{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T09:30:50Z","timestamp":1768901450099,"version":"3.49.0"},"reference-count":38,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2019,2,15]],"date-time":"2019-02-15T00:00:00Z","timestamp":1550188800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2019,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Suszko\u2019s problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski\u2019s structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called <jats:italic>mixed consequence<\/jats:italic>, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this article we give a more systematic perspective on Suszko\u2019s problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that (intersective) mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those theorems to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos, and Wansing, for logics with various structural properties (reflexivity, transitivity, none, or both). We strengthen these results into exact rank results for <jats:italic>nonpermeable<\/jats:italic> logics (roughly, those which distinguish the role of premises and conclusions). We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call <jats:italic>regular connectives<\/jats:italic>, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties.<\/jats:p>","DOI":"10.1017\/s1755020318000503","type":"journal-article","created":{"date-parts":[[2019,2,15]],"date-time":"2019-02-15T06:33:43Z","timestamp":1550212423000},"page":"736-767","source":"Crossref","is-referenced-by-count":18,"title":["SUSZKO\u2019S PROBLEM: MIXED CONSEQUENCE AND COMPOSITIONALITY"],"prefix":"10.1017","volume":"12","author":[{"given":"EMMANUEL","family":"CHEMLA","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"PAUL","family":"\u00c9GR\u00c9","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2019,2,15]]},"reference":[{"key":"S1755020318000503_ref37","doi-asserted-by":"publisher","DOI":"10.1007\/BF02123806"},{"key":"S1755020318000503_ref34","first-page":"30","volume-title":"Logic, Semantics, Metamathematics","author":"Tarski","year":"1930"},{"key":"S1755020318000503_ref32","first-page":"87","article-title":"Remarks on \u0141ukasiewicz\u2019s three-valued logic","volume":"3\u20134","author":"Suszko","year":"1975","journal-title":"Bulletin of the Section of Logic"},{"key":"S1755020318000503_ref31","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(09)70230-7"},{"key":"S1755020318000503_ref10","first-page":"2193","article-title":"Characterizing logical consequence in many-valued logic","volume":"27","author":"Chemla","year":"2017","journal-title":"Journal of Logic and Computation"},{"key":"S1755020318000503_ref3","doi-asserted-by":"publisher","DOI":"10.18778\/0138-0680.46.3.4.05"},{"key":"S1755020318000503_ref19","first-page":"288","article-title":"Investigations into logical deduction","volume":"1","author":"Gentzen","year":"1964","journal-title":"American Philosophical Quarterly"},{"key":"S1755020318000503_ref27","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/025\/0363802"},{"key":"S1755020318000503_ref35","doi-asserted-by":"publisher","DOI":"10.1023\/A:1005020217249"},{"key":"S1755020318000503_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02218674"},{"key":"S1755020318000503_ref28","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511565687"},{"key":"S1755020318000503_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s11229-019-02344-0"},{"key":"S1755020318000503_ref38","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-6942-2"},{"key":"S1755020318000503_ref21","doi-asserted-by":"publisher","DOI":"10.1007\/BF00183072"},{"key":"S1755020318000503_ref33","doi-asserted-by":"publisher","DOI":"10.1007\/BF02120672"},{"key":"S1755020318000503_ref6","unstructured":"Caleiro C. , Carnielli W. A. , Coniglio M. E. , & Marcos J. (2003). Dyadic semantics for many-valued logics. Preprint. 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