I usually see "vacuous truth" in the context of logical implication. But I guess it results whenever you have:
- a mental model of something
- a corresponding, isomorphic formal model
- the freedom, in the mental model, to leave truth values undefined
- the necessity, in the formal model, to define truth values
Some other examples would be good. What other operators/connectives/etc. generate vacuous truth?
I guess the concept "vacuous falsity" also exist, though, for some reason, vacuous truth is more prevelant.
...connections to the concept of closure
...like 0!=1
--Ryguasu
- That 0!=1 is actually quite logical. (I assume you mean 0 factorial) -- Tarquin
The whole thing sounds pretty vacuous to me (see talk:surrealism). --Ed Poor
- No, this is mathematics. It's real. There's just parts of it that only make sense after the imbibing of certain quantities of alcohol... -- Tarquin
- I'd like to better understand the comment. Is Ed Poor suggesting the article "aims at producing irrational fantasies or hallucinatory and dream-like effects" (from surrealism)? If so, I guess it is a form of criticism, because, whether or not vacuous truth is confusing, an article about it ideally should not be. Unfortunately, I think the concept is hard to pin down in its full glory, so it might take several iterations to get it right. --Ryguasu
I think Ed is just kidding... ;-) Are the two concepts of "false implies anything" and "anything is true for the empty set" really that closely related? -- Tarquin
- I confess that I was just clowning around: the surrealism article seemed so surrealistic that when I saw vacuous truth right beside it on Recent Changes, I couldn't resist. Frankly, I don't understand either article, maybe washing my brain with alcohol would help? ;-) --Ed Poor
- Ok, I'm glad we're just laughing at one another, rather than anything too serious. As for the relatedness of those concepts, Tarquin, I can say at least this much: they both have the quality that you could legitimately "legislate in several manners". That is, there is certainly a reasonable point of view from which "nothing is true for the empty set", rather than everything. This seems remarkably parallel to the point of view from which "false implies nothing". I guess I need to think some more about whether "vacuous truth" is the best name for what is in common between them.
- Then there's the standard connection between logical implication and set theory: possible world semantics. Here, A -> B is isomorphic to "the set of conceivable worlds in which A holds" is a subset of "the set of conceivable worlds in which B holds". This gives a justification for why 3>3 implies anything; the set of conceivable worlds where 3>3 is said to be the null set, and the null set is a subset of every set. Therefore, by the isomorphism, 3>3 implies anything. I'm not sure exactly how this is related, but I can't help feel that it is. --Ryguasu