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Easy Proofs of Löwenheim-Skolem Theorems by Means of Evaluation Games [article]

Jacques Duparc
2015 arXiv   pre-print
We propose a proof of the downward L\"owenheim-Skolem that relies on strategies deriving from evaluation games instead of the Skolem normal forms.  ...  the one of an evaluation game.  ...  Löwenheim-Skolem theorem.  ... 
arXiv:1507.03665v1 fatcat:sp3b3jedjfdk5i24sipoox75ui

Lectures on Jacques Herbrand as a Logician [article]

Claus-Peter Wirth, Joerg Siekmann, Christoph Benzmueller, Serge Autexier
2014 arXiv   pre-print
Besides Herbrand's Fundamental Theorem and its relation to the Loewenheim-Skolem-Theorem, we carefully investigate Herbrand's notion of intuitionism in connection with his notion of falsehood in an infinite  ...  We sketch Herbrand's two proofs of the consistency of arithmetic and his notion of a recursive function, and last but not least, present the correct original text of his unification algorithm with a new  ...  of this paper.  ... 
arXiv:0902.4682v5 fatcat:pgxxo7vjcbeyvhjej4svsv6lqe

On synthesizing Skolem functions for first order logic formulae [article]

S. Akshay, Supratik Chakraborty
2022 arXiv   pre-print
While every Skolemized formula in first-order logic makes use of Skolem constants and/or functions, not all such Skolem constants and/or functions admit effectively computable interpretations.  ...  In this paper, we investigate the computability of Skolem functions and their automated synthesis in the full generality of first order logic.  ...  Löwenheim-Skolem theorem.  ... 
arXiv:2102.07463v2 fatcat:aafbqeoj3zhy3cugbey4gvdqdq

The Development of Mathematical Logic from Russell to Tarski, 1900–1935 [chapter]

Paolo Mancosu, Richard Zach, Calixto Badesa
2009 The Development of Modern Logic  
This is the Löwenheim-Skolem theorem.  ...  Herbrand's Theorem can thus be seen as giving finitary meaning to the Löwenheim-Skolem theorem. Let us now give a brief outline of the theorem.  ... 
doi:10.1093/acprof:oso/9780195137316.003.0029 fatcat:ygiplet5oraf7fza6fsmf5j5ly

First-order logic with self-reference [article]

Reijo Jaakkola, Antti Kuusisto
2022 arXiv   pre-print
We also argue for the naturalness and benefits of the investigated approach to recursion and self-reference by, for example, relating the new logics to Lindstrom's Second Theorem.  ...  For example, the validity problems of the two-variable fragments of the logics are shown coNexpTime-complete, which is in stark contrast with the high undecidability of two-variable logic extended with  ...  In the case of SCL it turns out that we can adapt the standard proof that FO has the countable downwards Löwenheim-Skolem property. Theorem 6.2. Let ϕ be a sentence of SCL and suppose that A |= ϕ.  ... 
arXiv:2207.07397v1 fatcat:rxfau7u7sffcbm4mxtwlmlyche

A Finitary Analogue of the Downward Löwenheim-Skolem Property [article]

Abhisekh Sankaran
2017 arXiv   pre-print
We present a model-theoretic property of finite structures, that can be seen to be a finitary analogue of the well-studied downward Löwenheim-Skolem property from classical model theory.  ...  We conclude by presenting a strengthening of L-EBSP, that asserts "logical self-similarity at all scales" for a suitable notion of scale.  ...  of the downward Löwenheim-Skolem theorem that is "intermediate" between the versions of this theorem by Skolem and Mal'tsev.  ... 
arXiv:1705.04493v1 fatcat:w646656oprfgdedxsard2ti2im

Quantified Constraints and Containment Problems

Barnaby Martin, Hubie Chen, Florent Madelaine, Martin Grohe
2015 Logical Methods in Computer Science  
We further show the exponent's bound |A|^|B| to be close to tight by giving a sequence of structures A together with a fixed B, |B|=2, such that there is a surjective homomorphism from A^r to B only when  ...  In the final part of the paper we ponder a notion of Q-core that is some canonical representative among the class of templates that engender the same QCSP.  ...  Acknowledgements We are very grateful to Arnaud Durand for supplying the undecidability proof of the entailment problem for positive FO.  ... 
doi:10.2168/lmcs-11(3:9)2015 fatcat:mgjtfhpasrewzm2dwh3kbtsl24

Quantified Constraints and Containment Problems

Hubie Chen, Florent Madelaine, Barnaby Martin
2008 Logic in Computer Science  
We note that this condition is already necessary to guarantee containment of the Π 2 restriction of QCSP, that is Π 2 -CSP(A) ⊆ Π 2 -CSP(B).  ...  That is, given two sentences ϕ and ψ of first-order logic with no instances of negation or disjunction, we give an algorithm that determines whether ϕ → ψ is true in all structures (models).  ...  Acknowledgements We are very grateful to Arnaud Durand for supplying the undecidability proof of the entailment problem for positive FO.  ... 
doi:10.1109/lics.2008.15 dblp:conf/lics/ChenMM08 fatcat:shnqvf2vyngyvivpbsz7oifloy

Equilibrium semantics of languages of imperfect information

Merlijn Sevenster, Gabriel Sandu
2010 Annals of Pure and Applied Logic  
In this approach, coined equilibrium semantics, the value of an IF sentence on a particular structure is determined by the expected utility of the existential player in any of the game's equilibriums.  ...  We prove that each rational number can be realized by an IF sentence. We also give a lower and upper bound on the expressive power of IF logic under equilibrium semantics.  ...  Acknowledgments We are much indebted to the reviewers for many excellent technical and nontechnical suggestions and to Jakub Szymanik for pointing out the recent work by Galliani.  ... 
doi:10.1016/j.apal.2009.07.019 fatcat:izg7yevkyfadjcvdoypqzrhenq

On modal logics of model-theoretic relations [article]

Denis I. Saveliev, Ilya B. Shapirovsky
2019 arXiv   pre-print
We prove a downward LöwenheimSkolem theorem for first-order language expanded with the modal operator for the extension relation between models.  ...  Given a class C of models, a binary relation R between models, and a model-theoretic language L, we consider the modal logic and the modal algebra of the theory of C in L where the modal operator is interpreted  ...  The work on this paper was supported by the Russian Science Foundation under grant 16-11-10252 and carried out at Steklov Mathematical Institute of Russian Academy of Sciences.  ... 
arXiv:1804.09810v4 fatcat:4ta7oth5zbdsblofpzdihiw6wy

On the relative consistency strength of determinacy hypotheses

Alexander S. Kechris, Robert M. Solovay
1985 Transactions of the American Mathematical Society  
A variant of the usual proof of the Skolem-Löwenheim theorem shows that X < sup< £: There is a surjection tr: R -> £ >. Theorem 2.3.  ...  So by Skolem-Löwenheim Vz3>> >rz3# < aAL»[y] •= ZFn + AC + _,OD-DET).  ... 
doi:10.1090/s0002-9947-1985-0787961-2 fatcat:am6k3agb4nelvmalr3ezefgssu

On Semantic Generalizations of the Bernays-Schönfinkel-Ramsey Class with Finite or Co-finite Spectra [article]

Abhisekh Sankaran, Supratik Chakraborty
2010 arXiv   pre-print
Our study provides alternative proofs of interesting results like the Loś-Tarski Theorem and the semantic subsumption of the Löwenheim class with equality by BSR.  ...  Motivated by model-theoretic properties of the BSR class, we present a family of semantic classes of FO formulae with finite or co-finite spectra over a relational vocabulary Σ.  ...  Then if Φ has models of unbounded domain-size -which is iff it has an infinite model -then by the previous statement and the Upward and Downward Löwenheim-Skolem theorems, it has models of all domain-  ... 
arXiv:1002.4334v2 fatcat:erzc5b3h45cnhnoxwbctygb3ou

Logical queries over views

James Bailey, Guozhu Dong, Anthony WIDJAJA To
2010 ACM Transactions on Computational Logic  
known as the Löwenheim class.  ...  By expanding the view definitions, every first-order view query can clearly be rewritten to eliminate the views.  ...  [If it is infinite, by the Löwenheim-Skolem theorem, we may assume that it is countable.]  ... 
doi:10.1145/1656242.1656243 fatcat:vv2spm7q6fgmdegb7kdbwwn3ve

Expressiveness of the modal mu-calculus on monotone neighborhood structures [article]

Sebastian Enqvist, Fatemeh Seifan, Yde Venema
2015 arXiv   pre-print
We characterize the expressive power of the modal mu-calculus on monotone neighborhood structures, in the style of the Janin-Walukiewicz theorem for the standard modal mu-calculus.  ...  Our main result shows that the monotone modal mu-calculus corresponds exactly to the fragment of this second-order language that is invariant for neighborhood bisimulations.  ...  Let κ be the smallest Löwenheim-Skolem number for µNML g , so that κ has the property described in the previous lemma.  ... 
arXiv:1502.07889v1 fatcat:gafhmhfk4fef3hr73rvrzqgpv4

Some classical model theoretic aspects of bounded shrub-depth classes [article]

Abhisekh Sankaran
2020 arXiv   pre-print
We show that this class satisfies an extension of the classical Löwenheim-Skolem property into the finite and for MSO.  ...  The proof for the last of these is via an adaptation of the proof of the classical Lindström's theorem characterizing FO over arbitrary structures.  ...  I also thank Anand Pillay for pointing me to [12] for the proof of Lindström's theorem.  ... 
arXiv:2010.05799v1 fatcat:fpco5dvxarfi3pegqogcgxbjym
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