Ockham on the paradoxes of entailment

While looking through an edition of Ockham, I came across this interesting text:

Other rules are given:
(10) From an impossibility anything follows.
(11) What is necessary follows from everything.
Therefore this follows: 'You are a donkey, therefore you are God'. This also follows: 'You are white, therefore God is triune'. But these consequences are not formal ones and they should be used much, nor, indeed are they used much. (Summa totius logicae III, III, C. XXXVI)

What is interesting to me is (a) that the consequences are not formal ones (don't follow in relevance logic?) and especially (b) that these rules shouldn't be used much. Why not?

Presumably, the only time you establish an impossibility in a sound argument is as part of a reductio, and if you do that, you don't use (10) next—you close the subproof and use reductio ad absurdum.

And using (11) leads to arguments that are not as perspicuous as they could be. For if you've established necessarily(p), it is more perspicuous to conclude p from necessarily(p) by axiom M of modal logic than to do something like:

12. necessarily(p).
13. 2+2=4.
14. p. (By (12), (13) and rule (11))

However, while (10) and (11) are useless considered as rules of inference, as true propositions they can be quite useful, and do in fact occur in philosophical discussion, for instance in providing counterexamples (suppose you say that x depends on y if and only if exists(x) entails exists(y); then if numbers are necessary beings, everything depends on the number 49, which may seem to be absurd). So we need to distinguish between (10) and (11) as truths and (10) and (11) as rules of inference.

This distinction is needed anyway for modus ponens; for if modus ponens is simply the universally quantified truth:

15. For all p and q, if (p is true and it is true that if p, then q), then q is true,

then to apply modus ponens given the truth of P and of if P then Q, you will need to do universal instantiation on (15) to get:

16. If (P is true and it is true that if P, then Q), then Q is true.

And then using the antecedent of (16) as a premise to get to the conclusion Q, one will have to use modus ponens, which lands one in a vicious regress. (This is an argument of Sextus Empiricus against the very idea of rules of logic. But Sextus confuses truths qua truths and rules. Not that I know exactly how to draw the distinction either.)

Eifelheim

I recently read, and very much enjoyed, Eifelheim (can be bought from amazon, or downloaded for free in pdf format from the author's literary agency), Michael Flynn's novel of aliens landing outside a Black Forest village in the 14th century. The best parts--from my point of view--of the novel were the interactions between the village's pastor, a man of formidable Parisian education (yes, this might seem unlikely, but there is a story there) and a friend of Ockham, and the aliens. Flynn captures cutting-edge 14th century scientific and philosophical thinking with great sympathy and surprising fidelity, and shows how a smart 14th century scholastic with Ockhamist tendencies would interact with 21st (or later) century science. We see the flexibility of scholastic categories, a flexibility that is going to be of interest to those of us still interested in a reconciliation between contemporary science and scholastic metaphysics.

I did not expect to read a novel that alludes to an Ockhamist argument against the possibility of nested accidents. (This is in support of an argument that the pastor gives against light having a velocity. The idea is that light is an accident of fire, so if light had a velocity, that velocity would be an accident of an accident, which is impossible.)

The medieval and alien characters are well-drawn and develop over the course of the novel. I was somewhat less happy with the portions set in the 21st century, but they were relatively briefer (and they may have been written first). All in all, quite an excellent piece of science fiction.