Multiple-Conclusion Logics

João Marcos

Center for Logic and Computation
Technical University of Lisbon - Portugal

Logic is about what-follows-from-what. The idea that an argument is to sanction the derivation of one conclusion from a set of premises is just as intuitive as, unfortunately, technically wanting. A more generous and more symmetric account of consequence and entailment should in fact allow for the derivation of a set of alternatives as the conclusion of the argument. As it has been argued by universal logicians, the right choice of framework is important from a methodological, a philosophical, and a mathematical point of view. And that choice is nothing but fundamental if logic is to deserve a distinguished place at an updated Bourbakian architecture of mathematics.
The aim of this tutorial is to present the advantages and purview of a multiple-conclusion approach to logic and metalogic. Such an approach helps not only to eliminate any still existing bias towards truth (when falsity is just as respectable), but also to help expressing and comparing different kinds of logical systems. The very basic, though lamentably still insufficiently well-known, general issues and techniques related to multiple-conclusion reasoning will be presented in the tutorial.
The first lecture will compare the single-conclusion and multiple-conclusion approaches, for a motivation, and will survey some standard approximations to the very concept of logic.
The second lecture will investigate what happens in such an environment with the general metalogical tools and concepts, both from an abstract and a semantical standpoints.
The third lecture will deal with specific applications, further illustrations, and more esoteric results.


A.Avron, “Simple consequence relations”, Information and Computation, 92 (1991), pp.105-139.

J.-Y.Béziau, “A survey of general abstract logic”, Preprint (2003).

N.Bourbaki, “The architecture of mathematics”, American Mathematical Monthly, 57 (1950), pp.221-232.

S.Read, “Harmony and autonomy in classical logic”, Journal of Philosophical Logic, 29 (2000), no.2, pp.123-154.

D.Scott, “Completeness and axiomatizability in many-valued logic”, in Proceedings of the Tarski Symposium, L.Henkin (ed.), American Mathematical Society, Providence, 1974, pp.411-435.

K.Segerberg, Classical Propositional Operators – an exercise in the foundations of logic, Oxford Logic Guides 5, The Clarendon Press, Oxford University Press, New York, 1982.

D.J.Shoesmith and T.J.Smiley, Multiple-Conclusion Logic, Cambridge University Press, Cambridge-New York, 1978.

A.Tarski, “Über den Begriff der logischen Folgerung”, Actes du Congrès International de Philosophie Scientifique, 7 (1936), pp.1-11.

R.Wójcicki, Theory of Logical Calculi, Kluwer, Dordrecht, 1988.

J.Zygmunt, An Essay in Matrix Semantics for Consequence Relations, Wydawnictwo Uniwersytetu Wroclawskiego, Wroclaw, 1984.

Lecture Notes.