Chapter 10

Sums in Natural Deduction

This chapter gives a brief description of those parts of natural deduction whose behaviour is not so pretty, although they show precisely the features which are most typical of intuitionism. For this fragment, our syntactic methods are frankly inadequate, and only a complete recasting of the ideas could allow us to progress. In terms of syntax, there are three connectors to put back: : , _ and 9. For : , it is common to add a symbol ? (absurdity) and interpet :A as A ) ?.

The rules are:

The variable must no longer be free in the hypotheses or the conclusion after use of the rule 9E. There is, of course, no rule ?I.

10.1Defects of the system

The introduction rules (two for _, none for ? and one for 9) are excellent! Moreover, if you mentally turn them upside-down, you will nd the same structure as ^1E, ^2E, 8E (in linear logic, there is only one rule in each case, since they are actually turned over).

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The elimination rules are very bad. What is catastrophic about them is the parasitic presence of a formula C which has no structural link with the formula which is eliminated. C plays the r^ole of a context, and the writing of these rules is a concession to sequent calculus.

In fact, the adoption of these rules (and let us repeat that there is currently no alternative) contradicts the idea that natural deductions are the \real objects" behind the proofs. Indeed, we cannot decently work with the full fragment without identifying a priori di erent deductions, for example:

Fortunately, this kind of identi cation can be written in an asymmetrical form as a \commuting conversion", satisfying Church-Rosser and strong normalisation. Nevertheless, even though the damage is limited, the need to add these supplementary rules reveals an inadequacy of the syntax. The true deductions are nothing more than equivalence classes of deductions modulo commutation rules.

What we would like to write in the case of _E for example, is

A _ B

AB

with two conclusions. Later, these two conclusions would have to be brought back together into one. But we have no way of bringing them back together, apart from writing _E as we did, which forces us to choose the moment of reuni cation. The commutation rules express the fact that this moment can fundamentally be postponed.

10.2Standard conversions

These are redexes of type introduction/elimination: