{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T14:10:41Z","timestamp":1699884641487},"reference-count":31,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T00:00:00Z","timestamp":1638403200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that monadic intuitionistic quantifiers admit the following temporal interpretation: \u201calways in the future\u201d (for<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline1.png\" \/><jats:tex-math>$\\forall $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>) and \u201csometime in the past\u201d (for<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline2.png\" \/><jats:tex-math>$\\exists $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>). It is well known that Prior\u2019s intuitionistic modal logic<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline3.png\" \/><jats:tex-math>${\\sf MIPC}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>axiomatizes the monadic fragment of the intuitionistic predicate logic, and that<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline4.png\" \/><jats:tex-math>${\\sf MIPC}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is translated fully and faithfully into the monadic fragment<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline5.png\" \/><jats:tex-math>${\\sf MS4}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>of the predicate<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline6.png\" \/><jats:tex-math>${\\sf S4}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>via the G\u00f6del translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline7.png\" \/><jats:tex-math>${\\sf TS4}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline8.png\" \/><jats:tex-math>${\\sf S4}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and provide a full and faithful translation of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline9.png\" \/><jats:tex-math>${\\sf MIPC}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>into<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline10.png\" \/><jats:tex-math>${\\sf TS4}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We compare this new translation of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline11.png\" \/><jats:tex-math>${\\sf MIPC}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>with the G\u00f6del translation by showing that both<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline12.png\" \/><jats:tex-math>${\\sf TS4}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline13.png\" \/><jats:tex-math>${\\sf MS4}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>can be translated fully and faithfully into a tense extension of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline14.png\" \/><jats:tex-math>${\\sf MS4}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, which we denote by<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline15.png\" \/><jats:tex-math>${\\sf MS4.t}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000496_inline16.png\" \/><jats:tex-math>${\\sf MS4.t}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered.<\/jats:p>","DOI":"10.1017\/s1755020321000496","type":"journal-article","created":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T09:23:04Z","timestamp":1638436984000},"page":"164-187","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["TEMPORAL INTERPRETATION OF MONADIC INTUITIONISTIC QUANTIFIERS"],"prefix":"10.1017","volume":"16","author":[{"given":"GURAM","family":"BEZHANISHVILI","sequence":"first","affiliation":[]},{"given":"LUCA","family":"CARAI","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,12,2]]},"reference":[{"key":"S1755020321000496_r25","first-page":"65","article-title":"Relations between intuitionistic modal logics and intermediate predicate logics","volume":"22","author":"Ono","year":"1988","journal-title":"Reports on Mathematical Logic"},{"key":"S1755020321000496_r19","first-page":"85","volume-title":"Advances in Modal Logic (Berlin, 1996)","volume":"1","author":"Grefe","year":"1998"},{"key":"S1755020321000496_r14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02121259"},{"key":"S1755020321000496_r4","first-page":"95","volume-title":"Advances in Modal Logic","volume":"13","author":"Bezhanishvili","year":"2020"},{"key":"S1755020321000496_r9","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198537793.001.0001","volume-title":"Modal Logic","author":"Chagrov","year":"1997"},{"key":"S1755020321000496_r3","doi-asserted-by":"publisher","DOI":"10.1023\/A:1005173628262"},{"key":"S1755020321000496_r7","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093958154"},{"key":"S1755020321000496_r12","first-page":"4","volume-title":"Intensional Logics and Logical Structure of Theories","author":"Esakia","year":"1988"},{"key":"S1755020321000496_r31","doi-asserted-by":"publisher","DOI":"10.1023\/A:1004218110879"},{"key":"S1755020321000496_r26","volume-title":"Time and Modality","author":"Prior","year":"1957"},{"key":"S1755020321000496_r30","doi-asserted-by":"publisher","DOI":"10.2307\/2272558"},{"key":"S1755020321000496_r18","volume-title":"Logics of Time and Computation","volume":"7","author":"Goldblatt","year":"1992"},{"key":"S1755020321000496_r1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1958-0094308-5"},{"key":"S1755020321000496_r16","volume-title":"Many-Dimensional Modal Logics: Theory and Applications","author":"Gabbay","year":"2003"},{"key":"S1755020321000496_r13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-12096-2"},{"key":"S1755020321000496_r27","volume-title":"The Mathematics of Metamathematics","author":"Rasiowa","year":"1963"},{"key":"S1755020321000496_r29","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1970.tb00429.x"},{"key":"S1755020321000496_r20","first-page":"217","article-title":"Algebraic logic. 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