Graded Entailment

Let L = (*L*, ≤, ∗) such that (*L*, ≤) is a complete meet-semilattice (with 1 denoting its upper bound) and ∗ is a binary operation on *L*. An L*-entailment system* (*Sign*, *Sen*, ") consists of a functor *Sen* : *Sign* → **Set** and a family "= ("<sup>Σ</sup> : P*Sen*(Σ) → *Sen*(Σ))Σ∈|*Sign*<sup>|</sup> such that the following axioms hold:


When L is just the binary Boolean algebra (with ∗ being ∧), L-entailment systems are just ordinary entailment systems [44,103]/*π*-institutions [104]. In the graded context, the binary entailment systems will also be called *crisp entailment systems*. Previous to [49], the idea of graded entailment has appeared in various different forms in works such as [50,94,102,105]; in [49], there is a brief analysis on the differences between these several variants, which are in fact rather slight. Depending on actual applications, graded entailments may be interpreted in various ways: as provability degree, as degree of confidence in proofs, or even as a(n inverse) measure for the complexity of a proof. Moreover, in [49], there are also temporal interpretations of graded proofs. An important technical aspect worth mentioning is the use of ∗ rather than ∧ in the *transitivity* axiom; in [49], it is shown that this choice is necessary for accommodating the semantic interpretations of graded entailment.

The result of [49] that the graded semantic consequence in an L-institution I yields an L-entailment system—called the *semantic entailment system of* I—seems to suggest that L-entailment systems are more abstract/general than L-institutions. However, at least when L is a complete residuated lattice, this is a wrong impression, because a result from [49] shows that each L-entailment system determines an L-institution whose semantic entailment is precisely the respective L-entailment system.
