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Source: p 120 Middle, Introducing Philosophy for Canadians: A Text with Integrated Readings (2011 1 ed).

  Here, we will begin by discussing the Ontological Argument. This is the most difficult of the three, for it is a purely logical proof─it attempts to argue from the idea of God to His necessary existence.

I know the definition and etymology of 'ontological', but are not all arguments for God's existence 'ontological'? If so, then 'ontological' is a hypernym. So the Ontological Argument should be called something else, perchance the Definitional or the Self-Evident or Self-Satisfying Argument (because you need only the definition of God as omnibenevolent, omnipotent, and omnipresent; and Deductive Logic; to deduce the conclusion)?

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This might be helpful:

The distinctive feature of the [ontological] arguments ‒ at least according to the traditional Kantian method of classification ‒ is that they proceed from premises which at least some defenders of the arguments allege can all be known a priori. Consequently, it would be most appropriate to call these arguments 'a priori arguments for the existence of God'. However, following Kant, it has been established practice to call these kinds of arguments "ontological arguments," and I see no urgent reason to depart from this tradition. (Ontological Arguments and Belief in God)

But note that not all arguments for the existence of God are ontological, as you suggest. By the above definition, for example, arguments from something like the complexity of life to the existence of God would not be ontological arguments, since they proceed from empirical facts.

Also, the names that you suggest might be too narrow. SEP lists several kinds of ontological arguments:

  1. definitional ontological arguments;
  2. conceptual (or hyperintensional) ontological arguments;
  3. modal ontological arguments;
  4. Meinongian ontological arguments;
  5. experiential ontological arguments;
  6. mereological ontological arguments;
  7. higher-order ontological arguments; and
  8. ‘Hegelian’ ontological arguments;
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    Correct ! The short answer is : a priori argument. – Mauro ALLEGRANZA Jun 25 '16 at 16:18
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Kant's title "the ontological argument" does reflect a specific use, among philosophers, of the term 'ontology'. Etymologically, ontology pertains to being, existence. As Kant (and many other philosophers) use this term, ontology does not pertain to questions of the form: what exists, but to questions of the form: what is existence, what are special properties of existing objects, as such. So the question "does God exist" is not, by itself, an ontological question. On the other hand, the question "is there a being which possesses necessary existence" is ontological. That's why the "ontological argument" to God's existence is indeed ontological, while other arguments to God's existence are not ontological.

Metaphysic, in the more limited acceptation of the term, consists of two parts . . . The former presents the system of all the conceptions and principles . . . which relate to objects in general, but not to any particular given objects (Ontologia) ... (Kant, Critique of Pure Reason "The Architectonic of Pure Reason")

The term ontology was apparently coined only in the 17th century. Yet it seems to reflect well Aristotle's original intent in the Metaphysics:

There is a kind of science whose remit is being qua being and the things pertaining to that which is per se. This science is not the same as any of the departmental disciplines. For none of these latter engages in this general speculation about that which is qua that which is ...

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