Multiplicative quantifiers in fuzzy and substructural logics
Libor Behounek, Petr Cintula
Institute of Computer Science, Academy of Sciences of the Czech Republic
{behounek | cintula}@cs.cas.cz

In substructural logics without contraction or weakening there are two sets of connectives: lattice-theoretical (“additive”) and group-theoretical (“multiplicative”). There is a long-standing problem of extending this distinction to quantifiers in first-order substructural logics. While the additive quantifiers can easily be defined (as the infimum or supremum in the lattice of truth values) and axiomatized (by Rasiowa’s [1] axioms for quantifiers in implicative logics), the same cannot easily be done for multiplicative quantifiers.
Deductive fuzzy logics can be viewed as substructural logics that enjoy the prelinearity property [2, 3]. We show how multiplicative quantifiers of various strength can be introduced in first-order fuzzy logic. Moreover we demonstrate that the additive quantifiers suffice for the development of sufficiently rich higher-order fuzzy logic [4], in which the multiplicative quantifiers become definable, and an internal theory of generalized quantifiers can be developed in its framework. The latter method can be extended to a broader class of substructural logics; the problem of multiplicative quantifiers can thus be bypassed through a detour to higher-order substructural logics based on additive quantifiers only.

[1]	Rasiowa H.: An Algebraic Approach to Non-Classical Logics. North-Holland 1974.
[2]	Cintula P.: Weakly implicative (fuzzy) logics I: Basic properties. Archive for Mathematical Logic 45:673–704, 2006.
[3]	Behounek L.: On the difference between traditional and formal fuzzy logic. Submitted.
[4]	Behounek L., Cintula P.: Fuzzy class theory. Fuzzy Sets and Systems 154:34–55, 2005.