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This generalised Kripke semantics is based on the<jats:italic>poset sum<\/jats:italic>construction, used in Bova and Montagna (Theoret Comput Sci 410(12):1143\u20131158, 2009). to show the decidability (and PSPACE completeness) of the quasiequational theory of commutative, integral and bounded GBL algebras. The main idea is that<jats:inline-formula><jats:alternatives><jats:tex-math>$$w \\Vdash \\psi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>w<\/mml:mi><mml:mo>\u22a9<\/mml:mo><mml:mi>\u03c8<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>\u2014which for<jats:bold>IL<\/jats:bold>is a relation between worlds<jats:italic>w<\/jats:italic>and formulas<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\psi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03c8<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and can be seen as a function taking values in the booleans<jats:inline-formula><jats:alternatives><jats:tex-math>$$(w \\Vdash \\psi ) \\in {{\\mathbb {B}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>w<\/mml:mi><mml:mo>\u22a9<\/mml:mo><mml:mi>\u03c8<\/mml:mi><mml:mo>)<\/mml:mo><mml:mo>\u2208<\/mml:mo><mml:mi>B<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>\u2014becomes a function taking values in the unit interval<jats:inline-formula><jats:alternatives><jats:tex-math>$$(w \\Vdash \\psi ) \\in [0,1]$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>w<\/mml:mi><mml:mo>\u22a9<\/mml:mo><mml:mi>\u03c8<\/mml:mi><mml:mo>)<\/mml:mo><mml:mo>\u2208<\/mml:mo><mml:mo>[<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. An appropriate monotonicity restriction (which we call<jats:italic>sloping functions<\/jats:italic>) needs to be put on such functions in order to ensure soundness and completeness of the semantics.<\/jats:p>","DOI":"10.1007\/s11225-020-09908-z","type":"journal-article","created":{"date-parts":[[2020,4,21]],"date-time":"2020-04-21T11:03:02Z","timestamp":1587466982000},"page":"313-339","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Kripke Semantics for Intuitionistic \u0141ukasiewicz Logic"],"prefix":"10.1007","volume":"109","author":[{"given":"A.","family":"Lewis-Smith","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P.","family":"Oliva","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"E.","family":"Robinson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,4,21]]},"reference":[{"key":"9908_CR1","doi-asserted-by":"crossref","unstructured":"Bova, S., and F. 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