In one of his very useful writings, Robert Harper writes the following thoughtful sentence:
But the upshot of Gödel’s Theorem is that as soon as we fix the concept of formal proof, it is immediate that it is not an adequate conception of proof simpliciter, because there are propositions that are true, which is to say have a proof, but have no formal proof according to the given rules.
With simpliciter Harper presumably (?) alludes to, and warns readers against, the logical fallacy a dicto secundum quid ad dictum simpliciter which was already known to, and warned against by, Kant.
A dicto secundum quid ad dictum simpliciter is Latin, whose literal translation is roughly from what-is-the-secondary-saying to the absolute saying, and can be rendered into more usual English as something like from a qualified claim to conclude an absolute claim. It means, it seems to me, the error of, from a statement-given-under-certain-qualifying-conditions, to conclude a statement-purported-to-be-true-absolutely.
A physical example of a dicto secundum quid ad dictum simpliciter often given is the seemingly unqualified statement Water boils at 100 degrees Celsius. This is a seeming dictum simpliciter which really is a dictum secundum that has forgotten (or hidden) a qualification: atmospheric pressure.
The connection to a dicto secundum quid ad dictum simpliciter not being explicitly made in loc. cit., to make a precise connection of a dicto secundum quid ad dictum simpliciter with Gödel’s first incompleteness theorem one of course now has to speculate: one attempt, in words, would be the following. One views Gödel’s first incompleteness theorem to show that a naive absolute conception of provability is itself a fallacy. For example, one could write, if only words are allowed, and abbreviating w.t.s.f. ‘within the same formalization’:
- Gödel’s first incompleteness theorem shows that to conclude from the fact that in special instances formal statements can be proved w.t.s.f., the absolute statement that every true formal statement can be proved w.t.s.f., is to commit a dicto secundum quid ad dictum simpliciter.
Readers who would like some in-depth-reading on simpliciter may like to read D.G.Walton, Logique & Analyse 129-130 (1990), 113-154.