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A short question about vacuously true.

(1)

I talk to every goblin every day.

(2)

I talked to every goblin yesterday.

Goblins don't exist. So universal claims like (1) and (2) are vacuously true, am I right?

vincentlin
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    How about you give the possible duplicate question and its answers a read (particularly the highst rated answer), and if it doesn't answer your questions or point you in the direction, let us know: [Why is it that the statement “All goblins are yellow” does not contradict the statement “All goblins are pink?”](https://philosophy.stackexchange.com/questions/75665/why-is-it-that-the-statement-all-goblins-are-yellow-does-not-contradict-the-st) – J D Nov 14 '20 at 19:55
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    Does this answer your question? [Why is it that the statement "All goblins are yellow" does not contradict the statement "All goblins are pink?"](https://philosophy.stackexchange.com/questions/75665/why-is-it-that-the-statement-all-goblins-are-yellow-does-not-contradict-the-st) – Conifold Nov 14 '20 at 21:38
  • To me, the answers seem to not address the situations I provided above. "All goblins are pink or yellow" is about an attribute of the goblin. However, "I talk to every goblin every day", the statement involves a person who said he talks to every members of a non-existent species. This translates to logic differently, I think. Would you shed light on this topic? – vincentlin Nov 15 '20 at 06:35
  • A [vacuous truth](https://en.wikipedia.org/wiki/Vacuous_truth) "is a conditional or universal statement that is only true because the antecedent cannot be satisfied". Thus the form must be "All goblins are red". You example must be rephrased "All being with which I talk every day are goblin". It is a different form. – Mauro ALLEGRANZA Nov 15 '20 at 09:29
  • @MauroALLEGRANZA I don't think my states are equivalent to "All being with which I talk every day are goblin". Your statement means goblins are the only ones that are talked to. However, my statements don't imply that, so they aren't supposed to be rephrased like that. – vincentlin Nov 15 '20 at 14:56
  • @MauroALLEGRANZA Do you mean that my statements have the meaning of existential claim? If "I talk to every goblin every day" implies that goblins exist, then yes, the statement is false. But I don't know how to translate my statements to logic, and I don't know if it has a meaning of existential claim. – vincentlin Nov 15 '20 at 15:24
  • Ok. If you equate it to "For every x (if x is a Goblin, then I Talked to x)", then yes: it is a vacuous truth. In this case, the form (as said above) is exactly those of "All Goblins are Red". – Mauro ALLEGRANZA Nov 15 '20 at 16:04
  • You are WRONG!!!! Vacously true statements must be universal statements. What you have is a slick way of saying SOME. So you OUGHT to rephrase your claim to Some days I talk to goblins. You are adding psychological things here trying to pass as philosophy. Deductive reasoning eliminates language like yours by requiring specific details which you are going way out to avoid. You ought to take modern languages & place it inside the deductive logical template --not the other way around. Then you would see your issue with paradox or semantics issues disappear. – Logikal Nov 18 '20 at 18:02

1 Answers1

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  • Set theory tells us that the empty set is a subset of every set, meaning that, whatever the set S may be, the empty set in included in S.

  • The set of goblins is identical to the empty set ( for there is no goblin).

  • So, the set of goblins is included in the set of people you talk, have talked or will talk to.

  • Now, why is the empty set included in any set whatever? The reason lies in the truth table of the " if ... then " operator ( material conditional). When the first sentence of an " if... then " statement is false, the conditional as a whole is automatically true. Consider an arbitrary object x , an arbitrary set S, and the (open)sentence

" if x belongs to the empty set, then x belongs to S"

The first sentence ( antecedent) is false ( for , by definition, no object belongs to the set that has no element); so the whole conditional is true. Since object x and set S were arbitrary, we are allowed to generalize and to say

"for all object x and all set S, if x belongs to the empty set, then x belongs to S"

That means that the conditions for set inclusion are fullfilled between the set " empty set" and any set S.

Floridus Floridi
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