Thomas Aquinas said that God was absolute perfect, God being pure actuality and had no potentiality. If there was a time where there wasn't an existence, then God would have a potentiality and not being pure actuality. Does this mean existence is essential for a God who is pure actuality?

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    Are you asking generically or specifically for Aquinas? Generically is too broad and opinion based for this SE. For Aquinas only is answerable. – virmaior Jun 1 '19 at 10:27
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    Is not Aquinas pointing out that God must transcend existence in order to explain it? God would be real and the source of existence. It seems obvious that there must be some Ultimate from which existence springs, for otherwise existence would be an intractable problem. . . – user20253 Jun 1 '19 at 11:54
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    Not only that, Aquinas literally says that the essence of God is his existence, and he is the only such entity, see IEP. – Conifold Jun 1 '19 at 18:56

Assuming the philosophy of Thomas Aquinas...

Yes, your reasoning is correct. If God is pure actuality, then he has no unrealized potential, or non-actual but possible attributes. The possibility of not existing would be such an unrealized potential that God cannot possess. So, God conceived of as pure actuality must exist necessarily and a fortiori can never not exist at any time.

In fact, that Aquinas believes it is essential to God to exist is made even more explicit in another argument of his for the existence of God, from the essence-existence distinction. Roughly, it says that for ordinary objects there is a distinction between what something is (its essence) and that it is (its existence). Consider a unicorn as (say) a being whose essence is a horse with a horn. Once we've understood its key properties and all it entails, we've understood the essence of a unicorn. But, it's still an open question whether there are in fact any existing unicorns. For ordinary existing objects, that existence is "added" to their essence and so are made existent needs to be caused by something already existing, since essences cannot just cause themselves to exist. It only pushes the question back if we explain the existence of ordinary objects using other objects whose essence doesn't contain their existence, since they too would need a cause of their existence outside themselves. If we think there cannot be an infinite regress of causes, the chain of causes must terminate in a being whose essence contains its existence, since such a being wouldn't require a cause (thereby ending the chain of causes). Such a being, whose essence contains its existence, is therefore self-explaining. We can make sense of this, by considering what happens if we ask "why does this being exist?". The answer can be given by inspecting its essence closely and seeing that it contains its existence and so cannot fail to exist, the same way inspecting the essence of a Euclidean triangle shows why the sum of its angles must be 180 degrees. Such a self-expaining being does not need a cause for its existence, since what it is implies that it is. For Aquinas, this being whose essence contains its existence is God.

In fact, as Conifold mentions in the comments, Aquinas doesn't just say that God's essence contains God's existence, he believes they are one and the same; God's existence is God's essence, which is all of God's other attributes. They are all identical, literally the same entity. This is called the Doctrine of Divine Simplicity (DDS). It has its critics, and some think it's completely unintelligible. For a really interesting contemporary analytic defense of the DDS using truthmakers, see Jeffrey Brower's Making Sense of Divine Simplicity, especially page 14, the section titled 3.2 The Constituent Interpretation onward.

For further reading, check out Edward Feser's Five Proofs of the Existence of God, especially the chapters on the Aristotelian Proof (from the act-potency distinction) and the Thomistic Proof (from the essence-existence distinction).

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