An interesting argument from Rosmini for the principle that only what is conceivable is possible:
The ideological evidence can also be analysed by reflection and reduced to an argument as follows: all that does not involve contradiction is conceivable. But that which involves contradiction implies annihilation of itself because one extreme of a contradiction annihilates the other. Every contradiction can, in fact, be represented by the formula: a – a = 0. But nothing cannot be, precisely because it is not being. Therefore, all that cannot be conceived, cannot exist.
This proof is founded on the breadth of thought and intelligence. In turn, this breadth arises because the proper, objective form of intelligence is being and, in the case of human intelligence, undetermined being, which has no limits. The essential unlimitedness of being and of undetermined being is, however, evident per se.
[Antonio Rosmini, Theosophy, Volume 1, Denis Cleary & Terence Watson, trs., Rosmini House (Durham: 2007), p. 396 (section 451).]
We have to be somewhat careful with Theosophy, since it was published posthumously without Rosmini having wholly revised it. But this is a very Rosminian argument. A potential worry about it is the first premise, "all that does not involve contradiction is conceivable"; but I take it that Rosmini regards this as implied by the close connection between the principle of noncontradiction and being as an object of intellect (hence the second paragraph).
The equational representation of contradiction is interesting, and I would guess is directly or indirectly influenced by Leibniz. It's worth comparing this to Boole's representation, x(1-x)=0, which, because of the equivalence of x^2 and x in Boole's logic is equivalent to Rosmini's; although, of course, there are reasons why Boole gives it in a multiplication-based form rather than an addition-based form.
