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25.3 Variable Switch (VS)

We learned in Chapter 21 that variables all have the same basic meaning, from a logical point of view.

x, y and z are interchangeable: Happy(x) and Happy(y) both mean “It is happy.” And ExHappy(x) and EyHappy(y) both say “Something is happy.”

Variable Switch (VS)
AxP(x) ⇔ AyP(y)
ExP(x) ⇔ EyP(y)

The Variable Switch (VS) equivalence says that you can uniformly substitute one variable for another:

AxP(x) ⇔ AyP(y)

ExP(x) ⇔ EyP(y)

But there are several key restrictions on how we can apply this equivalence.

  1. You must switch all occurrences of the variable that are connected to the same quantifier. If you have Ax(P(x)->Q(x)), you can’t write: Ay(P(y)->Q(x)).
  2. But if you have different quantifiers binding a variable, each is independent. You can switch one but not the other. AxP(x)&AxQ(x) ⇔ AxP(x)&AyQ(y) is correct.
  3. You cannot switch to a variable with scope overlap. If you have AxAyLikes(x,y), you cannot switch the y to an x, because it conflicts with the scope of the Ax.