{"title":"ANSELM’S ONTOLOGICAL ARGUMENT AND GRADES OF BEING","authors":"CHARLES McCARTY","doi":"10.1017/s1755020324000133","DOIUrl":"https://doi.org/10.1017/s1755020324000133","url":null,"abstract":"<p>Anselm described god as “something than which nothing greater can be thought” [1, p. 93], and Descartes viewed him as “a supreme being” [7, p. 122]. I first capture those characterizations formally in a simple language for monadic predicate logic. Next, I construct a model class inspired by Stoic and medieval doctrines of <span>grades of being</span> [8, 20]. Third, I prove the models sufficient for recovering, as internal mathematics, the famous ontological argument of Anselm, and show that argument to be, on this formalization, valid. Fourth, I extend the models to incorporate a modality fit for proving that any item than which necessarily no greater can be thought is also necessarily real. Lastly, with the present approach, I blunt the sharp edges of notable objections to ontological arguments by Gaunilo and by Grant. A trigger warning: every page of this writing flouts the old saw “Existence is not a predicate” and flagrantly.</p>","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
MARK J. SCHERVISH, TEDDY SEIDENFELD, JOSEPH B. KADANE, RUOBIN GONG, RAFAEL B. STERN
{"title":"WHEN NO PRICE IS RIGHT","authors":"MARK J. SCHERVISH, TEDDY SEIDENFELD, JOSEPH B. KADANE, RUOBIN GONG, RAFAEL B. STERN","doi":"10.1017/s1755020324000017","DOIUrl":"https://doi.org/10.1017/s1755020324000017","url":null,"abstract":"<p>In this paper, we show how to represent a non-Archimedean preference over a set of random quantities by a nonstandard utility function. Non-Archimedean preferences arise when some random quantities have no fair price. Two common situations give rise to non-Archimedean preferences: random quantities whose values must be greater than every real number, and strict preferences between random quantities that are deemed closer in value than every positive real number. We also show how to extend a non-Archimedean preference to a larger set of random quantities. The random quantities that we consider include real-valued random variables, horse lotteries, and acts in the theory of Savage. In addition, we weaken the state-independent utility assumptions made by the existing theories and give conditions under which the utility that represents preference is the expected value of a state-dependent utility with respect to a probability over states.</p>","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"182 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"ON THE STRUCTURE OF BOCHVAR ALGEBRAS","authors":"STEFANO BONZIO, MICHELE PRA BALDI","doi":"10.1017/s175502032400008x","DOIUrl":"https://doi.org/10.1017/s175502032400008x","url":null,"abstract":"<p>Bochvar algebras consist of the quasivariety <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240530141428462-0247:S175502032400008X:S175502032400008X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {BCA}$</span></span></img></span></span> playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar [4] in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem based on Płonka sums and investigate the lattice of subquasivarieties, showing that Bochvar (external) logic has only one proper extension (apart from classical logic), algebraized by the subquasivariety <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240530141428462-0247:S175502032400008X:S175502032400008X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {NBCA}$</span></span></img></span></span> of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240530141428462-0247:S175502032400008X:S175502032400008X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {BCA}$</span></span></img></span></span>. Furthermore, we address the problem of (passive) structural completeness ((P)SC) for each of them, showing that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240530141428462-0247:S175502032400008X:S175502032400008X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {NBCA}$</span></span></img></span></span> is SC, while <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240530141428462-0247:S175502032400008X:S175502032400008X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {BCA}$</span></span></img></span></span> is not even PSC. Finally, we prove that both <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240530141428462-0247:S175502032400008X:S175502032400008X_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {BCA}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240530141428462-0247:S175502032400008X:S175502032400008X_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {NBCA}$</span></span></img></span></span> enjoy the amalgamation property (AP).</p>","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"THE TEMPORAL CONTINUUM","authors":"MOHAMMAD ARDESHIR, RASOUL RAMEZANIAN","doi":"10.1017/s1755020324000078","DOIUrl":"https://doi.org/10.1017/s1755020324000078","url":null,"abstract":"<p>The continuum has been one of the most controversial topics in mathematics since the time of the Greeks. Some mathematicians, such as Euclid and Cantor, held the position that a line is composed of points, while others, like Aristotle, Weyl, and Brouwer, argued that a line is not composed of points but rather a matrix of a continued insertion of points. In spite of this disagreement on the structure of the continuum, they did distinguish the <span>temporal line</span> from the <span>spatial line</span>. In this paper, we argue that there is indeed a difference between the intuition of the spatial continuum and the intuition of the temporal continuum. The main primary aspect of the temporal continuum, in contrast with the spatial continuum, is the notion of <span>orientation</span>.</p><p>The continuum has usually been mathematically modeled by <span>Cauchy</span> sequences and the <span>Dedekind</span> cuts. While in the first model, each point can be approximated by rational numbers, in the second one, that is not possible constructively. We argue that points on the temporal continuum cannot be approximated by rationals as a temporal point is a <span>flow</span> that sinks to the past. In our model, the continuum is a collection of constructive <span>Dedekind</span> cuts, and we define two topologies for temporal continuum: 1. <span>oriented</span> topology and 2. the <span>ordinary</span> topology. We prove that every total function from the <span>oriented</span> topological space to the <span>ordinary</span> one is continuous.</p>","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"ARROW’S THEOREM, ULTRAFILTERS, AND REVERSE MATHEMATICS","authors":"BENEDICT EASTAUGH","doi":"10.1017/s1755020324000054","DOIUrl":"https://doi.org/10.1017/s1755020324000054","url":null,"abstract":"<p>This paper initiates the reverse mathematics of social choice theory, studying Arrow’s impossibility theorem and related results including Fishburn’s possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405151338335-0717:S1755020324000054:S1755020324000054_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathsf {RCA}}_0$</span></span></img></span></span>. We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405151338335-0717:S1755020324000054:S1755020324000054_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${mathsf {RCA}}_0$</span></span></img></span></span>. This approach yields a proof of Arrow’s theorem in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405151338335-0717:S1755020324000054:S1755020324000054_inline3.png\"><span data-mathjax-type=\"texmath\"><span>${mathsf {RCA}}_0$</span></span></img></span></span>, and thus in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405151338335-0717:S1755020324000054:S1755020324000054_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {PRA}$</span></span></img></span></span>, since Arrow’s theorem can be formalised as a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405151338335-0717:S1755020324000054:S1755020324000054_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$Pi ^0_1$</span></span></img></span></span> sentence. Finally we show that Fishburn’s possibility theorem for countable societies is equivalent to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405151338335-0717:S1755020324000054:S1755020324000054_inline6.png\"><span data-mathjax-type=\"texmath\"><span>${mathsf {ACA}}_0$</span></span></img></span></span> over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405151338335-0717:S1755020324000054:S1755020324000054_inline7.png\"><span data-mathjax-type=\"texmath\"><span>${mathsf {RCA}}_0$</span></span></img></span></span>.</p>","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"227 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"CONCEPTUAL DISTANCE AND ALGEBRAS OF CONCEPTS","authors":"MOHAMED KHALED, GERGELY SZÉKELY","doi":"10.1017/s1755020324000029","DOIUrl":"https://doi.org/10.1017/s1755020324000029","url":null,"abstract":"<p>We show that the conceptual distance between any two theories of first-order logic is the same as the generator distance between their Lindenbaum–Tarski algebras of concepts. As a consequence of this, we show that, for any two arbitrary mathematical structures, the generator distance between their meaning algebras (also known as cylindric set algebras) is the same as the conceptual distance between their first-order logic theories. As applications, we give a complete description for the distances between meaning algebras corresponding to structures having at most three elements and show that this small network represents all the possible conceptual distances between complete theories. As a corollary of this, we will see that there are only two non-trivial structures definable on three-element sets up to conceptual equivalence (i.e., up to elementary plus definitional equivalence).</p>","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"BEYOND LINGUISTIC INTERPRETATION IN THEORY COMPARISON","authors":"TOBY MEADOWS","doi":"10.1017/s1755020323000321","DOIUrl":"https://doi.org/10.1017/s1755020323000321","url":null,"abstract":"<p>This paper assembles a unifying framework encompassing a wide variety of mathematical instruments used to compare different theories. The main theme will be the idea that theory comparison techniques are most easily grasped and organized through the lens of category theory. The paper develops a table of different equivalence relations between theories and then answers many of the questions about how those equivalence relations are themselves related to each other. We show that Morita equivalence fits into this framework and provide answers to questions left open in Barrett and Halvorson [4]. We conclude by setting up a diagram of known relationships and leave open some questions for future work.</p>","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138825468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"PARACONSISTENT AND PARACOMPLETE ZERMELO–FRAENKEL SET THEORY","authors":"YURII KHOMSKII, HRAFN VALTÝR ODDSSON","doi":"10.1017/s1755020323000382","DOIUrl":"https://doi.org/10.1017/s1755020323000382","url":null,"abstract":"<p>We present a novel treatment of set theory in a four-valued <span>paraconsistent</span> and <span>paracomplete</span> logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell’s paradox and fulfil naive comprehension. Instead, we prioritise setting up a system with a clear ontology of non-classical sets, which can be used to reason informally about incomplete and inconsistent phenomena, and is sufficiently similar to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240222115540260-0904:S1755020323000382:S1755020323000382_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathrm {ZFC}}$</span></span></img></span></span> to enable the development of interesting mathematics.</p><p>We propose an axiomatic system <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240222115540260-0904:S1755020323000382:S1755020323000382_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${mathrm {BZFC}}$</span></span></img></span></span>, obtained by analysing the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240222115540260-0904:S1755020323000382:S1755020323000382_inline3.png\"><span data-mathjax-type=\"texmath\"><span>${mathrm {ZFC}}$</span></span></img></span></span>-axioms and translating them to a four-valued setting in a careful manner, avoiding many of the obstacles encountered by other attempted formalizations. We introduce the <span>anti-classicality axiom</span> postulating the existence of non-classical sets, and prove a surprising results stating that the existence of a single non-classical set is sufficient to produce any other type of non-classical set.</p><p>Our theory is naturally bi-interpretable with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240222115540260-0904:S1755020323000382:S1755020323000382_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${mathrm {ZFC}}$</span></span></img></span></span>, and provides a philosophically satisfying view in which non-classical sets can be seen as a natural extension of classical ones, in a similar way to the non-well-founded sets of Peter Aczel [1].</p><p>Finally, we provide an interesting application concerning Tarski semantics, showing that the classical definition of the satisfaction relation yields a logic precisely reflecting the non-classicality in the meta-theory.</p>","PeriodicalId":501566,"journal":{"name":"The Review of Symbolic Logic","volume":"219 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139919451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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