{"title":"On the algebraization of Henkin-type second-order logic","authors":"Miklós Ferenczi","doi":"10.1002/malq.202100057","DOIUrl":"10.1002/malq.202100057","url":null,"abstract":"<p>There is an extensive literature related to the algebraization of first-order logic. But the algebraization of full second-order logic, or Henkin-type second-order logic, has hardly been researched. The question arises: what kind of set algebra is the algebraic version of a Henkin-type model of second-order logic? The question is investigated within the framework of the theory of cylindric algebras. The answer is: a kind of cylindric-relativized diagonal restricted set algebra. And the class of the subdirect products of these set algebras is the algebraization of Henkin-type second-order logic. It is proved that the algebraization of a complete calculus of the Henkin-type second-order logic is a class of a kind of diagonal restricted cylindric algebras. Furthermore, the connection with the non-standard enlargements of standard complete second-order structures is investigated.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 2","pages":"149-158"},"PeriodicalIF":0.3,"publicationDate":"2022-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202100057","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74341249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Forcing axioms for λ-complete \u0000 \u0000 \u0000 μ\u0000 +\u0000 \u0000 $mu ^+$\u0000 -c.c.","authors":"Saharon Shelah","doi":"10.1002/malq.201900020","DOIUrl":"10.1002/malq.201900020","url":null,"abstract":"<p>We consider forcing axioms for suitable families of μ-complete <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>μ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <annotation>$mu ^+$</annotation>\u0000 </semantics></math>-c.c. forcing notions. We show that some form of the condition “<math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>p</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>p</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$p_1,p_2$</annotation>\u0000 </semantics></math> have a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mo>≤</mo>\u0000 <mi>Q</mi>\u0000 </msub>\u0000 <mi>-</mi>\u0000 <mi>lub</mi>\u0000 </mrow>\u0000 <annotation>$le _{{mathbb {Q}}}text{-}{rm lub}$</annotation>\u0000 </semantics></math> in <math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>${mathbb {Q}}$</annotation>\u0000 </semantics></math>” is necessary. We also show some versions are really stronger than others.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"6-26"},"PeriodicalIF":0.3,"publicationDate":"2022-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73389675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cut-conditions on sets of multiple-alternative inferences","authors":"Harold T. Hodes","doi":"10.1002/malq.202000032","DOIUrl":"10.1002/malq.202000032","url":null,"abstract":"<p>I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set-theoretic assumptions, to what I will call the Cut-for-Formulas to Cut-for-Sets Theorem: for a set <i>F</i> and a binary relation ⊢ on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mo>(</mo>\u0000 <mi>F</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {P}(F)$</annotation>\u0000 </semantics></math>, if ⊢ is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order-theoretic variant of the Tukey-Teichmüller Lemma. I then discuss relationships between various cut-conditions in the absence of finitariness or of monotonicity.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"95-106"},"PeriodicalIF":0.3,"publicationDate":"2022-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"119112159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cut‐conditions on sets of multiple‐alternative inferences","authors":"Harold T. Hodes","doi":"10.1002/malq.202000032","DOIUrl":"https://doi.org/10.1002/malq.202000032","url":null,"abstract":"I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set‐theoretic assumptions, to what I will call the Cut‐for‐Formulas to Cut‐for‐Sets Theorem: for a set F and a binary relation ⊢ on P(F)$mathcal {P}(F)$ , if ⊢ is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order‐theoretic variant of the Tukey‐Teichmüller Lemma. I then discuss relationships between various cut‐conditions in the absence of finitariness or of monotonicity.","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"31 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86420569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Strong Ambiguity","authors":"Panagiotis Rouvelas","doi":"10.1002/malq.202100067","DOIUrl":"10.1002/malq.202100067","url":null,"abstract":"<p>We examine the conditions under which a model of Tangled Type Theory satisfies the same sentences as a model of <math>\u0000 <semantics>\u0000 <mi>NF</mi>\u0000 <annotation>$mathsf {NF}$</annotation>\u0000 </semantics></math> (assuming we ignore type indices).</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"110-117"},"PeriodicalIF":0.3,"publicationDate":"2022-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75998096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A class of higher inductive types in Zermelo-Fraenkel set theory","authors":"Andrew W. Swan","doi":"10.1002/malq.202100040","DOIUrl":"10.1002/malq.202100040","url":null,"abstract":"<p>We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo-Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals. This class includes the example of unordered trees of any arity.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"118-127"},"PeriodicalIF":0.3,"publicationDate":"2022-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202100040","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77756649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sequential topologies and Dedekind finite sets","authors":"Jindřich Zapletal","doi":"10.1002/malq.202100013","DOIUrl":"10.1002/malq.202100013","url":null,"abstract":"<p>It is consistent with <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> set theory that the Euclidean topology on <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math> is not sequential, yet every infinite set of reals contains a countably infinite subset. This answers a question of Gutierres.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"107-109"},"PeriodicalIF":0.3,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81838794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Interpreting the weak monadic second order theory of the ordered rationals","authors":"John K. Truss","doi":"10.1002/malq.202100047","DOIUrl":"10.1002/malq.202100047","url":null,"abstract":"<p>We show that the weak monadic second order theory of the structure <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Q</mi>\u0000 <mo>,</mo>\u0000 <mo><</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$({mathbb {Q}}, <)$</annotation>\u0000 </semantics></math> is first order interpretable in its automorphism group.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"74-78"},"PeriodicalIF":0.3,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81524766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the effective universality of mereological theories","authors":"Nikolay Bazhenov, Hsing-Chien Tsai","doi":"10.1002/malq.202100016","DOIUrl":"10.1002/malq.202100016","url":null,"abstract":"<p>Mereological theories are based on the binary relation “being a part of”. The systematic investigations of mereology were initiated by Leśniewski. More recent authors (including Simons, Casati and Varzi, Hovda) formulated a series of first-order mereological axioms. These axioms give rise to a plenitude of theories, which are of great philosophical interest. The paper considers first-order mereological theories from the point of view of computable (or effective) algebra. Following the approach of Hirschfeldt, Khoussainov, Shore, and Slinko, we isolate two important computability-theoretic properties P (namely, degree spectra of structures, and effective dimensions), and consider the following problem: for a given mereological theory <i>T</i>, is it true that its models can realize every possible variant of the property P? If the answer is positive, then we say that the theory <i>T</i> is <math>\u0000 <semantics>\u0000 <mi>DSED</mi>\u0000 <annotation>$mathit {DSED}$</annotation>\u0000 </semantics></math>-universal. We obtain the following results about known mereological theories. Any theory <i>T</i> which is weaker than Extensional Closure Mereology (CEM) is <math>\u0000 <semantics>\u0000 <mi>DSED</mi>\u0000 <annotation>$mathit {DSED}$</annotation>\u0000 </semantics></math>-universal. A similar fact is true for the theory GM2. On the other hand, any theory stronger that CEM + (C) + (G) is not <math>\u0000 <semantics>\u0000 <mi>DSED</mi>\u0000 <annotation>$mathit {DSED}$</annotation>\u0000 </semantics></math>-universal. In particular, General Extensional Mereology is not <math>\u0000 <semantics>\u0000 <mi>DSED</mi>\u0000 <annotation>$mathit {DSED}$</annotation>\u0000 </semantics></math>-universal.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"48-66"},"PeriodicalIF":0.3,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83326702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sheaves of structures, Heyting-valued structures, and a generalization of Łoś's theorem","authors":"Hisashi Aratake","doi":"10.1002/malq.202000088","DOIUrl":"https://doi.org/10.1002/malq.202000088","url":null,"abstract":"<p>Sheaves of structures are useful to give constructions in universal algebra and model theory. We can describe their logical behavior in terms of Heyting-valued structures. In this paper, we first provide a systematic treatment of sheaves of structures and Heyting-valued structures from the viewpoint of categorical logic. We then prove a form of Łoś's theorem for Heyting-valued structures. We also give a characterization of Heyting-valued structures for which Łoś's theorem holds with respect to any maximal filter.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"67 4","pages":"445-468"},"PeriodicalIF":0.3,"publicationDate":"2021-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72362541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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