### A dissimilarity-based framework for generating inconsistency-tolerant logics

Ofer Arieli, Anna Zamansky
2013 Annals of Mathematics and Artificial Intelligence
Many commonly used logics, including classical logic and intuitionistic logic, are trivialized in the presence of inconsistency, in the sense that inconsistent premises cause the derivation of any formula. It is thus often useful to define inconsistency-tolerant variants of such logics, which are faithful to the original logic with respect to consistent theories but also allow for nontrivial inconsistent theories. A common way of doing so is by incorporating distance-based considerations for
more » ... crete logics. So far this has been done mostly in the context of two-valued semantics. Our purpose in this paper is to show that inconsistency-tolerance can be achieved for any logic that is based on a denotational semantics. For this, we need to trade distances for the more general notion of dissimilarities. We then examine the basic properties of the entailment relations that are obtained and exemplify dissimilarity-based reasoning in various forms of denotational semantics, including multi-valued semantics, nondeterministic semantics, and possible-worlds (Kripke-style) semantics. Moreover, we show that our approach can be viewed as an extension of several well-studied forms of reasoning in the context of belief revision, database integration, consistent query answering, and inconsistency maintenance in knowledge-based systems. In the sequel, L denotes a propositional language with a countable set Atoms = {p, q, r . . .} of atomic formulas and a (countable) set F L = {ψ, ϕ, σ, . . .} of wellformed formulas. A theory Γ is a finite set of formulas in F L . The atoms appearing in the formulas of Γ and the subformulas of Γ are denoted, respectively, by Atoms(Γ ) and SF(Γ ). The set of all theories of L is denoted by T L . Definition 1. Given a language L, a propositional logic for L is a pair ⟨L, ⊢⟩, where ⊢ is a (Tarskian) consequence relation for L, i.e., a binary relation between sets of formulas in F L and formulas in F L , satisfying the following conditions: A common (model-theoretical) way of defining logics for L is based on the notion of denotational semantics: Definition 2. A denotational semantics for a language L is a pair S = ⟨S, |= S ⟩, where S is a non-empty set (of 'interpretations'), and |= S (the 'satisfiability relation' of S) is a computable binary relation on S × F L . Example 1. The most common case of denotational semantics S = ⟨S, |= S ⟩ for L is classical logic. In this case, L is a propositional language, the elements of S are (two-valued) valuations, i.e., functions from F L to the set {t, f } of the classical truth values, and |= S is the ordinary satisfaction relation, defined by ν |= S Γ iff ν(ψ) = t for every ψ ∈ Γ . Standard generalizations of classical logic to multiple-valued logics can also be described in terms of denotational semantics and so are, e.g., the various kinds of Kripke-structures for modal and for intuitionistic logics (some of which are described in greater detail in Section 6 below), provided that the underlying satisfiability relation is computable. Definition 3. Let S = ⟨S, |= S ⟩ be a denotational semantics for L, ν ∈ S an interpretation, and ψ ∈ F L a formula. a) If ν |= S ψ, we say that ν satisfies ψ and call ν an S-model of ψ. b) The set of the S-models of ψ is denoted by mod S (ψ). When mod S (ψ) is the set S, ψ is called an S-tautology, and when mod S (ψ) is the empty set, ψ is called an S-contradiction. c) If ν satisfies every formula ψ in a theory Γ , it is called an S-model of Γ . The set of the S-models of Γ is denoted by mod S (Γ ). If mod S (Γ ) ̸ = ∅ we say that Γ is S-consistent, otherwise Γ is S-inconsistent. In what follows we shall sometimes omit the prefix S from the above notions. Definition 4. A denotational semantics S = ⟨S, |= S ⟩ is normal , if for each ν ∈ S there is a formula ψ, such that ν ̸ |= S ψ.