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Consider the string 'Let $y = f(x)$." Suppose that it occurs in some elementary context, such as when graphing the function $f$ using $x$/$y$ coordinates. How is this to be understood in predicate logic? We can't have either $x$ or $y$ be free variables, for consider the following:

Let $f:\mathbb{R}\rightarrow \mathbb{R}, \forall x,\ f(x)=2x$

Let $g:\mathbb{R}→\mathbb{R}, \forall x,\ g(x)=x+x$

Let $y = f(x)$

Let $z = g(x)$

$\therefore y = z$

Here the last line is clearly true, but would lack a truth value if either variable were a free variable.

However, if both variables are bound, we're stuck with permutations of quantifiers that mean the wrong things:

$\forall x,\forall y,y=f(x)$ [says the universe has cardinality 1]

$\forall x,\exists y,y=f(x)$ [says f's domain is the universe]

$\exists y,\forall x,y=f(x)$ [says f is a constant function]

$\exists x,\forall y,y=f(x)$ [says the universe has cardinality 1 and f is nonempty]

$\forall y,\exists x,y=f(x)$ [says f is onto the universe]

$\exists x,\exists y,y=f(x)$ [says f is not the empty function]

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    Saying that $y=z$ lacks a truth value is meaningless, the truth value for a formula is defined *only* in a given *model/interpretation* (and an interpretation will fix the value of each free variable to a fixed object in the model, and the formula will be true in that model if the values assigned to the two free variables are the same). I think what you intended to say is that the formula is not *valid*.2011-03-04

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