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Since we want to keep things simple we treat ◦ as a dummy operator and hence don’t attach any logical properties to ◦. As a lower limit logic we employ classical propositional logic CL equipped with ◦. 4 The semantics of CL◦ is like the semantics for CL, just besides the usual assignment function v that assigns to each propositional letter a truth value, we also use an enhanced assignment function v◦ that (independently from v) associates each well-formed formula with a truth-value. Truth in a model M is defined as usual for the classical operators: 4 In [9] we show that CL◦ gives rise to very simple ALs that represent the Rescher-Manor consequence relations [16].

Viewed in this way, we only have a ‘deductive’ logic in which we formally distinguish between two types of premises. The adaptive marking then handles which parts of the uncertain basis may be considered safe in specific inferences and retracts inferences that are based on unsafe assumptions. Let us demonstrate this with a familiar example. n} ∅ ∅ On the right side we use two premise introduction rules: PREM1 for the premises in the solid base Γ and PREM2 for the premises in the uncertain premise set Ω ¬ˇ.

In the last column we keep a record of the used “uncertain” premises. , deductive) inferences that stem from the lower limit logic. DS is disjunctive syllogism (we could have also just written RU since DS is valid in the lower limit logic enriched by the “checked connectives”). The question which parts of the uncertain premise set can be considered safe for a given inference is analogous to the determination of the marking of lines. For instance, according to the minimal abnormality strategy a line l with formula A and a record Δ ↓ Ω ¬ˇ is marked at stage s iff, (i) there is no ϕ ∈ Φs (Γ ) for which A similar distinction can be found for instance in the ASPIC+ -framework [27, 28] where we find an ‘ordinary’ knowledge base K p that is uncertain and an ‘axiomatic’ solid knowledge base Kn .