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On their approach, a conditional takes the value of its consequent whenever its antecedent is true, and the value Indeterminate otherwise. Here we deal with the problem of selecting an adequate notion of validity for this conditional. We show that all standard validity schemes based on de Finetti\u2019s table come with some problems, and highlight two ways out of the predicament: one pairs de Finetti\u2019s conditional () with validity as the preservation of non-false values (-validity), but at the expense of Modus Ponens; the other modifies de Finetti\u2019s table to restore Modus Ponens. In Part I of this paper, we present both alternatives, with specific attention to a variant of de Finetti\u2019s table () proposed by Cooper (<jats:italic>Inquiry 11<\/jats:italic>, 295\u2013320, 1968) and Cantwell (<jats:italic>Notre Dame Journal of Formal Logic 49<\/jats:italic>, 245\u2013260, 2008). In Part II, we give an in-depth treatment of the proof theory of the resulting logics,  and : both are connexive logics, but with significantly different algebraic properties.<\/jats:p>","DOI":"10.1007\/s10992-020-09549-6","type":"journal-article","created":{"date-parts":[[2020,8,20]],"date-time":"2020-08-20T10:04:00Z","timestamp":1597917840000},"page":"187-213","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["De Finettian Logics of Indicative Conditionals Part I: Trivalent Semantics and Validity"],"prefix":"10.1007","volume":"50","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9114-7686","authenticated-orcid":false,"given":"Paul","family":"\u00c9gr\u00e9","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1932-5484","authenticated-orcid":false,"given":"Lorenzo","family":"Rossi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0083-9685","authenticated-orcid":false,"given":"Jan","family":"Sprenger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,8,20]]},"reference":[{"key":"9549_CR1","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1080\/00201746508601430","volume":"8","author":"EW Adams","year":"1965","unstructured":"Adams, E.W. 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