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Negative Translations for Affine and Lukasiewicz Logic [article]

Rob Arthan, Paulo Oliva
2019 arXiv   pre-print
We find that in affine logic the translation schemes due to Kolmogorov and G\"odel both satisfy Troelstra's criteria for a negative translation.  ...  On the other hand, the schemes of Glivenko and Gentzen both fail for affine logic, but for different reasons: one can extend affine logic to make Glivenko work and Gentzen fail and vice versa.  ...  We would also like to thank George Metcalfe and Isabel Ferreirim for helpful correspondence.  ... 
arXiv:1912.00012v1 fatcat:cwt5i3lvsndrbjxztk3nry3d3q

On Affine Logic and Łukasiewicz Logic [article]

Rob Arthan, Paulo Oliva
2014 arXiv   pre-print
Nonetheless, we show that all the usual negative translations are already correct for intuitionistic Łukasiewicz logic, where only the limited amount of contraction given by [CWC] is allowed.  ...  This is in contrast with affine logic for which we show, by appeal to results on semantics proved in a companion paper, that both the Gentzen and the Glivenko translations fail.  ...  classical variants of affine logic and Lukasiewicz logic.  ... 
arXiv:1404.0570v2 fatcat:hx7j2fbx4ncczpctfylg3md63q

On a Convex Logic Fragment for Learning and Reasoning [article]

Francesco Giannini, Michelangelo Diligenti, Marco Gori, Marco Maggini
2018 arXiv   pre-print
In this paper we introduce the convex fragment of Łukasiewicz Logic and discuss its possible applications in different learning schemes.  ...  Indeed, the provided theoretical results are highly general, because they can be exploited in any learning framework involving logical constraints.  ...  whose formulas are restricted to disjunctive clauses and translated by the Lukasiewicz t-norm and t-conorm.  ... 
arXiv:1809.06778v1 fatcat:ioyw6wnd55eohfragg26w4azie

Double Negation Semantics for Generalisations of Heyting Algebras

Rob Arthan, Paulo Oliva
2020 Studia Logica: An International Journal for Symbolic Logic  
This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel,  ...  We view these as variant semantics and present a semantic formulation of Troelstra's syntactic criteria for a satisfactory negative translation.  ...  In logical terms these correspond to the conjunctionimplication fragment of intuitionistic affine logic and what we call intuitionistic Lukasiewicz logic, respectively.  ... 
doi:10.1007/s11225-020-09909-y fatcat:rdtujdrwdbfofjojk4zx7azimm

Łukasiewicz Games

Enrico Marchioni, Michael Wooldridge
2015 ACM Transactions on Computational Logic  
By expressing goals as formulae of Łukasiewicz logics, we can express a much richer class of utility functions for players than is possible using classical Boolean logic: we can express every continuous  ...  In particular, we present a logical characterisation of the existence of Nash equilibria in finite and infinite Łukasiewicz games.  ...  the possibility of allowing a user to specify a game in Łukasiewicz logic, and then automatically translate this specification into disjunctive logic programming (or vice versa).  ... 
doi:10.1145/2783436 fatcat:tdgcdwy3mfauridrdfvt55wjfe

Unifying Functional Interpretations: Past and Future [article]

Paulo Oliva
2014 arXiv   pre-print
Our goal in the present paper is twofold: (1) to look back and single out the main lessons learnt so far, and (2) to look forward and list several open questions and possible directions for further research  ...  interpretations including G\"odel's dialectica interpretation, its Diller-Nahm variant, Kreisel modified realizability, Stein's family of functional interpretations, functional interpretations "with truth", and  ...  , we can first translate A into affine logic, and then apply the interpretation with ∀x ≺ a A :≡ ∀xA.  ... 
arXiv:1410.4364v1 fatcat:p2q5nxafnnarxptyj3wqlfxxh4

Carnap and the Members of the Lvov–Warsaw School. Carnap's Warsaw Lectures (1930) in the Polish context [chapter]

Anna Brożek
2021 Veröffentlichungen des Instituts Wiener Kreis  
This was an opportunity for him to meet such members the Lvov-Warsaw School as Jan Łukasiewicz, Stanisław Leśniewski, Tadeusz Kotarbiński, and others.  ...  AbstractIn March 1930, Alfred Tarski visited Vienna and delivered few lectures which presented the achievements of the logical branch of the Lvov-Warsaw School.  ...  (Carnap 1963, 13) Carnap and Łukasiewicz also had the same negative opinion of existing philosophy (especially the one dominating in Germany).  ... 
doi:10.1007/978-3-030-58251-7_10 fatcat:p3gz4fl56rfvfogmb5yil44csy

Conditions for Unnecessary Logical Constraints in Kernel Machines [chapter]

Francesco Giannini, Marco Maggini
2019 Lecture Notes in Computer Science  
In particular, we discuss the case of logical constraints expressed by Lukasiewicz logic, where both inferential and algebraic arguments can be considered.  ...  scheme of learning from constraints, where possibly several constraints are considered, some of them may turn out to be unnecessary with respect to the learning optimization, even if they are active for  ...  Logic connectives and their algebraic semantics for the Lukasiewicz logic.  ... 
doi:10.1007/978-3-030-30484-3_49 fatcat:u4qgswhwczfbvlxdjhbav7bqwu

Saving the Truth Schema from Paradox

Hartry Field
2002 Journal of Philosophical Logic  
Keeping these things requires a weakening of classical logic; I suggest a logic based on the strong Kleene truth tables, but with → as an additional connective, and where the effect of classical logic  ...  a truth theory; Section 3 investigates the logical laws that result from this; and Section 4 provides some philosophical commentary.  ...  epistemological lines (perhaps with affinities to [1] , but based on conditional degrees of belief appropriate to a Kleene-style logic or to FDE).  ... 
doi:10.1023/a:1015063620612 fatcat:hwltsocl7zhrjkmiyx3a53cm5e

Page 14 of Mathematical Reviews Vol. , Issue 82a [page]

1982 Mathematical Reviews  
{English translation: Algebra and Logic 17 (1978), no. 4, 247- 263 (1979).} Tucker, J. V. Computability and the algebra of fields: some affine constructions. 82a:03027 J.  ...  Then, interpre- ting the connective © as the product in the {~,-—>}-reduct of the &,-valued Lukasiewicz matrix, he proves a completeness theorem for this logic.  ... 

The complexity of admissible rules of Lukasiewicz logic

E. Jerabek
2012 Journal of Logic and Computation  
We investigate the computational complexity of admissibility of inference rules in infinite-valued Łukasiewicz propositional logic (Ł).  ...  Acknowledgements I would like to thank the referee for useful suggestions.  ...  Admissible rules of Lukasiewicz logic were investigated by Jeřábek [13, 14] .  ... 
doi:10.1093/logcom/exs007 fatcat:jwopddmyqfbljbk7n2sbpkukqi

Three characterizations of strict coherence on infinite-valued events

Tommaso Flaminio
2019 The Review of Symbolic Logic  
In particular, we will make use of geometrical, measure-theoretical and logical methods to provide three characterizations of strict coherence on formulas of infinite-valued Łukasewicz logic  ...  Hosni and F. Montagna. It is meant to contribute to the logical foundations of probability theory on many-valued events and, specifically, to a deeper understanding of the notion of strict coherence.  ...  For, if the bookmaker publishes a book with positive expectation (for him) a logically infallible gambler will choose negative stakes and inflict a sure loss on him, that is to say, a sure financial loss  ... 
doi:10.1017/s1755020319000546 fatcat:uybzzv2kdjf5rhg4zv67h3x6fe

Unification in Łukasiewicz Logic with a Finite Number of Variables [chapter]

Marco Abbadini, Federica Di Stefano, Luca Spada
2020 Communications in Computer and Information Science  
We prove that the unification type of Lukasiewicz logic with a finite number of variables is either infinitary or nullary.  ...  , and a homotopytheoretic argument.  ...  The standard references for Lukasiewicz logic are [7, 18] . The unification type of Lukasiewicz logic is known to be the worst possible: Theorem 1 ([17] ).  ... 
doi:10.1007/978-3-030-50153-2_46 fatcat:kogad36marhibkbn22otk6qqrq

Hoops, Coops and the Algebraic Semantics of Continuous Logic [article]

Rob Arthan, Paulo Oliva
2012 arXiv   pre-print
Hoops provide a natural algebraic semantics for a class of substructural logics that we think of as intuitionistic analogues of the widely studied Łukasiewicz logics.  ...  In this paper, we define the notion of continuous hoop, or coop for short, and show that coops provide a natural algebraic semantics for continuous logic.  ...  A similar approach for Lukasiewicz logic was developed in [7] where classical affine logic was taken as the starting point.  ... 
arXiv:1212.2887v1 fatcat:xnqqxhahb5c2ti6ymp2ilqdh7e

On Pocrims and Hoops [article]

Rob Arthan, Paulo Oliva
2014 arXiv   pre-print
Pocrims and suitable specialisations thereof are structures that provide the natural algebraic semantics for a minimal affine logic and its extensions.  ...  Hoops comprise a special class of pocrims that provide algebraic semantics for what we view as an intuitionistic analogue of the classical multi-valued Łukasiewicz logic.  ...  ; George Metcalfe for encouraging remarks and for pointers to the literature; and Isabel Ferreirim for helpful correspondence about the theory of hoops.  ... 
arXiv:1404.0816v2 fatcat:2ptuwx3mbjegtlal2usliedixy
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