$A[t/x]$ in formal logic

Question

I have been trying to understand some first-order logic, but it's basically seeming like randomly-generated nonsense.

$ A \equiv ( \exists y \enspace \lnot Q(x,y)\lor \forall z Q(x,z))$ where $t\equiv c,$

so $A[t/x] \equiv ( \exists y \enspace \lnot Q(c,y)\lor \forall z Q(c,z)).$

$ A \equiv ( \exists y \enspace \lnot Q(x,y)\lor \forall z Q(x,z))$ where $t\equiv y,$

so $A[t/x] \equiv ( \exists y \enspace \lnot Q(x,y)\lor \forall z Q(y,z)).$

Why did we arbitrarily choose to substitute in for t in some parts of the formulae but not in others?

Because the term $y$ is not "free for" the first occurrence of the variable $x$ in the wff $A$, because that occurrence of $x$ is within the scope of a $y$ quantifier, but it is free for the second occurrence of $x$ -- whereas the term $c$ (since it has no variables) is free for both occurrences of $x$ in $A$. Your logic text should have something about this -- it's technical but necessary so that (roughly) the new wff says the same thing about $t$ that the original said about $x$. — Ned, Oct 04 '21 at 03:45
Does this answer your question? Definition of "substitutable" in Mathematical Logic — Mauro ALLEGRANZA, Oct 04 '21 at 10:33
Substitution is the process of putting a "name" in the empty place of a statement that is represented with a (free) variable. If we have the "pseudo-statement" "x is a Philosopher", we replace the name "Socrates" in the empty place to get the complete statement: "Socrates is a Philosopher". — Mauro ALLEGRANZA, Oct 04 '21 at 10:35
A quantified variable does not represent an empty place: the statement "Everyone is a Philosopher" has no empty places. Thus, to replace the x of "for every x (x is a Philosopher)" with the name Socrates leave the statement unaffected. — Mauro ALLEGRANZA, Oct 04 '21 at 10:38

ryang · Accepted Answer · 2021-10-04T05:41:44.687

$ A \equiv ( \exists y \enspace \lnot Q(x,y)\lor \forall z Q(x,z))$ where $t\equiv c,$
so $A[t/x] \equiv ( \exists y \enspace \lnot Q(c,y)\lor \forall z Q(c,z)).$

$ A \equiv ( \exists y \enspace \lnot Q(x,y)\lor \forall z Q(x,z))$ where $t\equiv y,$
so $A[t/x] \equiv ( \exists y \enspace \lnot Q(x,y)\lor \forall z Q(y,z)).$

Why did we arbitrarily choose to sub in for t in some parts but not others?

Now, $A[t/x]$ typically denotes the result of replacing all free occurrences of the variable $x$ in formula $A$ with term $t.$ As such, the third occurrence of $x$ in the above exerise would technically have been changed from $$\exists y \enspace \lnot Q(x,y)$$ to $$\exists y \enspace \lnot Q(y,y),$$ even if it turns out that $t$ (here $y$) is not freely substitutable for $x$ in $A.$

Since the author declined to make the third substitution, they must be defining $A[t/x]$ as the result of replacing every free occurrence of the variable $x$ in formula $A$ with term $t$ in which such a replacement results in no variable in $t$ becoming bound.
If the goal is to get rid of every occurrence of $x$ without causing any trouble, we can write:

$$ A \equiv ( \exists y \enspace \lnot Q(x,y)\lor \forall z Q(x,z))\\A[y/x] \equiv ( \exists z \enspace \lnot Q(y,z)\lor \forall z Q(y,z)).$$

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