Dylan Bellier, Massimo Benerecetti, Dario Della Monica, Fabio Mogavero
{"title":"交替(In)依赖友好逻辑","authors":"Dylan Bellier, Massimo Benerecetti, Dario Della Monica, Fabio Mogavero","doi":"10.1016/j.apal.2023.103315","DOIUrl":null,"url":null,"abstract":"<div><p>Hintikka and Sandu originally proposed <em>Independence Friendly Logic</em> (<figure><img></figure>) as a first-order logic of <em>imperfect information</em> to describe <em>game-theoretic phenomena</em> underlying the semantics of natural language. The logic allows for expressing independence constraints among quantified variables, in a similar vein to Henkin quantifiers, and has a nice <em>game-theoretic semantics</em> in terms of <em>imperfect information games</em>. However, the <figure><img></figure> semantics exhibits some limitations, at least from a purely logical perspective. It treats the players asymmetrically, considering only one of the two players as having imperfect information when evaluating truth, <em>resp.</em>, falsity, of a sentence. In addition, truth and falsity of sentences coincide with the existence of a uniform winning strategy for one of the two players in the semantic imperfect information game. As a consequence, <figure><img></figure> does admit undetermined sentences, which are neither true nor false, thus failing the law of excluded middle. These idiosyncrasies limit its expressive power to the existential fragment of <em>Second Order Logic</em> (<figure><img></figure>). In this paper, we investigate an extension of <figure><img></figure>, called <em>Alternating Dependence/Independence Friendly Logic</em> (<figure><img></figure>), tailored to overcome these limitations. To this end, we introduce a novel <em>compositional semantics</em>, generalising the one based on trumps proposed by Hodges for <figure><img></figure>. The new semantics (i) allows for meaningfully restricting both players at the same time, (ii) enjoys the property of game-theoretic determinacy, (iii) recovers the law of excluded middle for sentences, and (iv) grants <figure><img></figure> the full descriptive power of <figure><img></figure>. We also provide an equivalent <em>Herbrand-Skolem semantics</em> and a <em>game-theoretic semantics</em> for the prenex fragment of <figure><img></figure>, the latter being defined in terms of a determined infinite-duration game that precisely captures the other two semantics on finite structures.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Alternating (In)Dependence-Friendly Logic\",\"authors\":\"Dylan Bellier, Massimo Benerecetti, Dario Della Monica, Fabio Mogavero\",\"doi\":\"10.1016/j.apal.2023.103315\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Hintikka and Sandu originally proposed <em>Independence Friendly Logic</em> (<figure><img></figure>) as a first-order logic of <em>imperfect information</em> to describe <em>game-theoretic phenomena</em> underlying the semantics of natural language. The logic allows for expressing independence constraints among quantified variables, in a similar vein to Henkin quantifiers, and has a nice <em>game-theoretic semantics</em> in terms of <em>imperfect information games</em>. However, the <figure><img></figure> semantics exhibits some limitations, at least from a purely logical perspective. It treats the players asymmetrically, considering only one of the two players as having imperfect information when evaluating truth, <em>resp.</em>, falsity, of a sentence. In addition, truth and falsity of sentences coincide with the existence of a uniform winning strategy for one of the two players in the semantic imperfect information game. As a consequence, <figure><img></figure> does admit undetermined sentences, which are neither true nor false, thus failing the law of excluded middle. These idiosyncrasies limit its expressive power to the existential fragment of <em>Second Order Logic</em> (<figure><img></figure>). In this paper, we investigate an extension of <figure><img></figure>, called <em>Alternating Dependence/Independence Friendly Logic</em> (<figure><img></figure>), tailored to overcome these limitations. To this end, we introduce a novel <em>compositional semantics</em>, generalising the one based on trumps proposed by Hodges for <figure><img></figure>. The new semantics (i) allows for meaningfully restricting both players at the same time, (ii) enjoys the property of game-theoretic determinacy, (iii) recovers the law of excluded middle for sentences, and (iv) grants <figure><img></figure> the full descriptive power of <figure><img></figure>. We also provide an equivalent <em>Herbrand-Skolem semantics</em> and a <em>game-theoretic semantics</em> for the prenex fragment of <figure><img></figure>, the latter being defined in terms of a determined infinite-duration game that precisely captures the other two semantics on finite structures.</p></div>\",\"PeriodicalId\":50762,\"journal\":{\"name\":\"Annals of Pure and Applied Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-07-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Pure and Applied Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0168007223000726\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pure and Applied Logic","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168007223000726","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"LOGIC","Score":null,"Total":0}
Hintikka and Sandu originally proposed Independence Friendly Logic () as a first-order logic of imperfect information to describe game-theoretic phenomena underlying the semantics of natural language. The logic allows for expressing independence constraints among quantified variables, in a similar vein to Henkin quantifiers, and has a nice game-theoretic semantics in terms of imperfect information games. However, the semantics exhibits some limitations, at least from a purely logical perspective. It treats the players asymmetrically, considering only one of the two players as having imperfect information when evaluating truth, resp., falsity, of a sentence. In addition, truth and falsity of sentences coincide with the existence of a uniform winning strategy for one of the two players in the semantic imperfect information game. As a consequence, does admit undetermined sentences, which are neither true nor false, thus failing the law of excluded middle. These idiosyncrasies limit its expressive power to the existential fragment of Second Order Logic (). In this paper, we investigate an extension of , called Alternating Dependence/Independence Friendly Logic (), tailored to overcome these limitations. To this end, we introduce a novel compositional semantics, generalising the one based on trumps proposed by Hodges for . The new semantics (i) allows for meaningfully restricting both players at the same time, (ii) enjoys the property of game-theoretic determinacy, (iii) recovers the law of excluded middle for sentences, and (iv) grants the full descriptive power of . We also provide an equivalent Herbrand-Skolem semantics and a game-theoretic semantics for the prenex fragment of , the latter being defined in terms of a determined infinite-duration game that precisely captures the other two semantics on finite structures.
期刊介绍:
The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.