i.e. there is a necessary connection between the antecedent and consequent) with the indicative conditional (i.e. the connection between the antecedent and consequent is vague, and is merely reasonable or probable at best).

Logicians are still confused about how to classify indicative conditionals and how to draw up accurate truth tables for them, so I wouldn't hold my breath. Aristotle recognized paradoxes of material conditionals over 2,000 years ago, and we're still trying to figure out how to resolve them. Postulating the indicative conditional is the furthest we've gone. At any rate, most sentences in natural languages are not material conditionals. A majority of philosophers, logicians, and mathematicians would say that only analytic statements are material conditionals (e.g. If you're a bachelor, then you're unmarried). Even the laws of physics cannot be considered material conditionals because it is nonetheless logically conceivable that the laws of physics could be otherwise. In contrast to those who argue that only analytic statements are material conditionals, a Kantian may argue that supposedly synthetic statements, such as the following, are material conditionals "If 1+1, then 2". However, I'll put aside that discussion, including Quine's critique of the analytic/synthetic distinction. The point is this: material conditionals are especially rare and correctly determining their truth-values is only useful for understanding the most mundane statements (in a natural language).

Anyway, to answer your question, yes, the statement you initially gave is easily translatable from English to Japanese and finally to formal logic. Nevertheless, and this may be pointing out the obvious, we should be aware that people can reasonably disagree over the correct translation at all stages of the process.

magamo Wrote:In your example, P is "Bob is a great dancer," and Q is "pigs can fly." Because this is supposed to be taken from everyday conversation, this statement assumes a connection between P and Q. Now Q is so obviously false, by the assumed connection between P and Q, people conclude that P is as obviously false as Q is. This is the reason why people think Bob is a bad dancer.

The reasoning I use to conclude that Bob is a bad dancer is the following:

i) If Bob is a great dancer, then pigs can fly.
ii) Pigs cannot fly.
∴ Bob is not a great dancer.

That's just reductio ad absurdum; no indicative conditionals.

What you're describing, Magamo, is this:

i) If Bob is a great dancer, then pigs can fly.
∴ Pigs' ability to fly is related to Bob's dancing skills.

The most common phrase is "When pigs can fly." Which is a hyperbole - an exaggerated statement or claim not meant to be taken literally. In the context of "when pigs can fly" it is used to describe something that is impossible. It is an integral part of language called a figure of speech.

There are many different variations on that phrase such as "..., and pigs can fly", "When pigs fly" and "And pigs might fly."

With proverbs there seems to be an entire phrase that is needed. Such as 一石二鳥 being "Kill to birds with one stone" in English. 木を見て森を見ず as "Can't see the forest for the trees." Or, even 見ざる聞かざる言わざる which is commonly written 見猿聞か猿言わ猿 being "See no evil, hear no evil, say no evil" in English.

As can be seen their are variations in the terms. Each demonstrates a unique difference in the terms. Even 百年河清を俟つ has differences from the original question. It isn't intuitive, but just like Kanji the meaning has to be independently learned.

Other figure of speech such include Onomatopoeia are different between languages. Linguistics have already done the job of compiling lists. It just take time to find the most suitable equivalence and its correct usage.

Maybe ANKI or something else could be used to learn the various Japanese ことわざ and other figure of speeches for those who are adventurous.

Flying pig

Another definition at urban dictionary

iSoron Wrote:The reasoning I use to conclude that Bob is a bad dancer is the following:

i) If Bob is a great dancer, then pigs can fly.
ii) Pigs cannot fly.
∴ Bob is not a great dancer.

I believe you just proved Magamo's point. When using the language naturally you said 'Bob is a bad dancer'. When using the language purely analytically, you said 'Bob is not a great dancer'.

If the phrase is taken to abide by the same rules as a material conditional, then we do not conclude that Bob is a bad dancer. We can only conclude that Bob's dancing ability is somewhere between non-existent and very good, but not good enough to be called 'great'.

(Assuming that there is no skill better than 'great'; since we can talk about 'the greatest in the world' and so on, it is reasonable to assume that 'great' includes a range from 'better than merely good' on up to 'the best possible'. In making that assumption, of course, we are tacitly accusing people who use phrases like 'beyond great' of making a logical fallacy as they bestow praise.)

I don't know why anyone is disagreeing with this point, honestly. The whole reason that propositional calculus has all those funny symbols to describe logic with is that natural language does -not- express logic precisely. Clearly (to me at least), 'If Bob is a great dancer, then pigs can fly' expresses that Bob is a terrible dancer for the reasons magamo outlined.

Consider, "If Bob is a great dancer, then he won Dancing with the Stars"
(where our Bob is not any particular Bob that -did- win Dancing with the Stars, if any did. Or if you can even 'win' on that show. If not, assume some dance contest that you can win... ahem, anyway, the 'then' part is known to be false.)

By pure logic, then, we can assume only that Bob's skill is somewhere between non-existent and good enough to come in second in a national dance competition, but not to win it, very similar to 'when pigs fly.'

But the natural language strongly suggests Bob is a good enough dancer to at least consider showing off his talents on nationwide television, even if he proved to be not that great in the end. (Well, it's sort of a contrived example, admittedly, but I think it makes the point, if it needed making, that what's being communicated by these conditionals is -not- strictly logical.)

Quote:i) If Bob is a great dancer, then pigs can fly.
∴ Pigs' ability to fly is related to Bob's dancing skills.

Actually, what magamo said is more like,
i) in natural language, the 'if' and 'then' clauses are related.
ii) if bob is a great dancer, then pigs can fly
iii) it is ludicrous to suggest that pigs can fly
iv) a reasonable conclusion to satisfy i,ii, and iii could be 'it is equally ludicrous to suggest that bob can dance'

There is no ∴ (or even 'therefore') because it's not really a logic problem at all, and there might be other reasonable conclusions.