We're covering Prepositional Logic in our class and I'm a little confused as to how the entire concept actually works. My understanding is that prepositional logic is used to study ways of combining or altering statements or propositions to form more complicated statements or propositions. Therefore when we have logical expression, we have to show that the statement is true or false.
Let's say I want to show that the following logical statement is true:
$$(\exists x. P_1(x) \to P_2(x)) \to ((\forall y.P_1(y)) \to \exists z.P_2(z))$$
- Assumption ($A1$): $(\exists x. P_1(x) \to P_2(x))$
- To Show ($S1$): $((\forall y.P_1(y)) \to \exists z.P_2(z))$
- Assumption ($A2$): $((\forall y.P_1(y))$
- To Show ($S2$): $\exists z.P_2(z))$
Now I know that at this point I should choose $x$ as arbitrary but fixed in A1, but I don't know the exact reason for this, instead of lets says choosing $z \equiv x$.
- Assumption ($A3$): $P_1(x) \to P_2(x)$
- Choose $y \equiv x$ in A2.
- Assumption ($A4$): $P_1(x)$
After this point, I'm uncertain with what to do.
PS - I know that "To Show" and "Assumption" may not be the correct English terms but they're the closest translation from German "Annahme" and "Zu Zeigen" respectively.