Sufficient triangular norms in many-valued logics with standard negation (Q2573723)

In this paper, a t-norm is a binary function \(T: [0, 1]^2\to [0, 1]\) that is commutative, associative, monotone and fulfils the boundary condition \(T(x, b)=x\), \(x\in [0, 1]\). Recall that t-norms provide a natural basis for the construction of semantics of various kinds of fuzzy logics. When bound to continuous t-norms, completed by standard negation \(N(x)=1-x\) and by extending operations to countable arity, we may derive all measurable logic functions. The question remains wheter there are other t-norms with this property, and also whether all strict t-norms have this property. The paper gives a full answer together with convenient sufficient conditions. Several counterexamples are also provided and consequences for the characterization of tribes based on strict t-norms are discussed.