Translating English to FO Logic

Is this an accurate paraphrase of the sentence "No one has more than three grandmothers" into the predicate calculus?

∀u∀w∀x∀y∀z(((G(w,u)&G(x,u))&(G(y,u)&G(z,u)))->(((w=xvw=y)vw=z)v((x=yvx=z)vy=z)))

Where G(x,y) is a propositional function meaning "x is the grandmother of y," and the domain for all variable consists of the set of all people.

I figured I would write the negation of the statement "Someone has more than three grandmothers," i.e.,

~∃u∃w∃x∃y∃z(((Gwu&Gxu)&(Gyu&Gzu))&(((w≠x&w≠y)&w≠z) &((x≠y&x≠z)&y≠z)))

and then just move the ~ in until I got the above universal quantification. But is there a simpler way to express the proposition?

I think your parentheses might mess things up a bit. How about this:




This expression is not fully parenthesized, I understand. If your teacher is picky about that, then just go with your original statement, which is correct as far as I can see. My version takes advantage of the associativity of AND and OR.

I don't think there are any FOL translations of the sentence that are much clearer than this.

Quote:

---

Originally Posted by Ackbeet

I think your parentheses might mess things up a bit. How about this:




This expression is not fully parenthesized, I understand. If your teacher is picky about that, then just go with your original statement, which is correct as far as I can see. My version takes advantage of the associativity of AND and OR.

I don't think there are any FOL translations of the sentence that are much clearer than this.

---

I don't know of this is "legal", but maybe we could use indices, which would help for a statement such as "No one has more than thirty grandmothers".

My guess is that he wants to do it in the pure predicate calculus with identity without use of mathematics such as you've used.