Philosophers, even the best of them, sometimes take too much for granted and try to build ambitious structures on weak foundations. Purporting to secure their conclusions by rigorous reasonings, they actually take for granted far more than the available information warrants.

An instructive example of this is afforded by the following passage from David Hume's Dialogue Concerning National Religion:

Close scrutiny shows that Hume's contention is very problematic.

To elucidate Hume's line of reasoning, let us press a few items of symbolic machinery into service for the sake of precision in formulation and analysis. Let “p,” “q,” etc., serve as propositional variables, let “∼” represent negation, “&” conjunction, “→” logical entailment, and let “ξ” represent the impossible (or self-contradictory). Furthermore, let “E (p)” serve as an abbreviation for “ ‘p’ asserts existence,” “C (p)” for “ ‘p’ is conceivable,” and “D (p)” for “ ‘p’ is demonstrable.” Hume's argument can now be formulated as follows:

• Premiss 1: Nothing is demonstrable, unless the contrary implies a contradiction (i.e., is impossible).

  (P1) D (p) → (∼p → ξ)

• Premiss 2: Nothing, that is distinctly [coherently] conceivable, implies a contradiction (i.e., is impossible).

  (P2) C (p) → (∼p → ξ)

• Premiss 3: Whatever we conceive as existent, we can also conceive as nonexistent.

  (P3) C (E (p)) → C (∼E (p))

Given these premisses, can one demonstrate Hume's thesis that there is no (coherently conceivable) being whose existence is demonstrable, namely:

We now reason as follows.

By (P1) we have

By substituting E (p) for p, we obtain

To obtain Hume's conclusion (H ) it now suffices to show that:

But we already have

So what still we need is: