Logics for approximate and strong entailments

We consider two kinds of similarity-based reasoning and formalise them in a logical setting. In one case, we are led by the principle that conclusions can be drawn even if they are only approximately correct. This leads to a graded approximate entailment, which is weaker than classical entailment. In the other case, we follow the principle that conclusions must remain correct even if the assumptions are slightly changed. This leads to a notion of a graded strong entailment, which is stronger than classical entailment. We develop two logical calculi based on the notions of approximate and of strong entailment, respectively. © 2011 Elsevier B.V.The authors acknowledge partial support of the bilateral Austrian-Spanish project HA2008-0017 and the Eurocores-LogICCC ESF project LoMoReVI. Esteva and Godo also acknowledge partial support of the Spanish project FFI2008-03126-E/FILO and Rodrıguez acknowledges the projects CyT-UBA X484 and the research CONICET program PIP 12-200801-02543 2009-2011. Finally, Esteva, Godo and Rodrıguez also acknowledge partial support of the MaToMUVI project (PIRSES-GA-2009- 247584).Peer Reviewe

The McKinsey-Tarski theorem for locally compact ordered spaces

We prove that the modal logic of a crowded locally compact generalized ordered space is S4. This provides a version of the McKinsey–Tarski theorem for generalized ordered spaces. We then utilize this theorem to axiomatize the modal logic of an arbitrary locally compact generalized ordered space

Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3