Research:

On the Origins of Transductive Inference

'In logic, statistical inference, and supervised learning, transduction or transductive inference is reasoning from observed, specific (training) cases to specific (test) cases. In contrast, induction is reasoning from observed training cases to general rules, which are then applied to the test cases....

Transduction was introduced by Vladimir Vapnik in the 1990's, motivated by his view that transduction is preferable to induction since, according to him, induction requires solving a more general problem (inferring a function) before solving a more specific problem (computing outputs for new cases): "When solving a problem of interest, do not solve a more general problem as an intermediate step. Try to get the answer that you really need but not a more general one."' (Wikipedia).

Although Vapnik is undoubtedly responsible for introducing the term and concept as it is used in the statistical learning literature, I am interested if anyone else used the term prior to him, or if the concept existed before (unknown to him).

I believe that the following passage from Bertrand Russell's 1912 book, The Problems of Philosophy, is an early, oblique reference to what Vapnik calls transductive inference, without using that terminology. In the first paragraph, Russell states that reasoning from the particular to the particular is a form of induction (what Vapnik calls transduction). In the second paragraph he goes on to state that reasoning from cases A,B,C to the case D is more secure (probabilistically) than reasoning from A,B,C to a general rule, and then deducing D from the general rule. This, I believe, is very close to Vapnik's principle of transduction. Here is the extract:

The Problems of Philosophy
Bertrand Russell (1912)

CHAPTER VII
ON OUR KNOWLEDGE OF GENERAL PRINCIPLES
.
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"The fact is that, in simple mathematical judgements such as 'two and two are four', and also in many judgements of logic, we can know the general proposition without inferring it from instances, although some instance is usually necessary to make clear to us what the general proposition means. This is why there is real utility in the process of _deduction_, which goes from the general to the general, or from the general to the particular, as well as in the process of _induction_, which goes from the particular to the particular, or from the particular to the general. It is an old debate among philosophers whether deduction ever gives _new_ knowledge. We can now see that in certain cases, at least, it does do so. If we already know that two and two always make four, and we know that Brown and Jones are two, and so are Robinson and Smith, we can deduce that Brown and Jones and Robinson and Smith are four. This is new knowledge, not contained in our premisses, because the general proposition, 'two and two are four', never told us there were such people as Brown and Jones and Robinson and Smith, and the particular premisses do not tell us that there were four of them, whereas the particular proposition deduced does tell us both these things.

But the newness of the knowledge is much less certain if we take the stock instance of deduction that is always given in books on logic, namely, 'All men are mortal; Socrates is a man, therefore Socrates is mortal.' In this case, what we really know beyond reasonable doubt is that certain men, A, B, C, were mortal, since, in fact, they have died. If Socrates is one of these men, it is foolish to go the roundabout way through 'all men are mortal' to arrive at the conclusion that _probably_ Socrates is mortal. If Socrates is not one of the men on whom our induction is based, we shall still do better to argue straight from our A, B, C, to Socrates, than to go round by the general proposition, 'all men are mortal'. For the probability that Socrates is mortal is greater, on our data, than the probability that all men are mortal. (This is obvious, because if all men are mortal, so is Socrates; but if Socrates is mortal, it does not follow that all men are mortal.) Hence we shall reach the conclusion that Socrates is mortal with a greater approach to certainty if we make our argument purely inductive than if we go by way of 'all men are mortal' and then use deduction."

The whole text is avaliable online:
Russell - Problems of Philosphy

I would be interested to hear from anyone who agrees or disagrees with me that this passage from Russell does correspond closely to the concept of transductive inference, as introduced by Vapnik. I would also be interested to know if there are any other texts referring to a similar logical concept before Vapnik.

Other definitions

Transductive reasoning - The belief two events correlating in time have a cause-effect relationship with each other.

and:

However, during the preoperational period children make no distinction between the general and the particular. They use a type of logic that seems to fall somewhere between deductive and inductive logic. It is called "transductive logic" and consists of reasoning from the particular to the particular.

The former is referring to some usage in the hypnosis/NLP literature. The latter is from child psychology and the theories of Piaget. I would be very grateful to hear from anyone who knows whether the term (or the concept) has been known/used in logic before Vapnik's definition. Please drop me an email.