# Finite Models for Non-Commutative Ambiguity Logics

## Christian Wurm

We speak of ambiguity (in the linguistic sense) if one sign (word, phrase, sentence) has two or more clearly distinct meanings. Recent work on logics with an explicit representation of ambiguity (see [2,3]) has shown that there are two kinds of logics: those who are closed under transitive inference or more generally under the rule (cut), and those which are closed under uniform substitution of atoms. Having both properties (which are both basic requirements of abstract logics in the sense of Tarski) – together with the most basic properties of ambiguity – leads to triviality of the logic.

Moreover, every ambiguity logic comes in two versions: a) with commutative ambiguity (basically, readings can be conceived of as sets), and b) with non-commutative ambiguity (readings can be conceived of as an ordered list). Two ambiguity logics have received a particular interest (in both variants): DAL (introduced in [1], closed under (cut)) and and TAL (see [2], closed under uniform substitution), with their commutative variants cDAL and cTAL. From a semantic point of view, cDAL and cTAL are considerably simpler: there is a straightforward semantics for both based on finite classical models, which allows to prove two key results (for proofs see [1,3]):

- both cDAL, cTAL are decidable (via Finite Model Property)
- cDAL is the inner logic of cTAL (The inner logic of a non-transitive logic is roughly speaking its largest transitive fragment.)

In this talk, we pesent a sound and complete semantics with finite models for non-commutative DAL and TAL, which allows to transfer the two results to this case. The models are sets of strings with certain language-theoretic closure properties.

## Bibliography

- Jan van Eijck, Jan Jaspars,
*Ambiguity and Reasoning*,*Manuscript* - Christian Wurm,
*Reasoning with Ambiguity*,,vol.30 (1), pp.139-206.*Journal of Logic, Language and Information* - Christian Wurm,
*The Family of Ambiguity Logics*,*Submitted*