How does Dummett see deductive inference extending our knowledge?

Question

Michael Dummett writes (page 195)

Once the justification of deductive inference is perceived as philosophically problematic at all, the temptation to which most philosophers succumb is to offer too strong a justification: to say, for instance, that when we recognize the premisses of a valid inference as true, we have thereby already recognized the truth of the conclusion. If that were correct, all that deductive inference would accomplish would be to render explicit knowledge that we already possessed: mathematics would be merely a matter of getting things down on paper, since, as soon as we had acknowledged the truth of the axioms of a mathematical theory, we should thereby know all the theorems. Obviously, this is nonsense: deductive reasoning has here been justified at the expense of its power to extend our knowledge and hence of any genuine utility.

I assume that the temptation he is referring to is to believe that all we have to do is verify the premises to accept a validly derived conclusion. We've justified the deductive inference so strongly that we somehow lose the power of the deductive inference to "extend our knowledge".

I can see how knowledge is not "getting things down on paper". We have to know what is on that paper for the data written there to become knowledge. However, I think he is referring to something besides knowing what is written down that we should be doing so that deductive inference can extend our knowledge. Perhaps it is to understand the proof as well? Or is Dummett referring to something else?

---

Dummett, M. (1991). The logical basis of metaphysics. Harvard university press.

Answer 1

For Dummett, there is a tension between the certainty of deductive inference (the guarantee of the truth of the conclusion licensed by the truth of the premisses) and its usefulness or fruitfulness, that is, the ability of deductive arguments to yield new information.

We have to consider the context :

• Wittgenstein with the well-know Tractarian slogan about the tautologueness of logic (4.461 Propositions show what they say: tautologies and contradictions show that they say nothing. A tautology has no truth-conditions, since it is unconditionally true [...] Tautologies and contradictions lack sense.)

• The neo-positivist Conception of analytic truth : a statement whose tuth depends upon the meanings of its constituent terms, sometimes : true by convention.

• an back to Kant's theory of analytic judgement, where "the predicate B belongs to the subject A as something that is contained in this concept A."

According to the above points of view, deductive inference is "trivial" : deduction is limited to unwind the "content" already implicitly present in the axiom.

See also A.J.Ayer, Language, Truth, and Logic (1936) :

the truths of logic and mathematics are analytic propositions or tautologies. [...] we say that analytic propositions are devoid of factual content, and consequently that they say nothing, we are not suggesting that they are senseless in the way that metaphysical utterances are senseless.

But everyone that have tried to study e.g. Euclid's Elements is aware of the fact that the simple reading of many of Euclid's propositions, without a careful reading of their proofs, give us no clues about how they can be "contained" into the axioms.

This fact was already discussed (contra Kant) by Frege : "logic is not purely formal, from Frege's point of view, but rather can provide substantive knowledge of objects and concepts.."

Dummett's point of view is extensively explained into his The Justification of Deduction.

This must be linked to - according to Dummett - the theory of Meaning; see also (for further development) Proof-Theoretic Semantics.