Formal fuzzy logic

Necessity logic

Assume that the deduction apparatus of classical first order logic is presented by a suitable set la of logical axioms, by the MP-rule and the Generalization rule and denote by $\vdash$ the related consequence relation. Then a fuzzy deduction system is obtained by considering as fuzzy subset of logical axioms the characteristic function of la and as fuzzy inference rules the extension of MP obtained by assuming that ʘ is the minimum operator $\wedge$. Moreover, an extension of the Generalization Rule is obtained by assuming that if we prove α at degree λ then we obtain $\forall$xα(x) at the same degree λ. Assume that D is the deduction operator of such a fuzzy logic and that s is a fuzzy theory. Then D(s)(α) = 1 for every logically true formula α and, otherwise,

D(s)(α) = Sup{s(α₁)$\wedge ...\wedge$s(α_n) : α₁,..., α_n $\vdash$α}.

By recalling that the existential quantifier is interpreted by the supremum operator, such a formula arises from a multivalued valuation of the (metalogical) claim: "α is a consequence of the fuzzy subset s of axioms provided there are formulas α₁, ...,α_n in s able to prove $\alpha$". In such a case the vagueness originates from s, i.e., from the notion of "hypothesis". Moreover s(α) is not a truth degree but rather a degree of "preference" or "acceptability" for α. For example, let T be a system of axioms for set theory and assume that the choice axiom CA does not depend on T. Then we can consider the fuzzy subset of axioms s defined by setting

s(α) = 1 if α є T,

s(α) = 0.8 if α = CA ,

s(α) = 0 otherwise.

A simple calculation shows that:

D(s)(α) = 1 if α is a theorem of T,

D(s)(α) = 0.8 if we cannot prove α from T but α is a theorem of T + CA,

D(s)(α) = 0 otherwise .

Then, despite the fact that no vague predicate is considered in set theory, in the metalanguage we can consider a vague meta-predicate as "is acceptable" and to represent it by a suitable fuzzy subset s.