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Axiomatizations of t-norm based logics – A survey

S. Gottwald
2000 Soft Computing - A Fusion of Foundations, Methodologies and Applications  
In many-valued logic, the Lukasiewicz systems, the Gödel sytems, and also the product logic all are t-norm based systems.  ...  The main emphasis in the present paper is on propositional logic.  ...  lattice with a continuous t-norm, cf  ... 
doi:10.1007/s005000000048 fatcat:byjasgtervbcdhkaur6lnkzdpq

On the Logic of Left-Continuous t-Norms and Right-Continuous t-Conorms [chapter]

Lluís Godo, Martín Sócola-Ramos, Francesc Esteva
2020 Communications in Computer and Information Science  
Our aim is to axiomatise the logic whose intended semantics are commutative and integral double residuated structures on [0, 1], that are induced by an arbitrary left-continuous t-norm, an arbitrary right-continuous  ...  by left-continuous t-norms.  ...  Esteva and Godo acknowledge partial support by the Spanish MINECO/FEDER project RASO (TIN2015-71799-C2-1-P).  ... 
doi:10.1007/978-3-030-50153-2_48 fatcat:m3h2a3ofrndalemlgzwa5bxlhi

Defining Homomorphisms and Other Generalized Morphisms of Fuzzy Relations in Monoidal Fuzzy Logics by Means of BK-Products [article]

Ladislav J. Kohout
2003 arXiv   pre-print
This was generalized later by Kohout to relations based on fuzzy Basic Logic systems (BL) of H\'ajek and also for relational systems based on left-continuous t-norms.  ...  The present paper is based on monoidal logics, hence it subsumes as special cases the theories of generalized morphisms (etc.) based on the following systems of logics: BL systems (which include the well  ...  Residuated Lattices and Monoidal Logics BL systems were based on the idea that many important theorems of Zadeh's logics on [0, 1] would still hold when min is replaced by any continuous t-norm & T and  ... 
arXiv:math/0310175v1 fatcat:li67sifsbfbpln2ocsuf7s5zia

Tomonoid extensions: the key for the construction of t-norms

Thomas Vetterlein
2013 Proceedings of the 8th conference of the European Society for Fuzzy Logic and Technology  
Triangular norms, or t-norms for short, play an important role for the semantics of fuzzy logics.  ...  By means of a detailed review of an example case, we show how the approach can be applied to the construction of left-continuous t-norms in a straightforward way.  ...  Let a t-norm monoid be given and let F be one of its filters. Then the quotient of the t-norm monoid induced by F can be determined in the following way.  ... 
doi:10.2991/eusflat.2013.7 dblp:conf/eusflat/Vetterlein13 fatcat:7ck2h4jsqbchrkvxmgp4l3vsha

Fuzzy sets and formal logics

Lluís Godo, Siegfried Gottwald
2015 Fuzzy sets and systems (Print)  
The paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics.  ...  Our focus is on the historical side of these developments.  ...  Godo acknowledges partial support by the Spanish projects EdeTRI (TIN2012-39348-C02-01) and 2014 SGR 118.  ... 
doi:10.1016/j.fss.2015.06.021 fatcat:jstlgpnzrze3pkikat2rcbeeh4

Page 3255 of Mathematical Reviews Vol. , Issue 96f [page]

1996 Mathematical Reviews  
The most frequent structures we encounter in the literature are given by Heyting algebras, MV-algebras and semigroup structures on the real unit interval (so-called t-norms).  ...  Starting from general prop- erties of integral, commutative, residuated /-monoids we give some mainstream examples including Lindenbaum algebras as well as t-norms.  ... 

Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics

Carles Noguera, Francesc Esteva, Lluís Godo
2010 Information Sciences  
This paper focuses on the issue of how generalizations of continuous and leftcontinuous t-norms over linearly ordered sets should be from a logical point of view.  ...  Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate  ...  Acknowledgments We want to thank Vicenzo Marra for his helpful comments on group theory and the anonymous referees for their many comments and remarks that helped us improving much the first  ... 
doi:10.1016/j.ins.2009.12.011 fatcat:gt2fmpoj7jgcxnhot7d4asmwee

Page 7291 of Mathematical Reviews Vol. , Issue 2003j [page]

2003 Mathematical Reviews  
03B52 Hajek, Petr [Hajek, Petr'] (CZ-AOS-IC; Prague) Observations on the monoidal t-norm logic.  ...  (English summary) Possibility theory and fuzzy logic. Fuzzy Sets and Systems 132 (2002), no. 1, 107-112. The paper presents interesting remarks on extensions of MTL (monoidal t-norm logic).  ... 

On strong standard completeness in some MTL $$_\Delta $$ Δ expansions

Amanda Vidal, Félix Bou, Francesc Esteva, Lluís Godo
2016 Soft Computing - A Fusion of Foundations, Methodologies and Applications  
These will be extensions of the monoidal t-norm based logic, MTL, the logic of prelinear, bounded, commutative and integral residuated lattices [3], expanded with rational truth-constants and with an arbitrary  ...  It is known that MTL is strongly complete with respect to the class of all standard algebras based of left-continuous t-norms [2] .  ...  This research has been partially supported by the Spanish project EdeTRI (TIN2012-39348-C02-01).  ... 
doi:10.1007/s00500-016-2338-0 fatcat:jdad4mujkzdvvbr5yifuqwj3ri

Decidability of Cancellative Extension of Monoidal T-norm Based Logic

R. HorCIk
2006 Logic Journal of the IGPL  
It is known that the monoidal t-norm based logic (MTL) and many of its schematic extensions are decidable.  ...  Typical examples of such extensions are the product logic and the cancellative extension of MTL (ΠMTL) because the only finite algebras belonging to the corresponding varieties are finite Boolean algebras  ...  Acknowledgement The work of the author was partly supported by the grant No.  ... 
doi:10.1093/jigpal/jzl012 fatcat:mcncfgy2zvhazozhrlwqwvgp54

A categorical semantics for fuzzy predicate logic

Lawrence Neff Stout
2010 Fuzzy sets and systems (Print)  
We see how the structure of the various kinds of algebras shows up in the categorical logic, giving a variant on natural deduction for these logics.  ...  To apply the approach to fuzzy logic we want a setting with a monoidal structure and we may want to work either internally as in 1181 (which aimed at an internal higher order fuzzy logic) or in a more  ...  fact that T is an identity for &: which is natural since commutativity of the t-norm is an axiom and it gives rise to commutativity of the monoidal structure on the categories of predicates. • A4: (¢&  ... 
doi:10.1016/j.fss.2009.09.007 fatcat:yke56p7pf5enrajcc6chdnzw4i

Computational complexities of axiomatic extensions of monoidal t-norm based logic

Moataz Saleh El-Zekey, Wafik Boulos Lotfallah, Nehad Nashaat Morsi
2008 Soft Computing - A Fusion of Foundations, Methodologies and Applications  
We study the computational complexity of some axiomatic extensions of the monoidal t-norm based logic (MTL), namely NM corresponding to the logic of the socalled nilpotent minimum t-norm (due to Fodor  ...  The set of 1-tautologies of SMTL is only shown to be co-NP-hard, and it remains open if this set is in co-NP. Also, some results on the relations between these sets are obtained.  ...  Acknowledgments We thank the three learned referees for their apt and expansive advice, which was instrumental in correcting and improving this article.  ... 
doi:10.1007/s00500-008-0382-0 fatcat:c33v2f2h35c4feqghqqekqdxla

The construction of left-continuous t-norms: a geometric approach in two dimensions

Thomas Vetterlein
2014 Fuzzy sets and systems (Print)  
Cayley tomonoids provide an easily manageable, yet versatile tool for the construction of t-norms.  ...  Each t-norm can be identified with its Cayley tomonoid, which consists of pairwise commuting order-preserving functions from the real unit interval to itself.  ...  We then observe that the correspondence of t-norm monoids with their Cayley tomonoids generalises to arbitrary quantic, negative, commutative tomonoids.  ... 
doi:10.1016/j.fss.2013.11.007 fatcat:472odqtx3bgyhhjd4bavzwj23q

Totally Ordered Monoids Based on Triangular Norms

Thomas Vetterlein
2015 Communications in Algebra  
A t-norm is a binary operation used in fuzzy logic for the interpretation of the conjunction. Given a tomonoid of the indicated type, we consider the chain of its quotients induced by filters.  ...  Finally, a number of examples from the area of t-norms are included for illustration.  ...  I am deeply indepted to the anonymous referees, whose thorough reviews led to a substantial improvement of this paper.  ... 
doi:10.1080/00927872.2014.891599 fatcat:hqickffhjfdsplltpehgaai6sy

Formalization of implication based fuzzy reasoning method

Daowu Pei
2012 International Journal of Approximate Reasoning  
The current paper delves further into the TI method, and a sound logical foundation is set for the TI method based on the monoidal t-norm based logical system MTL.  ...  The full implication triple I method (a method only based on implication, TI method for short) for fuzzy reasoning is proposed in 1999 to improve the popular CRI method (a hybrid method based on implication  ...  Acknowledgements The authors would like to thank the referees and the area editor for their valuable comments and recommendations.  ... 
doi:10.1016/j.ijar.2012.01.007 fatcat:yyslnw72azfg5k4yzy4fduu22u
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