There are arguably three versions of the principle of non-contradiction to be found in Aristotle:
“It is impossible for the same thing to belong and not to belong at the same time to the same thing and in the same respect.” This is the ontological version. It is a statement about being itself.
“It is impossible to hold (suppose) the same thing to be and not to be.” This has to do with the mind and its conformity to reality.
The third is that “opposite assertions cannot be true at the same time.” This one is really just a variant of the first, but makes “less of a claim” about reality at any one moment. Only that at that moment, its opposite cannot occur.
Let’s look at a square circle…ok, let’s not, for it cannot be done. The definition of a square is contradictory to that of a circles. Something cannot be both a shape with no corners and one bent line with all points equally distant from the center and also be a shape with four corners of 90 degrees each and four equal sides. Yet we can say “square circle, square circle, square circle” all we want. It’s a funny thing about our reasoning ability. We can make statements about things that exist, things that don’t but in fact could exist, and even about things that cannot possibly exist.
Which brings us to female priests (in Catholicism, not paganism, which I am not addressing here), pro-abortion catholics, and other such absurdities. You see, in the real world in which we live, the law of non-contradiction applies to all of reality. Therefore, since by definition to be a Catholic priest includes being male, we are simply stating an absurdity when we make claims otherwise. Being Catholic includes, by definition, that we are pro-life. Making a statement to the contrary is by definition, likewise, another absurdity.
We can go around calling ourselves or others “pro-abortion catholics” or “female priests” or “living inorganic object” or purple solid-black object” or “dry-water” or even point to something and call it a “square circle.” We may even convince a few others to use the same absurd terms when referring to such objects. Yet part of sanity, of what it is to be a functioning rational being, is to recognize such absurdities and then point them out for the benefit of others.
God bless you,