0

I need help.

I refer to the book Introduction to Mathematical Logic (Fourth Edition), by Elliott Mendelson, page 95.

Is it the theorem (A7), substitutivity of equality, an open formula?

Could the formula be closed? That is, could it be written as below?

$ (Ɐx)(Ɐy) ((x, y) \Rightarrow (B(x,x) \Rightarrow B(x,y)))$

Mauro ALLEGRANZA
  • 94,169
Paulo Argolo
  • 4,210

1 Answers1

0

Yes, formula (A7):

$x= y \to (\mathcal B(x,x) \to \mathcal B(x,y))$

is an open one. Variables $x$ and $y$ are free.

But we can equivalently write it as follows:

$(\forall x) \ (\forall y) \ [x= y \to (\mathcal B(x,x) \to \mathcal B(x,y))].$

The equivalence is licensed by the (Gen) rule (page 70): $(\forall x_i) \ \mathcal B$ follows from $\mathcal B$, and logical axiom (A4).

Mauro ALLEGRANZA
  • 94,169