Relations and Multiple Quantification:Translation

1a.  (x)(Sx Tx)
OR
1b.  (x)Sx (y)Ty
are equivalent.

Let S = "has a location in space"
and T = "has a location in time."

The examples above thus read as follows:

1a.  For each thing, it has a location in space and time.

1b.  For each thing, it has a location in space, and for each thing, it has a location in time.

Now let's look at something different:

2a.  (x)(Mx v Px)
IS NOT EQUIVALENT TO
2b.  (x)Mx v (y)Py.

Let M = "mental" and P = "physical."

We now have:

2a.  For each thing, it is either mental or physical.

2b.  Either everything is physical or everything is mental.

Thus, with a universal quantifier, it doesn't matter where we put a sign of conjunction, but it does matter where we put a sign of disjunction.

3a.  (x)(Hx Cx)
IS NOT EQUIVALENT TO
3b.  (x)Hx (y)Cy.

Let H = "hot" and C = "cold."

We now have:

3a.  Something is hot and cold.

3b.  Something is hot and something is cold.

Now, let's look at:

4a.  (x)(Hx v Cx)
IS EQUIVALENT TO
4b.  (x)Hx v (y)Cy.