All my children in the yard are playing soccer.

It is vacuously true if the speaker has no child.

But if I parse the statement, is it "there are all my children in the yard and they are playing soccer"?

I don't understand why it isn't false.

• You parsing changed the wording and changed the meaning. Surely you can see that. Jan 16 at 1:07
• Does this answer your question? How can I understand vacuous truth? Jan 16 at 7:03
• if you are "clever/tricky/cunning" (then logic problems are foreseeable, but let me help you...;) it would be also "vacuous truth", when you have no yard, or have children and they are not in the yard, or have children and have no yard ...."vacuous truth" is "useful" in hardware/software/formal languages/nature ... but it's still "lying" in "natural language", and should only be used by "logicians"! Jan 16 at 11:00
• the "good things" about "vacuous truths" in (time-critical) "logic systems": their evaluation (time) can be skipped (saved)! Jan 16 at 11:04
• when "you are unlogic (time-critic!?;) system", you can only waste time on it! ;) Jan 16 at 11:05

It's a technicality. If you have n children and they are all in the yard playing soccer, you can express that by saying that the number of children you have is the same as the number of your children playing soccer in the yard. If n is zero, it is still true that the number of children you have is the same as the number of children you have playing soccer in the yard. In everyday English the use of the expression 'all my children' gives the impression you have more than zero children, but technically 'all' can refer to zero.

"All my children in the yard are playing soccer" is false if one of your children is in the yard and not playing soccer. But there is not one of your children in the yard and not playing soccer, so "All my children in the yard are playing soccer" is not false.

So it must be true. But truth of this proposition normally requires that some of your children are in the yard and playing soccer and that's false as well. So it looks as if "All my children in the yard are playing soccer." is neither true nor false.

But that contradicts the law of excluded middle, which says that every proposition is either true or false. (See Wikipedia - Law of Excluded Middle for more details.

So it is classified as true, with the rider "vacuous".

By the way, "All my children in the yard are playing soccer" is not identical in meaning to "There are all my children in the yard and they are playing soccer". "All my children in the yard are playing soccer" allows for the possibility that some of your children are not playing in the yard. "There are all my children in the yard and they are playing soccer" requires that all your children are in the yard.

I don't understand why it isn't false.

"there are all my children in the yard" also is true if you have no children: "all my children" is an empty collection, and no one being in the yard satisfies it.

On the other hand, I would not call that vacuous truth, which is rather truth as a consequence of a false assumption (as in ex falso quodlibet). There are no false assumptions in that statement.

• No. You're correct, sets don't play soccer, but nor does nothing, right? Feb 16 at 0:04
• You are just playing word salad, but my comments get simply deleted so I guess someone else knows better... Feb 16 at 12:07
• Your comments get deleted? That shouldn't happen. But I'm not playing word games. Feb 16 at 19:34