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In trying to understand some predicate calculus notes, let the domain for $x,y$ be buildings, and the domain for $z$ be cities.

Let $H(x,y)$ signify that $x$ is higher than $y$

Let $O(x,y)$ stand for $x$ and $y$ are not the same building

Let $L(x,z)$ stand for $x$ is in $z$,

Then my prof states:

"A building is higher than every other building only if this building is in Toronto"

He writes: this can be formulated as:

$1) \forall x (\forall y(O(x,y) \implies H(x,y)) \implies L(x, "Toronto"))$,

But it cannot be formulated as:

$2) \forall x \forall y ((O(x,y) \implies H(x,y)) \implies L(x, "Toronto"))$

In trying to understand this, I am trying to come up with an example where $1$, $2$ have different truth values for the same $x,y$ values since I believe this is the strategy to show that they are not logically equivalent, but I am stumped and need a hint to continue.

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In the second formulation, we are considering all of the pairs of buildings. What it says, is that for any two buildings in which one ($x$) is higher than the other ($y$), then the former building must be in Toronto.

In ($1$) however, we are restricting to the case where $x$ is a particular building that is higher than every other building.

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  • $\begingroup$ OK! That makes perfect sense so the second could be true for a building x larger than a building y, but which isn't larger than ALL buildings y? $\endgroup$ – IntegrateThis Jan 22 at 8:30
  • $\begingroup$ Exactly. All that's left to show is that there exist buildings outside of Toronto ;) $\endgroup$ – GossipM Jan 22 at 8:36
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See Prenex Normal Form: $(\forall x\phi )\rightarrow \psi$ is equivalent to $\exists x(\phi \rightarrow \psi )$ under the assumption that $x$ is not free in $\psi$.

In your example, $y$ is not free in $L(x , \text {Toronto})$.

Why the above equivalence (in classical logic)?

Transform $(\forall x\phi )\rightarrow \psi$ into $\lnot (\forall x\phi )\lor \psi$ and then in $(\exists x \lnot \phi )\lor \psi$.

If $x$ is not free in $\psi$ we have $\exists x (\lnot \phi \lor \psi)$ and this in turn is $\exists x (\phi \to \psi)$.

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  • $\begingroup$ Interesting. Why is $y$ not free in $L(x, Toronto)$ since it seems $L$ only depends on $x$ here? $\endgroup$ – IntegrateThis Jan 22 at 8:40

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