{
"id": "2006.16226",
"version": "v1",
"published": "2020-06-29T17:45:04.000Z",
"updated": "2020-06-29T17:45:04.000Z",
"title": "On Matrix Consequence (Extended Abstract)",
"authors": [
"Alexei Muravitsky"
],
"categories": [
"math.LO"
],
"abstract": "These results are a contribution to the model theory of matrix consequence. We give a semantic characterization of uniform and couniform consequence relations. These properties have never been treated individually, at least in a semantic manner. We consider these notions from a purely semantic point of view and separately, introducing the notion of a uniform bundle/atlas and that of a couniform class of logical matrices. Then, we show that any uniform bundle defines a uniform consequence; and if a structural consequence is uniform, then its Lindenbaum atlas is uniform. Thus, any structural consequence is uniform if, and only if, it is determined by a uniform bundle/atlas. On the other hand, any couniform set of matrices defines a couniform structural consequence. Also, the Lindenbaum atlas of a couniform structural consequence is couniform. Thus, any structural consequence is couniform if, and only if, it is determined by a couniform bundle/atlas. We then apply these observations to compare structural consequence relations that are defined in different languages when one language is a primitive extension of another. We obtain that for any structural consequence defined in a language having (at least) a denumerable set of sentential variables, if this consequence is uniform and couniform, then it and the \\emph{ W\\'{o}jcicki's consequence} corresponding to it, which is defined in any primitive extension of the given language, are determined by one and the same atlas which is both uniform and couniform.",
"revisions": [
{
"version": "v1",
"updated": "2020-06-29T17:45:04.000Z"
}
],
"analyses": {
"subjects": [
"03G27"
],
"keywords": [
"matrix consequence",
"extended abstract",
"couniform structural consequence",
"uniform bundle/atlas",
"lindenbaum atlas"
],
"note": {
"typesetting": "TeX",
"pages": 0,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}