TITLE: Categories of logical formulae
QUESTION [8 upvotes]: Consider the set of formulas of a logic. If there was only one sort of "unary" deduction $\phi \Rightarrow \psi$ - like $(\forall x)\phi(x) \Rightarrow \phi(a)$ - we would immediately have a category of formulas (with deductibility $\Rightarrow$ as morphism). But alas, there are other ("higher") deduction rules, e.g. 
$\lbrace p, q \rbrace \Rightarrow p \wedge q$ (rule of conjunction)
$\lbrace p\rightarrow q, p \rbrace\Rightarrow q$ (modus ponens)

(How) can formulas of "classical logics"
  (propositional and FO) be made into a
  category despite of those other
  relations (= rules)?

REPLY [2 votes]: I suspect the notion you are looking for is either a multicategory or a polycategory.  Multicategories generalise categories by allowing more than one object in the domain of an arrow.  These were introduced by Lambek specifically for the study of logical derivations.  See

J. Lambek (1969) "Deductive Systems
  and Categories (II)" in Category
  Theory, Homology Theory and their
  Applications I, Springer Lecture Notes
  in Mathematics 87, R. J. Hilton (ed.)

Polycategories generalise multicategories by allowing multiple objects in the codomain of an arrow.  See 

M. E. Szabo (1975) "Polycategories",
  Communications in Algebra 3.

Of course there is a rich history of looking at categories whose objects are formula and whose arrows are proofs, coming mostly from the computer science literature.  Lambek and Scott's book may be a good place to start, or for a more modern take try this introductory article http://arxiv.org/abs/1102.1313   However, for classical logics such categories always reduce to boolean algebras, so people tend to work with intuitionistic or linear logics.