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We view these as<jats:italic>variant semantics<\/jats:italic>and present a semantic formulation of Troelstra\u2019s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we demonstrate that failure via specific finite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstra\u2019s criteria in the corresponding substructural logical setting.<\/jats:p>","DOI":"10.1007\/s11225-020-09909-y","type":"journal-article","created":{"date-parts":[[2020,5,25]],"date-time":"2020-05-25T20:03:07Z","timestamp":1590436987000},"page":"341-365","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Double Negation Semantics for Generalisations of Heyting Algebras"],"prefix":"10.1007","volume":"109","author":[{"given":"Rob","family":"Arthan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paulo","family":"Oliva","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,5,25]]},"reference":[{"issue":"2\u20133","key":"9909_CR1","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1007\/s000120050156","volume":"43","author":"WJ Blok","year":"2000","unstructured":"Blok, W.\u00a0J., and I.\u00a0M.\u00a0A. Ferreirim, On the structure of hoops, Algebra Universalis 43(2\u20133): 233\u2013257, 2000.","journal-title":"Algebra Universalis"},{"issue":"4","key":"9909_CR2","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1007\/BF01195727","volume":"32","author":"WJ Blok","year":"1994","unstructured":"Blok, W.\u00a0J., and D.\u00a0Pigozzi, On the structure of varieties with equationally definable principal congruences. III, Algebra Universalis 32(4): 545\u2013608, 1994.","journal-title":"Algebra Universalis"},{"key":"9909_CR3","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1006\/jabr.1996.6834","volume":"190","author":"Willem J Blok","year":"1997","unstructured":"Blok, W.\u00a0J., and J.\u00a0G. 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