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I've recently been wondering about vacuous truths. I know a statement like "I've never been beaten in a race" is true if I've never been in a race, but what I'm wondering is if the following statements are true:

"Every time I've bought a lottery ticket, I won the jackpot" (assuming I've never bought a lottery ticket)

"Whenever Dave comes to one of my parties, he always makes a fool of himself" (Assuming Dave has never come to one of my parties)

I feel like these statements are technically true, but there's been some disagreement when it comes up with others, so I wanted to know for sure.

zyx
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StephenTG
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    Yes.$;;;;!!!!$ – goblin GONE Aug 26 '13 at 16:28
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    If some variant of this were used in an ad, then I would hope the advertiser would be prosecuted. The "vacuous truth" argument of the lawyer would be laughed out of court. The "translations" of English sentences into "logical" form often do violence to the actual connotations of the sentence. – André Nicolas Aug 26 '13 at 16:31
  • Because... English is not totally "logical"? – peterwhy Aug 26 '13 at 16:33
  • @peterwhy, more like because we assume people's motivations follow certain patterns. Like if I say, "whenever Dave comes to one of my parties, he always makes a fool of himself" you'd probably assume I was motivated to say this because he did indeed come to one or more of my parties. – goblin GONE Aug 26 '13 at 16:36
  • Because real English and the stripped down language of the "logical version" are very different languages, with the stripped down version being incomparably poorer. – André Nicolas Aug 26 '13 at 16:36
  • @AndréNicolas, but easier to manipulate in a logically correct way. They both have their places. – goblin GONE Aug 26 '13 at 16:37
  • I think the problems for ads lies in the fact that, people expect a consistent future or a prediction from this kind of statement, while the statements itself only gives the "historical" or "empirical" aspects. – peterwhy Aug 26 '13 at 16:38
  • @AndréNicolas You may be interested in pragmatics, particularly Grice's theory of conversational implicatures. See here. Grice is in section 2.1.2. – Potato Aug 27 '13 at 04:57
  • @Potato: Thank you for the link. Am copying to a file. – André Nicolas Aug 27 '13 at 05:00
  • This interpretation used in logic is called "material implication" ... there is a funny little essay by Lewis Carroll on this. Too bad I can't remember where. – GEdgar Aug 27 '13 at 14:44

4 Answers4

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If you judge them by the standards of formal logic, they are all vacuously true. Normal, everyday English does not operate by those standards, however. Normal discourse is also to a considerable extent governed by pragmatic considerations and the cooperative principle and Gricean maxims, which are violated by such vacuously true utterances. Thus, most people will understand such a statement to imply that you have done $X$ at least once, and probably more than once, and that $Y$ has happened on each of those occasions, and this understanding is perfectly reasonable outside of a formal mathematical setting.

Brian M. Scott
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  • I'm well aware that these can be taken the wrong way (In fact, I can't think of any real use for them other than to be taken the wrong way). I just wanted confirmation that, technically speaking, they are "True" – StephenTG Aug 26 '13 at 16:37
  • If you are in an introudctory logic course, the answer that will we marked as correct in a T/F test is T. – André Nicolas Aug 26 '13 at 17:20
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Further to Brian M. Scott's answer, another often hidden assumption in everyday usage of implication is the notion that the antecedent somehow "causes" the consequent. In mathematics, there is no notion of cause and effect.

In mathematics, we simply define $X \implies Y \equiv\neg(X\land \neg Y)$. Both are true if $X$ is false.

It may help to imagine that, in mathematics, you are simply describing patterns of symbols in a great book, patterns that are unchanging over time. Suppose it is the case that, if you find the letter p in this book, then the next letter will be q. Then you wouldn't say that finding the letter p "causes" the next letter to be q. But you could say that it is never the case that you find the letter p and the next letter is not q. You could also truthfully say this even if the letter p never occurred in the book.

Dan Christensen
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The discomfort in accepting these statements as true can be taken as evidence that classical propositional logic can diverge from human intuitive or verbal reasoning about implication (whether with a causal narrative or not).

In this case the constructive [as in "constructive mathematics" or "constructive/intuitionistic formal logic"] meaning of $X \implies Y$ is closer to the mark: from a proof of $X$ we can construct a proof of $Y$, or less formally, knowledge of $X$ can be correctly transformed into knowledge of $Y$. It is not equivalent to the classical forms with negation, such as "(not $X$) or $Y$", or "not ($X$ true and $Y$ false)", in the absence of excluded middle or some other nonconstructive reasoning principles. The second form means the implication is unfalsified and the conversational intuition about the $X/Y$ statement is more that it could be falsified by doing $X$, than that it must be equivalent to the implication being true (simply because classical rules for manipulating the words AND, OR and NOT would say so; maybe the rules don't apply or we feel that they don't).

There are interpretations in modal logic (which is also related to interpretations of excluded-middle free logics) as the statements are time-dependent or contingent on the past doings and not-doings of $X$. I am not familiar enough with modal logic to say whether this would be more convincing than the constructive interpretation.

It seems correct conversationally to call never instantiated statements (vacuously) unfalsified or untested, but not vacuously true.

The statement would be vacuously true in cases where the status of $Y$ is known to be independent of knowledge about $X$. For example,

"every time I went to sleep, the sun rose the next morning"

should be true and not only unfalsified, if we have reasons to accept that the sun will rise in the morning no matter whether particular persons sleep or not during the night. This holds in any non-classical logic that I can think of, and it makes perfect sense conversationally: an implication is vacuously true or vacuously false if its premise is irrelevant to drawing the conclusion.

zyx
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  • In short, non-classical logics better model (or modal) our intuition about these types of statements. – zyx Aug 29 '13 at 19:05
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Are statements like "Every time I've done X, Y has happened" (vacuously) true if I've never done X?

Yes: given that $X(t)$ is universally false, the sentence $$\forall t\,\Big(X(t)\to Y(t)\Big)$$ must, by definition, be vacuously true.

"Every time I've bought a lottery ticket, I've won the jackpot" (assuming I've never bought a lottery ticket)

Yes, vacuously true; here,

  • $X(t)\overset{\:\text{def}}{\iff}$ at time $t$, I've bought a lottery ticket
  • $Y(t)\overset{\:\text{def}}{\iff}$ at time $t$, I've won the jackpot.

"Whenever Dave comes to one of my parties, he makes a fool of himself" (assuming Dave has never come to one of my parties)

Yes, vacuously true; here,

  • $X(t)\overset{\:\text{def}}{\iff}$ at period $t$, Dave comes to one of my parties
  • $Y(t)\overset{\:\text{def}}{\iff}$ at period $t$, Dave makes a fool of himself.
ryang
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