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$P(x,y)$ is a predicate function meaning that person $x$ has read person $y$'s book.

Now I want to say there exists only 3 people having read person $t$'s book .

How can I state it with quantifiers?

Is this ok? : $\exists! x\exists! y\exists! z(P(x,t)\land P(y,t)\land P(z,t))$

amWhy
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fmatt
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  • $$\exists x \exists y \exists z \Big(x\neq y \land y\neq z \land x\neq z \land P(x, t) \land P(y, t) \land P(z, t) \land \forall q(P(q, t) \to q = x \lor q = y \lor q = z)\Big)$$ – amWhy Mar 13 '20 at 15:47
  • @amWhy Is what I added to my question in recent edit also possible? – fmatt Mar 13 '20 at 15:48
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    @fmatt, no, as written your statement actually expresses that there is a single person who has read the book. – Mees de Vries Mar 13 '20 at 15:51
  • fmatt The uniqueness quantifier designates the existence of one and only one ... Here we are interested in three, and only three, persons have read t's book. – amWhy Mar 13 '20 at 15:54
  • @amWhy But doesn't the statement you wrote mean that :" at least three different people exist" ??so it isn't three and only three – fmatt Mar 13 '20 at 15:58
  • Yes it is only three as well, that's where the $\forall q(P(q, t) \to (q=x \lor q=y \lor q = z))$ comes in. That is, for any q, if q reads t, then q is either x, or y, or z. We've already established the existence of three readers (x, y, z), and want to rule out all others. – amWhy Mar 13 '20 at 16:01
  • @amWhy thanks a million devoting your time to help me with my simple question – fmatt Mar 13 '20 at 16:05
  • Glad to help, @fmatt!! – amWhy Mar 13 '20 at 16:05

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