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The work resolves two longstanding meta-theoretical problems left open since its introduction: compactness and the computational status of validity. It proves the logic is neither compact nor recursively axiomatizable, and it also shows a violation of Craig interpolation. Results are further extended to inquisitive team logic.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/inquisitive-first-order-logic/505057/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/inquisitive-first-order-logic/505057.png","ImageObject",300,407,{"name":92,"@type":93},"4398046744996","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-18","2026-10-04",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What makes inquisitive first-order logic different from classical first-order logic?","Question",{"text":112,"@type":113},"It extends classical first-order logic so formulas can regiment not only declarative statements but also first-order questions such as whether, which, and what example objects satisfy a property.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What two main meta-theoretical questions are answered in this work?",{"text":117,"@type":113},"The paper addresses compactness and the computational status of validity, showing both properties fail for inquisitive first-order logic.",{"name":119,"@type":110,"acceptedAnswer":120},"Which additional first-order signature property does the logic violate?",{"text":121,"@type":113},"It violates Craig interpolation, in addition to failing compactness and not being recursively axiomatizable.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},505057,1791276266,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":66,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":133,"file_id":134,"file_url":135,"file_type":136,"file_size":137,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":138,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":26},4398046744996,"arXiv : 2603 .20845v2 [math .LO] 14 Sep 2026  \nInquisitive first-order logic is neither compact nor  \nrecursively axiomatizable ∗ Ivano Ciardelli† and Juha Kontinen‡  \nSeptember 15, 2026  \nAbstract  \nInquisitive first-order logic is an extension of classical first-order logic with formulas regimenting first-order questions, such as “whether all objects are P”,“which objects are P”, and “what is one object that is P”. Since it was first developed in 2009, two major meta-theoretical questions about this logic have remained open, in spite of significant efforts. The first concerns compactness:  \nif a conclusion follows from a set of premises, does it always follow from some finite subset? The second concerns the computational status of validity: is theset of validities recursively enumerable, or equivalently, does the logic admit arecursive axiomatization?  \nWe settle both questions in the negative, showing that inquisitive first-order logic is neither compact nor recursively axiomatizable. Furthermore, we prove that it violates another signature property of first-order logic, namely, Craig interpolation. We discuss the significance of our results, and show how to extend them to a closely related logic, viz., inquisitive team logic.  \n1 Introduction  \nTraditionally, logic is concerned with relations between statements—declarative sentences such as “Alice is a student” or “all students passed the test”. Inquisitive logic (Ciardelli et al. , 2018a; Ciardelli, 2023) is an approach that aims to extend the scope of logic to cover not only statements, but also questions—interrogative sentences such as “whether Alice is a student” and “which students passed the test”. This extension is interesting for several reasons: it makes it possible to regard interesting logical relations such as answerhood and dependency as facets of the key logical notion of entailment, and to deploy the tools traditionally developed for entailment  \n∗ The first author gratefully acknowledges funding from the European Research Council under the EU Horizon Europe research and innovation programme (Grant Agreement No. 101116774) . The second author was partially supported by a Research Council of Finland grant No. 375116.  \nNo use of AI was made in this work, either in obtaining the results or in writing the paper.†University of Padua. Email: [ivano.ciardelli@unipd.it. ORCID:](ivano.ciardelli@unipd.it. ORCID:) 0000-0001-6152-3401.  \n‡[University of Helsinki. Email: juha.kontinen@helsinki.fi. ORCID: 0000-0003-0115-5154](University of Helsinki. Email: juha.kontinen@helsinki.fi. ORCID: 0000-0003-0115-5154) .  \nto study these notions (Ciardelli, 2016 , 2023); it allows for a uniform account of the contribution of connectives and quantifiers across statements and questions (Ciardelliet al. , 2018a); it enables us to perform inferences with formulas denoting information types, rather than specific pieces of information (Ciardelli, 2018; Puncoch´aˇr and Noguera, 2026); it allows for the definition of new modal operators that capture interesting modal notions (Ciardelli and Roelofsen, 2015; Ciardelli, 2025 , 2026) . Since its development, inquisitive logic has found applications in different domains, including natural language semantics (see, among many others, Coppock and Brochhagen, 2013; AnderBois, 2014; Roelofsen and Farkas, 2015; Szabolcsi, 2015; Ciardelli et al. , 2018b; Booth, 2021; Roelofsen and Dotlaˇcil, 2022), cognitive psychology (Koralus and Mascarenhas, 2013; Mascarenhas and Koralus, 2015; Mascarenhas and Picat, 2019), formal epistemology (Cohen, 2021) and decision theory (Dever and Schiller, 2020) .  \nSyntactically, inquisitive logics are typically obtained by taking an existing logic of statements, and extending it with question-forming operators. Semantically, they are interpreted in standard possible-world models for intensional logics. In order to give a uniform account of statements and questions, however, the basic semantic notion of truth at a world ","cbCaiaxG418i4DXh","https://ap.wps.com/l/cbCaiaxG418i4DXh","pdf",495890,24,"English","# Abstract\n# 1 Introduction","[{\"question\":\"What makes inquisitive first-order logic different from classical first-order logic?\",\"answer\":\"It extends classical first-order logic so formulas can regiment not only declarative statements but also first-order questions such as whether, which, and what example objects satisfy a property.\"},{\"question\":\"What two main meta-theoretical questions are answered in this work?\",\"answer\":\"The paper addresses compactness and the computational status of validity, showing both properties fail for inquisitive first-order logic.\"},{\"question\":\"Which additional first-order signature property does the logic violate?\",\"answer\":\"It violates Craig interpolation, in addition to failing compactness and not being recursively axiomatizable.\"}]","Inquisitive first-order logic | PDF",1791087643]