http://oldmfl.ricerca.di.unimi.it/api.php?action=feedcontributions&user=Jmfont&feedformat=atomMathfuzzlog - User contributions [en]2021-05-18T16:19:11ZUser contributionsMediaWiki 1.27.1http://oldmfl.ricerca.di.unimi.it/index.php?title=Abstract_Algebraic_Logic_-_An_Introductory_Chapter&diff=3160Abstract Algebraic Logic - An Introductory Chapter2016-08-01T14:34:56Z<p>Jmfont: </p>
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<div>Forthcoming</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Abstract_Algebraic_Logic_-_An_Introductory_Chapter&diff=3159Abstract Algebraic Logic - An Introductory Chapter2016-08-01T14:33:26Z<p>Jmfont: Blanked the page</p>
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<div></div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Abstract_Algebraic_Logic_-_An_Introductory_Textbook&diff=3158Abstract Algebraic Logic - An Introductory Textbook2016-08-01T14:32:41Z<p>Jmfont: </p>
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<div>{{Book|author=Josep Maria Font|title=Abstract Algebraic Logic - An Introductory Chapter|series=Studies in Logic|volume=60|publisher=College Publications|city=London|year=2016}}<br />
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== Abstract ==<br />
Abstract algebraic logic is the more general and abstract side of algebraic logic, the branch of mathematics that studies the connections between logics and their algebra-based semantics. This emerging subfield of mathematical logic consolidated since the 1980s, and is considered as the algebraic logic of the twenty-first century; as such it is increasingly becoming an indispensable tool to approach the algebraic study of any (mainly sentential) logic in a systematic way. <br />
<br />
This book is an introductory textbook on abstract algebraic logic, and takes a bottom-up approach, treating first logics with a simpler algebraic study, such as Rasiowa's implicative logics, and then guides readers, by means of successive steps of generalization and abstraction, to meet more and more complicated algebra-based semantics. An entire chapter is devoted to Blok and Pigozzi's theory of algebraizable logics, proving the main theorems and incorporating later developments by other scholars. After a chapter with the basics of the classical theory of matrices, one chapter is devoted to an in-depth exposition of the semantics of generalized matrices. There are also two more avanced chapters providing introductions to the two hierachies that organize the logical landscape according to the criteria of abstract algebraic logic, the Leibniz hierarchy and the Frege hierarchy. All throughout the book, particular care is devoted to the presentation and classification of dozens of examples of particular logics, among which, needless to say, many-valued logics and fuzzy logics.<br />
<br />
The book is addressed to mathematicians and logicians with little or no previous exposure to algebraic logic. Some acquaintance with examples of non-classical logics is desirable in order to appreciate the extremely general theory. The book is written with students (or beginners in the field) in mind, and combines a textbook style in its main sections, including more than 400 carefully graded exercises, with a survey style in the exposition of some research directions. The book includes scattered historical notes, numerous bibliographic references, and a set of six comprehensive indices.<br />
<br />
You can view the Tables of Contents, Introduction and Bibliography of the book here: http://www.ub.edu/grlnc/members/jmfont/publs.html.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Abstract_Algebraic_Logic_-_An_Introductory_Textbook&diff=3157Abstract Algebraic Logic - An Introductory Textbook2016-08-01T14:22:52Z<p>Jmfont: </p>
<hr />
<div>{{Book|author=Josep Maria Font|title=Abstract Algebraic Logic - An Introductory Textbook|series=Studies in Logic|volume=60|publisher=College Publications|city=London|year=2016}}<br />
<br />
<br />
<br />
[[Category:Books]]<br />
<br />
Abstract algebraic logic is the more general and abstract side of algebraic logic, the branch of mathematics that studies the connections between logics and their algebra-based semantics. This emerging subfield of mathematical logic is increasingly becoming an indispensable tool to approach the algebraic study of any (mainly sentential) logic in a systematic way.<br />
<br />
This book takes a bottom-up approach and guides readers, by means of successive steps of generalization and abstraction, to meet more and more complicated algebra-based semantics. An entire chapter is devoted to Blok and Pigozzi's theory of algebraizable logics, proving the main theorems and incorporating later developments by other scholars. After a chapter on the basics of the classical theory of matrices, one chapter is devoted to an in-depth exposition of the semantics of generalized matrices. Two more avanced chapters provide introductions to the two hierachies that organize the logical landscape according to the criteria of abstract algebraic logic, the Leibniz hierarchy and the Frege hierarchy. Dozens of examples of particular logics are presented and classified, among which, needless to say, many-valued logics. The properties of particular logics are coveniently summarized in an Appendix.<br />
<br />
The book is addressed to mathematicians and logicians with little or no previous exposure to algebraic logic. Some acquaintance with non-classical logics is desirable. The book is written with students (or beginners in the field) in mind, and combines a textbook style in its main sections, including more than 400 carefully graded exercises, with a survey style in the discussion of newer research directions. The book includes some historical notes, numerous bibliographic references, and a set of six comprehensive indices, one of them being an index of logics.<br />
<br />
'''Table of Contents'''<br />
<br />
A letter to the reader<br />
<br />
Introduction and Reading Guide<br />
<br />
'''1''' Mathematical and logical preliminaries<br />
<br />
'''2''' The first steps in the algebraic study of a logic<br />
<br />
'''3''' The semantics of algebras<br />
<br />
'''4''' The semantics of matrices<br />
<br />
'''5''' The semantics of generalized matrices<br />
<br />
'''6''' Introduction to the Leibniz hierarchy<br />
<br />
'''7''' Introduction to the Frege hierarchy<br />
<br />
'''Appendix''' Summary of properties of particular logics<br />
<br />
Bibliography<br />
<br />
Indices</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Abstract_Algebraic_Logic_-_An_Introductory_Textbook&diff=3156Abstract Algebraic Logic - An Introductory Textbook2016-08-01T14:10:37Z<p>Jmfont: Created page with "{{Book|author=Josep Maria Font|title=Abstract Algebraic Logic - An Introductory Textbook|series=Studies in Logic|volume=60|publisher=College Publications|city=|year=2016}} ..."</p>
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<div>{{Book|author=Josep Maria Font|title=Abstract Algebraic Logic - An Introductory Textbook|series=Studies in Logic|volume=60|publisher=College Publications|city=|year=2016}}<br />
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[[Category:Books]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Logics_preserving_degrees_of_truth_from_varieties_of_residuated_lattices&diff=3155Logics preserving degrees of truth from varieties of residuated lattices2016-08-01T11:11:36Z<p>Jmfont: </p>
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<div>{{Paper|<br />
author=Félix Bou|author2=Francesc Esteva|author3=Josep Maria Font|author4=Àngel J. Gil|author5=Lluís Godo|author6=Antoni Torrens|author7=Ventura Verdú|<br />
title=Logics preserving degrees of truth from varieties of residuated lattices|<br />
journal=Journal of Logic and Computation|<br />
volume=19|<br />
number=6|<br />
pages=1031-1069|<br />
year=2009}}<br />
<br />
== Abstract ==<br />
Let K be a variety of (commutative, integral) residuated lattices. The substructural logic usually associated with K is an algebraizable logic that has K as its equivalent algebraic semantics, and is a logic that preserves truth, i.e. 1 is the only truth value preserved by the inferences of the logic. In this article, we introduce another logic associated with K, namely the logic that preserves degrees of truth, in the sense that it preserves lower bounds of truth values in inferences. <br />
<br />
We study this second logic mainly from the point of view of abstract algebraic logic. We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth. We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus and by a set of conditions on its closure operator. Concerning the relation between the two logics, we prove that the truth-preserving logic is the extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective.<br />
<br />
There is a '''Corrigendum''' for this paper in volume 22 (2012), pages 661–665, of the same journal. In it, a wrong argument in the proof of the last implication of Theorem 4.4 is corrected. The result itself remains true. The right proof incorporates the basic ideas in the originally alleged proof, but in a more restricted construction.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Belnap%27s_four-valued_logic_and_De_Morgan_lattices&diff=3154Belnap's four-valued logic and De Morgan lattices2016-08-01T10:57:29Z<p>Jmfont: Created page with "{{Paper| author=Josep Maria Font| title=Belnap's four-valued logic and De Morgan lattices| journal=Logic Journal of the IGPL| volume=5| number=| pages=413-440| year=1997}} ==..."</p>
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<div>{{Paper|<br />
author=Josep Maria Font|<br />
title=Belnap's four-valued logic and De Morgan lattices|<br />
journal=Logic Journal of the IGPL|<br />
volume=5|<br />
number=|<br />
pages=413-440|<br />
year=1997}}<br />
<br />
== Abstract ==<br />
This paper contains some contributions to the study of Belnap's four-valued logic from an algebraic point of view. We introduce a finite Hilbert-style axiomatization of this logic, along with its well-known semantical presentation, and a Gentzen calculus that slightly differs from the usual one in that it is closer to Anderson and Belnap's formalization of their ''logic of first-degree entailments''. <br />
<br />
We prove several Completeness Theorems and reduce every formula to an equivalent normal form. The Hilbert-style presentation allows us to characterize the Leibniz congruence of the matrix models of the logic, and to find that the class of algebraic reducts of its reduced matrices is strictly smaller than the variety of De Morgan lattices. This means that the links between the logic and this class of algebras cannot be fully explained in terms of matrices, as in more classical logics. It is through the use of generalized matrices (here called abstract logics) as models that we are able to confirm that De Morgan lattices are indeed the algebraic counterpart of Belnap's logic, in the sense of Font and Jansana's theory of full generalized models for sentential logics. <br />
<br />
Among other characterizations, we prove that its full generalized models are the abstract logics that are finitary, do not have theorems, and satisfy the metalogical properties of Conjunction, Disjunction, Double Negation, and Weak Contraposition. As a consequence, we find that the Gentzen calculus presented at the beginning is strongly adequate for Belnap's logic and is algebraizable in the sense of Rebagliato and Verd\'{u}, having the variety of De Morgan lattices as its equivalent algebraic semantics.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=An_abstract_algebraic_logic_view_of_some_mutiple-valued_logics&diff=3153An abstract algebraic logic view of some mutiple-valued logics2016-08-01T10:35:25Z<p>Jmfont: </p>
<hr />
<div>{{Book chapter|<br />
author=Josep Maria Font|<br />
chapter=An abstract algebraic logic view of some mutiple-valued logics|<br />
title=Beyond Two: Theory and Applications of Multiple-Valued Logic|<br />
series=|<br />
volume=|<br />
pages=25-58|<br />
publisher=Physica-Verlag|<br />
editor=Melvin Fitting|<br />
editor2=Eva Orlowska|<br />
city=Heidelberg-Berlin-New York|<br />
year=2003}}<br />
<br />
== Abstract ==<br />
Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by Blok, Pigozzi and Czelakowski, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. <br />
<br />
I will show how the idea (already explored by Wójckici and Nowak) of defining logics using a scheme of "preservation of degrees of truth" (as opposed to the more usual one of "preservation of truth") characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by Jansana and myself) can give some interesting results. After the general theory is explained, I apply it to an infinite family of logics defined in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. <br />
<br />
Moreover, in the finite case the logics so obtained are protoalgebraic, which implies they have a "strong version" defined from their Leibniz filters; again, the general theory helps in showing that it is the logic defined from the same subalgebra by the truth-preserving scheme, that is, the corresponding finite-valued logic in the most usual sense. However, for infinite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can finally show that this logic too has a strong version, and that it coincides with the ordinary infinite-valued logic of Lukasiewicz.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=An_abstract_algebraic_logic_view_of_some_mutiple-valued_logics&diff=3152An abstract algebraic logic view of some mutiple-valued logics2016-08-01T10:34:12Z<p>Jmfont: </p>
<hr />
<div>{{Book chapter|<br />
author=Josep Maria Font|<br />
chapter=An abstract algebraic logic view of some mutiple-valued logics|<br />
title=Beyond Two: Theory and Applications of Multiple-Valued Logic|<br />
series=|<br />
volume=|<br />
pages=25-58|<br />
publisher=Physica-Verlag|<br />
editor=Melvin Fitting|<br />
editor2=Eva Orlowska|<br />
city=Heidelberg-Berlin-New York|<br />
year=2003}}<br />
<br />
== Abstract ==<br />
Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by Blok, Pigozzi and Czelakowski, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. <br />
<P><br />
I will show how the idea (already explored by Wójckici and Nowak) of defining logics using a scheme of "preservation of degrees of truth" (as opposed to the more usual one of "preservation of truth") characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by Jansana and myself) can give some interesting results. After the general theory is explained, I apply it to an infinite family of logics defined in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. <br />
<P><br />
Moreover, in the finite case the logics so obtained are protoalgebraic, which implies they have a "strong version" defined from their Leibniz filters; again, the general theory helps in showing that it is the logic defined from the same subalgebra by the truth-preserving scheme, that is, the corresponding finite-valued logic in the most usual sense. However, for infinite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can finally show that this logic too has a strong version, and that it coincides with the ordinary infinite-valued logic of Lukasiewicz.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Francesc_d%27A._Sales_Vall%C3%A8s&diff=3151Francesc d'A. Sales Vallès2016-04-29T17:09:20Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Francesc d'A.<br />
|surname=Sales Vallès<br />
}}<br />
Francesc d'A. Sales Vallès (1914-2005) was a professor of Probability and Statistics at the Faculty of Mathematics of the University of Barcelona. During the 1970s and 1980s he promoted, besides his original field of research, the study of ordered algebraic structures (lattices, Boolean algebras, etc.) and the study and teaching of mathematical logic in the mathematics curriculum. Several dissertations on related topics were done under his supervision, and this eventually led to the establishment of several research groups in fuzzy set theory, many-valued logic, and algebraic logic. These groups have continued to expand up to this date, mainly at the University of Barcelona, at the Polytechnic University of Catalonia, and at the Research Institute on Artificial Intelligence (IIIA-CSIC) based in Bellaterrra (Barcelona, Spain) and currently produce a lot of research in these and neighbouring areas.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Category:Students_of_Francesc_d%27A._Sales_Vall%C3%A8s&diff=3150Category:Students of Francesc d'A. Sales Vallès2016-04-29T16:57:19Z<p>Jmfont: </p>
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<div>The following members of MathFuzzLog were students of Francesc d'A. Sales Vallès (University of Barcelona)</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Francesc_d%27A._Sales_Vall%C3%A8s&diff=3149Francesc d'A. Sales Vallès2016-04-29T16:56:40Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Francesc d'A.<br />
|surname=Sales Vallès<br />
}}<br />
Francesc d'A. Sales Vallès was a professor of Probability and Statistics at the Faculty of Mathematics of the University of Barcelona. From the late 1960s to the 1980s he promoted the study of ordered algebraic structures (lattices, Boolean algebras, etc.) and the study and teaching of mathematical logic in the mathematics curriculum of the university. Many dissertations on related topics were done under his supervision, and this eventually led to the establishment of several research groups in fuzzy set theory, many-valued logic, and algebraic logic. These groups have continued to expand up to this date and currently produce a lot of research in these and neighbouring areas, mainly at the University of Barcelona, at the Polytechnic University of Catalonia, and at the Research Institute on Artificial Intelligence (IIIA-CSIC) based in Bellaterrra (Barcelona, Spain).</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Siegfried_Gottwald&diff=3148Siegfried Gottwald2016-04-29T16:55:39Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Siegfried<br />
|surname=Gottwald<br />
|birth=<br />
|advisor=<br />
|page=http://www.uni-leipzig.de/~logik/gottwald/<br />
|affiliation=<br />
}}<br />
Siegfried Gottwald died on 20 September 2015.<br />
<br />
The (almost) complete list of his publications is on [http://www.uni-leipzig.de/~logik/gottwald his web page].<br />
<br />
[[Category:Members|Gottwald, Siegfried]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Francesc_d%27A._Sales_Vall%C3%A8s&diff=3147Francesc d'A. Sales Vallès2016-04-29T16:50:50Z<p>Jmfont: Created page with "{{Researcher |name=Francesc d'A. |surname=Sales Vallès }} Francesc d'A. Sales Vallès was a professor of Probability and Statistics at the Faculty of Mathematics of the Unive..."</p>
<hr />
<div>{{Researcher<br />
|name=Francesc d'A.<br />
|surname=Sales Vallès<br />
}}<br />
Francesc d'A. Sales Vallès was a professor of Probability and Statistics at the Faculty of Mathematics of the University of Barcelona. From the late 1960s to the 1980s he promoted the study of ordered algebraic structures (lattices, Boolean algebras, etc.) and the study and teaching of mathematical logic in the mathematics curriculum of the university. Many dissertations on related topics were done under his supervision, and this eventually led to the establishment of several research groups in fuzzy set theory, many-valued logic, and algebraic logic, which continue to flourish up to this date and to produce a lot of research in these and neighbouring areas, mainly at the University of Barcelona, at the Polytechnic University of Catalonia, and at the Research Institute on Artificial Intelligence (IIIA-CSIC) based in Bellaterrra (Barcelona, Spain).</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Francesc_Esteva&diff=3146Francesc Esteva2016-04-29T16:32:44Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Francesc<br />
|surname=Esteva<br />
|advisor=Francesc d'A. Sales Vallès<br />
|page=http://www.iiia.csic.es/~esteva/<br />
|affiliation=IIIA - CSIC<br />
}}<br />
<br />
<br />
<br />
[[Category:Members|Esteva, Francesc]]<br />
[[Category:LoMoReVIans|Esteva, Francesc]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Antoni_Torrens&diff=3145Antoni Torrens2016-04-29T16:32:09Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Antoni<br />
|surname=Torrens<br />
|birth=<br />
|advisor=Francesc d'A. Sales Vallès<br />
|page=<br />
|affiliation=Department of Mathematics and Computer Engineering, University of Barcelona<br />
}}<br />
<br />
<br />
[[Category:Members|Torrens, Antoni]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Category:Students_of_Francesc_d%27A._Sales_Vall%C3%A8s&diff=3144Category:Students of Francesc d'A. Sales Vallès2016-04-29T16:31:37Z<p>Jmfont: Created page with "The following members of MathFuzzLog were students of Francesc D'A. Sales Vallès (University of Barcelona)"</p>
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<div>The following members of MathFuzzLog were students of Francesc D'A. Sales Vallès (University of Barcelona)</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Ventura_Verd%C3%BA&diff=3143Ventura Verdú2016-04-29T16:30:34Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Ventura<br />
|surname=Verdú<br />
|birth=<br />
|advisor=Francesc d'A. Sales Vallès<br />
|page=http://www.ub.edu/plie/personal_PLiE/verdu_HTML(2)/<br />
|affiliation=Department of Mathematics and Computer Engineering, University of Barcelona<br />
}}<br />
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<br />
<br />
[[Category:Members|Verdú, Ventura]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Ventura_Verd%C3%BA&diff=3142Ventura Verdú2016-04-29T16:29:42Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Ventura<br />
|surname=Verdú<br />
|birth=<br />
|advisor=Francesc Sales<br />
|page=http://www.ub.edu/plie/personal_PLiE/verdu_HTML(2)/<br />
|affiliation=Department of Mathematics and Computer Engineering, University of Barcelona<br />
}}<br />
<br />
<br />
<br />
[[Category:Members|Verdú, Ventura]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Antoni_Torrens&diff=3141Antoni Torrens2016-04-29T16:29:17Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Antoni<br />
|surname=Torrens<br />
|birth=<br />
|advisor=Francesc Sales<br />
|page=<br />
|affiliation=Department of Mathematics and Computer Engineering, University of Barcelona<br />
}}<br />
<br />
<br />
[[Category:Members|Torrens, Antoni]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Category:Employees_of_Department_of_Mathematics_and_Computer_Engineering,_University_of_Barcelona&diff=3140Category:Employees of Department of Mathematics and Computer Engineering, University of Barcelona2016-04-29T16:28:32Z<p>Jmfont: Created page with "Researchers affiliated with the Department of Mathematics and Computer Engineering, University of Barcelona"</p>
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<div>Researchers affiliated with the [[Department of Mathematics and Computer Engineering, University of Barcelona]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Joan_Gispert&diff=3139Joan Gispert2016-04-29T16:26:43Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Joan<br />
|surname=Gispert<br />
|birth=<br />
|advisor=Antoni Torrens<br />
|page=<br />
|affiliation=Department of Mathematics and Computer Engineering, University of Barcelona<br />
}}<br />
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[[Category:Members|Gispert, Joan]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Department_of_Mathematics_and_Computer_Engineering,_University_of_Barcelona&diff=3138Department of Mathematics and Computer Engineering, University of Barcelona2016-04-28T17:08:13Z<p>Jmfont: Created page with "*Employees of the department (categorized at this website) * [http://ma..."</p>
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<div>*[[:Category:Employees of Department of Mathematics and Computer Engineering, University of Barcelona|Employees of the department]] (categorized at this website)<br />
<br />
* [http://mat.ub.edu/queoferim/en/index/dep.htm Website of the department]<br />
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[[Category:Research centers]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Josep_Maria_Font&diff=3137Josep Maria Font2016-04-28T17:03:52Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Josep Maria<br />
|surname=Font<br />
|birth=<br />
|advisor=Francesc d'A. Sales Vallès<br />
|page=http://www.ub.edu/grlnc/members/jmfont/<br />
|affiliation=Department of Mathematics and Computer Engineering, University of Barcelona<br />
}}<br />
I am a professor at the Faculty of Mathematics of the University of Barcelona. <br />
<br />
You will find more details about my career, and a full publication list, in my own web page linked on the right.<br />
== RESEARCH INTERESTS == <br />
<br />
Abstract algebraic logic.<br />
Algebraic logic.<br />
Non-classical logics.<br />
<br />
== OTHER SCIENTIFIC INTERESTS ==<br />
<br />
Applications of lattice theory and universal algebra to logic.<br />
History of logic.<br />
Foundations of mathematics.<br />
History of mathematics.<br />
Programming in (La)TeX.<br />
Editing and publishing mathematics.<br />
<br />
<br />
<br />
<br />
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[[Category:Members|Font, Josep Maria]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=On_substructural_logics_preserving_degrees_of_truth&diff=3136On substructural logics preserving degrees of truth2016-04-27T09:52:48Z<p>Jmfont: Created page with "{{Paper| author=Josep Maria Font| title=On substructural logics preserving degrees of truth| journal=Bulletin of the Section of Logic| volume=36| number=| pages=117–130| yea..."</p>
<hr />
<div>{{Paper|<br />
author=Josep Maria Font|<br />
title=On substructural logics preserving degrees of truth|<br />
journal=Bulletin of the Section of Logic|<br />
volume=36|<br />
number=|<br />
pages=117–130|<br />
year=2007}}<br />
<br />
== Abstract ==<br />
The purpose of this paper is to discuss how some ideas coming from the many-valued logic world can be introduced in a sensible way into the world of substructural logic; namely, the ideas around what does it mean for a logic to say that it preserves degrees of truth. These two subject areas are by their origin rather far apart; I would like to exemplify how the recent evolution of research in the field of substructural logics, with the application of some central techniques from abstract algebraic logic, has facilitated the investigation of such borderline issues.<br />
<br />
Degrees of truth are of course ubiquitous in the literature on many-valued and fuzzy logic, as one of the interpretations of non-classical truth-values. However, in general logics admitting semantics with more than two degrees of truth are cast to preserve just one of them, absolute truth. Less discussed is the idea of a logic preserving degrees of truth. It was considered, in algebraic terms, by Nóvak and by Wójcicki. Both assume an ordering relation between degrees of truth, so that the idea appears as related only to ordered algebras. Here I would like to give it a broader spectrum of application.<br />
<br />
The idea of a logic preserving degrees of truth is presented as opposed to that of a truth-preserving logic ; I will try to demonstrate that this is only apparent. The idea of a truth-preserving logic is related to the classical conception of logical consequence, according to which "a conclusion follows logically from some premises if and only if, whenever the premises are true, the conclusion is also true".<br />
<br />
I think that the difference between the ideas of preserving truth and preserving degrees of truth does not lie in the acceptation or rejection of this conception of consequence, but in applying it to a different conception of truth, which may consequently change the interpretation of the "whenever" in the sentence ending the previous paragraph.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Consequence_and_degrees_of_truth_in_many-valued_logic&diff=3135Consequence and degrees of truth in many-valued logic2016-04-27T08:28:30Z<p>Jmfont: Created page with "{{Book chapter| author=Josep Maria Font| chapter=Consequence and degrees of truth in many-valued logic| title=Petr Hájek on Mathematical Fuzzy Logic| series= Outstanding Cont..."</p>
<hr />
<div>{{Book chapter|<br />
author=Josep Maria Font|<br />
chapter=Consequence and degrees of truth in many-valued logic|<br />
title=Petr Hájek on Mathematical Fuzzy Logic|<br />
series= Outstanding Contributions to Logic|<br />
volume=6|<br />
pages=117–142|<br />
publisher=Springer-Verlag|<br />
editor=Franco Montagna|<br />
city=Heidelberg-Berlin-New York|<br />
year=2015}}<br />
<br />
== Abstract ==<br />
I argue that the definition of a logic by preservation of all degrees of truth is a better rendering of Bolzano’s idea of consequence as truth-preserving when “truth comes in degrees”, as is often said in many-valued contexts, than the usual scheme that preserves only one truth value. I review some results recently obtained in the investigation of this proposal by applying techniques of abstract algebraic logic in the framework of Łukasiewicz logics and in the broader framework of sub- structural logics, that is, logics defined by varieties of (commutative and integral) residuated lattices. I also review some scattered, early results, which have appeared since the 1970’s, and make some proposals for further research.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Abstract_Algebraic_Logic_-_An_Introductory_Chapter&diff=3134Abstract Algebraic Logic - An Introductory Chapter2016-04-27T08:21:42Z<p>Jmfont: </p>
<hr />
<div>{{Book|author=Josep Maria Font|title=Abstract Algebraic Logic - An Introductory Chapter|series=Studies in Logic|volume=60|publisher=College Publications|city=London|year=2016}}<br />
<br />
== Abstract ==<br />
Abstract algebraic logic is the more general and abstract side of algebraic logic, the branch of mathematics that studies the connections between logics and their algebra-based semantics. This emerging subfield of mathematical logic consolidated since the 1980s, and is considered as the algebraic logic of the twenty-first century; as such it is increasingly becoming an indispensable tool to approach the algebraic study of any (mainly sentential) logic in a systematic way. <br />
<br />
This book is an introductory textbook on abstract algebraic logic, and takes a bottom-up approach, treating first logics with a simpler algebraic study, such as Rasiowa's implicative logics, and then guides readers, by means of successive steps of generalization and abstraction, to meet more and more complicated algebra-based semantics. An entire chapter is devoted to Blok and Pigozzi's theory of algebraizable logics, proving the main theorems and incorporating later developments by other scholars. After a chapter with the basics of the classical theory of matrices, one chapter is devoted to an in-depth exposition of the semantics of generalized matrices. There are also two more avanced chapters providing introductions to the two hierachies that organize the logical landscape according to the criteria of abstract algebraic logic, the Leibniz hierarchy and the Frege hierarchy. All throughout the book, particular care is devoted to the presentation and classification of dozens of examples of particular logics. <br />
<br />
The book is addressed to mathematicians and logicians with little or no previous exposure to algebraic logic. Some acquaintance with examples of non-classical logics is desirable in order to appreciate the extremely general theory. The book is written with students (or beginners in the field) in mind, and combines a textbook style in its main sections, including more than 400 carefully graded exercises, with a survey style in the exposition of some research directions. The book includes scattered historical notes and numerous bibliographic references.<br />
<br />
You can view the Tables of Contents, Introduction and Bibliography of the book here: http://www.ub.edu/grlnc/members/jmfont/publs.html.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Abstract_Algebraic_Logic_-_An_Introductory_Chapter&diff=3133Abstract Algebraic Logic - An Introductory Chapter2016-04-27T08:18:29Z<p>Jmfont: Created page with "{{Book|author=Josep Maria Font|title=Abstract Algebraic Logic - An Introductory Chapter|series=Studies in Logic|volume=60|publisher=College Publications|city=London|year=2016}..."</p>
<hr />
<div>{{Book|author=Josep Maria Font|title=Abstract Algebraic Logic - An Introductory Chapter|series=Studies in Logic|volume=60|publisher=College Publications|city=London|year=2016}}<br />
<br />
== Abstract ==<br />
Abstract algebraic logic is the more general and abstract side of algebraic logic, the branch of mathematics that studies the connections between logics and their algebra-based semantics. This emerging subfield of mathematical logic consolidated since the 1980s, and is considered as the algebraic logic of the twenty-first century; as such it is increasingly becoming an indispensable tool to approach the algebraic study of any (mainly sentential) logic in a systematic way. <br />
<br />
This book is an introductory textbook on abstract algebraic logic, and takes a bottom-up approach, treating first logics with a simpler algebraic study, such as Rasiowa's implicative logics, and then guides readers, by means of successive steps of generalization and abstraction, to meet more and more complicated algebra-based semantics. An entire chapter is devoted to Blok and Pigozzi's theory of algebraizable logics, proving the main theorems and incorporating later developments by other scholars. After a chapter with the basics of the classical theory of matrices, one chapter is devoted to an in-depth exposition of the semantics of generalized matrices. There are also two more avanced chapters providing introductions to the two hierachies that organize the logical landscape according to the criteria of abstract algebraic logic, the Leibniz hierarchy and the Frege hierarchy. All throughout the book, particular care is devoted to the presentation and classification of dozens of examples of particular logics. <br />
<br />
The book is addressed to mathematicians and logicians with little or no previous exposure to algebraic logic. Some acquaintance with examples of non-classical logics is desirable in order to appreciate the extremely general theory. The book is written with students (or beginners in the field) in mind, and combines a textbook style in its main sections, including more than 400 carefully graded exercises, with a survey style in the exposition of some research directions. The book includes scattered historical notes and numerous bibliographic references.<br />
<br />
<br />
<br />
[[Category:Books]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=On_the_Hierachy_of_t-norm_Based_Residuated_Fuzzy_Logics&diff=3132On the Hierachy of t-norm Based Residuated Fuzzy Logics2016-04-27T08:14:11Z<p>Jmfont: </p>
<hr />
<div>{{Book chapter|<br />
author=Francesc Esteva|author2=Lluís Godo|author3=Àngel García-Cerdaña|<br />
chapter=On the Hierachy of t-norm Based Residuated Fuzzy Logics|<br />
title=Beyond Two: Theory and Applications of Multiple-Valued Logic|<br />
series=|<br />
volume=|<br />
pages=251-272|<br />
publisher=Physica-Verlag|<br />
editor=Melvin Fitting|<br />
editor2=Eva Orlowska|<br />
city=Heidelberg-Berlin-New York|<br />
year=2003}}<br />
<br />
== Abstract ==<br />
In this paper we overview recent results, both logical and algebraic, about [0,1]-valued logical systems having a t-norm and its residuum as truth functions for conjunction and implication. We describe their axiomatic systems and algebraic varieties and show they can be suitably placed in a hierarchy of logics depending on their characteristic axioms. We stress that the most general variety generated by residuated structures in [0,1], which are defined by left-continuous t-norms, is not the variety of residuated lattices but the variety of pre-linear residuated lattices, also known as MTL-algebras. Finally, we also relate t-norm based logics to substructural logics to substructural logics, in particular to Ono’s hierarchy of extensions of the Full Lambek Calculus.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=An_abstract_algebraic_logic_view_of_some_mutiple-valued_logics&diff=3131An abstract algebraic logic view of some mutiple-valued logics2016-04-27T08:13:07Z<p>Jmfont: </p>
<hr />
<div>{{Book chapter|<br />
author=Josep Maria Font|<br />
chapter=An abstract algebraic logic view of some mutiple-valued logics|<br />
title=Beyond Two: Theory and Applications of Multiple-Valued Logic|<br />
series=|<br />
volume=|<br />
pages=25-58|<br />
publisher=Physica-Verlag|<br />
editor=Melvin Fitting|<br />
editor2=Eva Orlowska|<br />
city=Heidelberg-Berlin-New York|<br />
year=2003}}<br />
<br />
== Abstract ==<br />
Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by Blok, Pigozzi and Czelakowski, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. I will show how the idea (already explored by Wójckici and Nowak) of defining logics using a scheme of "preservation of degrees of truth" (as opposed to the more usual one of "preservation of truth") characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by Jansana and myself) can give some interesting results. After the general theory is explained, I apply it to an infinite family of logics defined in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the finite case the logics so obtained are protoalgebraic, which implies they have a "strong version" defined from their Leibniz filters; again, the general theory helps in showing that it is the logic defined from the same subalgebra by the truth-preserving scheme, that is, the corresponding finite-valued logic in the most usual sense. However, for infinite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can finally show that this logic too has a strong version, and that it coincides with the ordinary infinite-valued logic of Lukasiewicz.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=An_abstract_algebraic_logic_view_of_some_mutiple-valued_logics&diff=3130An abstract algebraic logic view of some mutiple-valued logics2016-04-27T08:11:59Z<p>Jmfont: </p>
<hr />
<div>{{Book chapter|<br />
author=Josep Maria Font|<br />
chapter=An abstract algebraic logic view of some mutiple-valued logics|<br />
title=Beyond Two: Theory and Applications of Multiple-Valued Logic|<br />
series=|<br />
volume=|<br />
pages=25-58|<br />
publisher=Physica-Verlag|<br />
editor=Melvin Fitting|<br />
editor2=Eva Orlowska|<br />
city=Heidelberg-Berlin-New York|<br />
year=2003}}<br />
<br />
== Abstract ==<br />
Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by Blok, Pigozzi and Czelakowski, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. I will show how the idea (already explored by Wójckici and Nowak) of defining logics using a scheme of ``preservation of degrees of truth'' (as opposed to the more usual one of ``preservation of truth'') characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by Jansana and myself) can give some interesting results. After the general theory is explained, I apply it to an infinite family of logics defined in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the finite case the logics so obtained are protoalgebraic, which implies they have a ``strong version'' defined from their Leibniz filters; again, the general theory helps in showing that it is the logic defined from the same subalgebra by the truth-preserving scheme, that is, the corresponding finite-valued logic in the most usual sense. However, for infinite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can finally show that this logic too has a strong version, and that it coincides with the ordinary infinite-valued logic of Lukasiewicz.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=An_abstract_algebraic_logic_view_of_some_mutiple-valued_logics&diff=3129An abstract algebraic logic view of some mutiple-valued logics2016-04-27T08:11:35Z<p>Jmfont: Created page with "{{Paper| journal=Studia Logica| volume=70| number=2| pages=157-182| year=2002}} {{Book chapter| author=Josep Maria Font| chapter=An abstract algebraic logic view of some muti..."</p>
<hr />
<div>{{Paper|<br />
journal=Studia Logica|<br />
volume=70|<br />
number=2|<br />
pages=157-182|<br />
year=2002}}<br />
<br />
{{Book chapter|<br />
author=Josep Maria Font|<br />
chapter=An abstract algebraic logic view of some mutiple-valued logics|<br />
title=Beyond Two: Theory and Applications of Multiple-Valued Logic|<br />
series=|<br />
volume=|<br />
pages=25-58|<br />
publisher=Physica-Verlag|<br />
editor=Melvin Fitting|<br />
editor2=Eva Orlowska|<br />
city=Heidelberg-Berlin-New York|<br />
year=2003}}<br />
<br />
== Abstract ==<br />
Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by Blok, Pigozzi and Czelakowski, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. I will show how the idea (already explored by Wójckici and Nowak) of defining logics using a scheme of ``preservation of degrees of truth'' (as opposed to the more usual one of ``preservation of truth'') characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by Jansana and myself) can give some interesting results. After the general theory is explained, I apply it to an infinite family of logics defined in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the finite case the logics so obtained are protoalgebraic, which implies they have a ``strong version'' defined from their Leibniz filters; again, the general theory helps in showing that it is the logic defined from the same subalgebra by the truth-preserving scheme, that is, the corresponding finite-valued logic in the most usual sense. However, for infinite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can finally show that this logic too has a strong version, and that it coincides with the ordinary infinite-valued logic of Lukasiewicz.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=On_Lukasiewicz%27s_Four-Valued_Modal_Logic&diff=3128On Lukasiewicz's Four-Valued Modal Logic2016-04-27T08:07:13Z<p>Jmfont: </p>
<hr />
<div>{{Paper|<br />
author=Josep Maria Font|author2=Petr Hájek|<br />
title=On Lukasiewicz's Four-Valued Modal Logic|<br />
journal=Studia Logica|<br />
volume=70|<br />
number=2|<br />
pages=157-182|<br />
year=2002}}<br />
<br />
== Abstract ==<br />
Lukasiewicz's four-valued modal logic is surveyed and analyzed, together with Lukasiewicz's motivations to develop it. A faithful interpretation of it into classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed under the light of the presented results, Lukasiewicz's own texts, and related literature.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=On_the_infinite-valued_Lukasiewicz_logic_that_preserves_degrees_of_truth&diff=3127On the infinite-valued Lukasiewicz logic that preserves degrees of truth2016-04-27T08:06:08Z<p>Jmfont: </p>
<hr />
<div>{{Paper|<br />
author=Josep Maria Font|author2=Àngel J. Gil|author3=Antoni Torrens|author4=Ventura Verdú|<br />
title=On the infinite-valued Lukasiewicz logic that preserves degrees of truth|<br />
journal=Archive for Mathematical Logic|<br />
volume=45|<br />
number=7|<br />
pages=839-868|<br />
year=2006}}<br />
<br />
== Abstract ==<br />
Lukasiewicz's infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Lukasiewicz's algebra by using a "truth- preserving" scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of "preservation of degrees of truth". We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a "graded" deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Taking_degrees_of_truth_seriously&diff=3126Taking degrees of truth seriously2016-04-27T07:32:51Z<p>Jmfont: /* Abstract */</p>
<hr />
<div>{{Paper|<br />
author=Josep Maria Font|<br />
title=Taking degrees of truth seriously|<br />
journal=Studia Logica|<br />
volume=91|<br />
number= |<br />
pages=383-406|<br />
year=2009}}<br />
<br />
== Abstract ==<br />
This is a contribution to the discussion on the role of truth degrees in many-valued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko's Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the particular cases of Lukasiewicz's many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Logics_preserving_degrees_of_truth_from_varieties_of_residuated_lattices&diff=3125Logics preserving degrees of truth from varieties of residuated lattices2016-04-27T07:32:13Z<p>Jmfont: </p>
<hr />
<div>{{Paper|<br />
author=Félix Bou|author2=Francesc Esteva|author3=Josep Maria Font|author4=Àngel J. Gil|author5=Lluís Godo|author6=Antoni Torrens|author7=Ventura Verdú|<br />
title=Logics preserving degrees of truth from varieties of residuated lattices|<br />
journal=Journal of Logic and Computation|<br />
volume=19|<br />
number=6|<br />
pages=1031-1069|<br />
year=2009}}<br />
<br />
== Abstract ==<br />
Let K be a variety of (commutative, integral) residuated lattices. The substructural logic usually associated with K is an algebraizable logic that has K as its equivalent algebraic semantics, and is a logic that preserves truth, i.e. 1 is the only truth value preserved by the inferences of the logic. In this article, we introduce another logic associated with K, namely the logic that preserves degrees of truth, in the sense that it preserves lower bounds of truth values in inferences. We study this second logic mainly from the point of view of abstract algebraic logic. We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth. We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus and by a set of conditions on its closure operator. Concerning the relation between the two logics, we prove that the truth-preserving logic is the extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Taking_degrees_of_truth_seriously&diff=3124Taking degrees of truth seriously2016-04-27T07:30:42Z<p>Jmfont: </p>
<hr />
<div>{{Paper|<br />
author=Josep Maria Font|<br />
title=Taking degrees of truth seriously|<br />
journal=Studia Logica|<br />
volume=91|<br />
number= |<br />
pages=383-406|<br />
year=2009}}<br />
<br />
== Abstract ==<br />
<br />
This is a contribution to the discussion on the role of truth degrees in many-valued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko's Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the particular cases of Lukasiewicz's many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Taking_degrees_of_truth_seriously&diff=3123Taking degrees of truth seriously2016-04-27T07:29:12Z<p>Jmfont: Created page with "{{Paper| author=Josep Maria Font| title=Takaing degrees of truth seriously| journal=Studia Logica| volume=91| pages=383-406| year=2009}} == Abstract == This is a contributio..."</p>
<hr />
<div>{{Paper|<br />
author=Josep Maria Font|<br />
title=Takaing degrees of truth seriously|<br />
journal=Studia Logica|<br />
volume=91|<br />
pages=383-406|<br />
year=2009}}<br />
<br />
== Abstract ==<br />
<br />
This is a contribution to the discussion on the role of truth degrees in many-valued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko's Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the particular cases of Lukasiewicz's many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties.</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=Josep_Maria_Font&diff=3122Josep Maria Font2016-04-27T07:25:58Z<p>Jmfont: </p>
<hr />
<div>{{Researcher<br />
|name=Josep Maria<br />
|surname=Font<br />
|birth=<br />
|advisor=Francesc d'A. Sales Vallès<br />
|page=http://www.ub.edu/grlnc/members/jmfont/<br />
|affiliation=Department of Mathematics and Computer Engineering, University of Barcelona<br />
}}<br />
<br />
<br />
[[Category:Members|Font, Josep Maria]]</div>Jmfonthttp://oldmfl.ricerca.di.unimi.it/index.php?title=User:Jmfont&diff=3121User:Jmfont2016-04-27T07:24:16Z<p>Jmfont: </p>
<hr />
<div>I am a mathematical logician, specialized in algebraic logic and in the algebraic study of non-classical logics, among which, of course, many-valued logics.</div>Jmfont