Here is a simple example of a conceptual ontological argument:

1. I conceive of an existent God. (Premise)

2. (Hence) God exists. (From 1)

This argument can, it seems, be paralleled to its discredit:

1. I conceive of an existent God. (Premise)

2. (Hence) God does not exists. (From 1)

However, it might be object that the two arguments are not really parallel. In order to decide whether they are, we need to decide on the sense that we are to give to the expression ‘conceive of’.

There are a number of different sense that can be given to the expression ‘conceive of’. Suppose that I am asked to conceive of - think about, form an idea of - the current president of the United States. There seem to be at least four different ways in which I can respond to this request: (i) I can consider the description ‘the current president of the United States’, but without making any commitment to the existence of anyone who conforms to that description; (ii) I can consider the description ‘the current president of the United States’, while being committed to the view that there is no one who conforms to that description; (iii) I can consider the description ‘the current president of the United States’ and take it that the description provides a correct characterization of a unique person who conforms to it, even if there is no independent characterization that I could give of that person; and (iv) I can consider the description ‘the current president of the United States’ and take it that the description provides a correct characterization of a person whom I can also characterize independently.