Posted: May 18, 2012 8:45 am
by VazScep
amkerman wrote: The proper formulation is: if God exists then god is a square circle (E!(x))->(x=y).
Do you really want an equals on the right hand side of that implication?

In classical (and free) logic, both sides of an equation should denote unique individuals. But the phrase "a square circle" does not. This is not simply because there are no square circles. Even if there were, the phrase wouldn't denote a unique individual. Similarly, the phrases "a large circle", "a right-angled triangle" and "a President" don't denote unique individuals. When you've got that indefinite article a, you can't be pointing to anything definite.

What you would normally assume is being expressed here is not an equation, but a predication or role assignment. When we say "God is a square circle", we are saying either that God belongs to the class of square circles, or that God has the property of being square and circular. This is what Thommo has done, by rendering the sentence as "S(God) ∧ C(God)".

So in free logic, you might have "(E!(God)) → S(God) ∧ C(God)", which would correctly formalise "if God exists, then God is a square circle." If you later concluded that there is nothing which is both square and circular, you'd be forced to conclude that God does not exist.

This entails E!(y).
It doesn't. You started with an implication. If the left hand side is false, it wouldn't entail much at all.