A short question about vacuously true.


I talk to every goblin every day.


I talked to every goblin yesterday.

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

  • 2
    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?”
    – J D
    Nov 14, 2020 at 19:55
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  • 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, 2020 at 6:35
  • A 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. Nov 15, 2020 at 9:29
  • @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, 2020 at 14:56

1 Answer 1

  • 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.

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