The existence of God can be proved a priori by a logical deduction from the concept of God, when we think about the idea of God we realise that real existence is an essential part of how we conceive the idea of God. God therefore necessarily exists so that it is impossible for him to not exist. I belive to be necessary to examine Anselm's version of the ontological argument, in order to answer the above question, namely whether God's existence can be proved a priori. Anselm states that a non believing fool (in the Psalms) can conceive the idea that God is 'a being than which none greater can be thought' because he understands it.
Anselm further adds that if this idea could exist not only in his understanding but also in reality then that would be greater. Futher more we cannot conceive of God as not existing, becasue that would be less greater than a being which necessarily exists. In addition to this anselm states that God cannot be thought of as non-existent because we cannot conceive of him a beginning or an end. Gaunilo however criticises and objects to Anselms argument that the existence of God can be proved a priori.
He states that he can conceive the existence of a 'wonderful lost island', which is perfect, just by thinking about the idea. However Anselms counter objects to Gaunilo's argument by saying that only the idea of god can be thought of as necessarily existing because it is unique. Aquinas makes his views known, and I believe it would be beneficial to identify them because they are relevant to the question. Aquinas states the God's existence may indeed be self-evident, but it is not self-evident to us because we do not understand the essen c of God. Furthermore we (man) may self-evidently desire happiness, and God may be man's happiness but we dont desire God through ignorance. Finally saying that fools like the one Anselm describes deny the first premise that there is 'a being then which none greater can be thought' Hume however states that we can clearly conceive the non-existence of any existent thing, and therefore non-existence is never a contradiction.
Thus Hume voices the normal empiricist view, that logic could never prove the existence of anything. However the most potent criticisms come from Kant, who i believe 'demolishes' the claim that the existence of God can be proved a pro iri by a logical deduction.