[Lambek & Scott: Introduction to higher order categorical logic](http://dl.acm.org/citation.cfm?id=7517)


says that 



> 
> "For example, categorists may be unhappy when we treat categories as special kinds of deductive systems and logicians may be unhappy when we insist that deductive systems need not be freely generated from axioms and rules of inference"
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> 
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I am curious about the latter: why a deductive system need not be freely generated from axioms and rules of inference?


In computer science, derivable true formulae are the ones derivable from axioms by a finite series of using inference rules, that is, an inductively defined set.


Why is this doubtful for a category theorist? 



Category theorists stance on deductive systems
This sentence reads to me like "we will treat Xs as if they were special kinds of of Ys, which will make both X-theorists and Y-theorists unhappy because it is not true". Reminds me of the old joke when a patient says "Doctor, it hurts when I do this" and the doctor says "Then don't do that!"


I think a more modern thing to say is that a deductive system describes a sort of algebraic theory, where the axioms and rules of inference are the operations. The inductively defined derivable judgments are the elements of the initial/free algebra for this algebraic theory; whereas there are of course also algebras that are not free. In particular, the deductive *system* is a *theory* that is not identified with any of its *algebras*, either the free ones or the non-free ones. Computer scientists may be most interested in the free algebras, whereas category theorists are interested in all the algebras (although of course the free ones play an important role).


