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Three of them were given by Kearns for\n                    <jats:bold>S5<\/jats:bold>\n                    ,\n                    <jats:bold>S4<\/jats:bold>\n                    ,\n                    <jats:bold>KT<\/jats:bold>\n                    . Two of them were given by Coniglio, Fari\u00f1as del Cerro, and Peron for\n                    <jats:bold>S4<\/jats:bold>\n                    ,\n                    <jats:bold>KTB<\/jats:bold>\n                    . To these five sets, we add three new ones for\n                    <jats:bold>S5<\/jats:bold>\n                    and\n                    <jats:bold>KT<\/jats:bold>\n                    . We show that these eight sets are the only ones of this kind that satisfy certain additional natural conditions. From these eight sets of tables, one can create sixteen non-deterministic matrices, depending on whether we distinguish only the necessary truth or both truths. From these matrices, one can create various semantics by adding to them an appropriate set of valuations. Kearns singled out a countably infinite decreasing sequence of sets of valuations of a given level. For\n                    <jats:bold>S5<\/jats:bold>\n                    ,\n                    <jats:bold>S4<\/jats:bold>\n                    ,\n                    <jats:bold>KT<\/jats:bold>\n                    , Kearns applied valuations with infinite level, which belong to all these sets (for these valuations, the choice of the set of distinguished values is not important). Omori and Skurt prove that Kearns\u2019 tables for\n                    <jats:bold>S5<\/jats:bold>\n                    and\n                    <jats:bold>S4<\/jats:bold>\n                    can be used with 1st-level valuations. We add the set of PL-valuations (which assign necessary truth to all substitution instances of classical tautologies), which generally contains the set of 1st-level valuations. We showed that only for Kearns\u2019 tables for\n                    <jats:bold>S5<\/jats:bold>\n                    , the sets of PL- and 1st-level valuations are equal. We prove that the remaining seven tables with two distinguished values and PL-valuations are adequate for\n                    <jats:bold>S0.5<\/jats:bold>\n                    and six very weak logics containing it. We show that three of these seven tables with 1st-level valuations are adequate for\n                    <jats:bold>S5<\/jats:bold>\n                    and\n                    <jats:bold>S4<\/jats:bold>\n                    . The remaining four sets of tables with the infinite level valuations are adequate for\n                    <jats:bold>S5<\/jats:bold>\n                    ,\n                    <jats:bold>KTB<\/jats:bold>\n                    ,\u00a0\n                    <jats:bold>KT<\/jats:bold>\n                    .\n                  <\/jats:p>","DOI":"10.1007\/s10992-026-09864-4","type":"journal-article","created":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T16:36:18Z","timestamp":1791477378000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Non-Deterministic Four-Valued Semantics for Some Normal and Very Weak T-Normal Logics"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9133-5081","authenticated-orcid":false,"given":"Andrzej","family":"Pietruszczak","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,10,8]]},"reference":[{"key":"9864_CR1","doi-asserted-by":"crossref","unstructured":"Avron, A., & Lev, I. 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