Inspect every piece of pseudoscience and you will find a security blanket, a thumb to suck, a skirt to hold. What does the scientist have to offer in exchange? Uncertainty! Insecurity! — Isaac Asimov I think people get it upside down when they say the unambiguous is the reality and the ambiguous is merely uncertainty about what is really unambiguous. Let’s turn it around the other way: the ambiguous is the reality and the unambiguous is merely a special case of it, where we finally manage to pin down some very special aspect. — David Bohm But ignorance of the different causes involved in the production of events, as well as their complexity, taken together with the imperfection of analysis, prevents our reaching the same certainty about the vast majority of phenomena. — Pierre-Simon Laplace Contents Contents i Preface iii 1 Introduction 1 2 Interval-valued structures5 2.1 Partially ordered sets......................................5 2.2 Triangularizations........................................7 2.3 Lattices..............................................9 2.4 Residuated lattices....................................... 20 2.5 Filters of residuated lattices.................................. 25 2.6 Interval-valued residuated lattices.............................. 36 3 Triangle algebras 39 3.1 Definition and elementary properties............................ 39 3.2 Connections with other algebraic structures........................ 44 3.3 The connection between triangle algebras and IVRLs.................. 47 3.4 The relevance of the diagonal and the value u u .................... 49 3.5 Connections between properties on triangle algebras∗ and on their diagonal.... 55 3.6 Filters of triangle algebras................................... 60 3.7 Decomposition theorem for pseudo-prelinear triangle algebras............ 68 4 Interval-valued fuzzy logics 77 4.1 Formal fuzzy logics....................................... 77 4.1.1 Monoidal logic..................................... 79 4.1.2 Monoidal t-norm based logic............................ 81 4.1.3 Basic logic........................................ 82 4.1.4 Łukasiewicz logic.................................... 83 4.1.5 Classical logic...................................... 84 4.1.6 Other fuzzy logics................................... 85 4.2 Definition and elementary properties............................ 86 4.3 Soundness and completeness................................. 88 4.3.1 Soundness........................................ 88 4.3.2 General and strong general completeness..................... 88 4.3.3 Pseudo-chain and strong pseudo-chain completeness.............. 93 i ii CONTENTS 4.3.4 Standard and strong standard completeness................... 93 4.4 Local deduction theorem................................... 101 4.5 Interpretation.......................................... 103 5 Conclusion 105 6 Samenvatting 109 Bibliography 113 Index 119 Preface Dear reader, Maybe you are interested in algebraic structures. Maybe you are interested in logic. Maybe you know me and you are curious about what I have been doing the last four and half years. Or maybe you are a student who has been ordered to summarize a PhD-dissertation, and you chose the one with the fewest number of pages. In the last case, allow me to warn you that this PhD-dissertation might not only be the one with the fewest number of pages, but also the one with the greatest density of formulas and proofs. Indeed, according to my promotor Etienne Kerre, this dissertation is the most compact he has ever supervised. This might not be so special, were it not that more than 25 students already have obtained their doctoral degree under his guidance. Unfortunately for students to come, he is about to retire soon. A very well deserved pension, if I may say so, even though in my eyes he can go on for at least another 10 students. Maybe – however, never say never – I am his last doctoral student, and as people say “the last mile is the longest" (they also tend to say “the last straw breaks the camel’s back", but let’s hope Etienne’s back just bended a little and did not break). However, he has been a great help for me, not only in giving guidance to my scientific research, but also in assisting me with all kinds of administrative work. I would really like to thank him, for all these efforts and for the patience he has had with me. Furthermore, I wish him all the best with his retirement! Not yet retired are my other two supervisors, Chris Cornelis and Glad Deschrijver. Neverthe- less, I wish them all the best as well. I am very grateful I could always count on them for checking proofs, helping out with LATEX, finding the right formulations and terminology, recommending in- teresting papers, suggesting approaches to tackle the problems I encountered, and so on. If they had not done their job so well, I might even have listened to them and leave Ghent for a stay abroad in another institution. However, I did meet researchers from other institutions at conferences and workshops. Some of them were of great help to me: we had fruitful discussions, they sent me papers, suggested methods or computer software I could use and/or helped me out when there was something I did not understand. In particular, I would like to thank Félix Bou, Petr Cintula, Didier Dubois, Francesc Esteva, Nick Galatos, Lluis Godo, Petr Hájek, Ulrich Höhle, Rostislav Horˇcík, Afrodita Iorgulescu, Sándor Jenei, Peter Jipsen, George Metcalfe, Carles Noguera and Hiroakira Ono. My colleagues in Ghent deserve an acknowledgement as well. Our conversations, the lunch iii iv PREFACE breaks, the poker games, the conferences we attended together, the other activities we did, it has always been pleasant spending time with them. Yes, even the expensive paintball afternoon. In particular, I want to thank Yun Shi, with whom I cooperated on some papers about fuzzy impli- cators. I wish her good luck finding a new job, so she can stay in Belgium. My parents should not be forgotten either. Without them I would not even be here, let alone write a PhD-dissertation. I cannot thank them enough for their everlasting support. Finally, I want to express my gratitude to the BOF (Bijzonder Onderzoeksfonds) and FWO (Research Foundation - Flanders) for supplying me with the financial support I needed for my research activities and an expensive paintball afternoon. Last but not least, I would like to thank you, reader of this preface, for reading also the re- mainder of this dissertation. Enjoy! Bart Van Gasse Labour Day/May Day/International Workers Day, 2010 CHAPTER 1 Introduction Classical logic is a two-valued logic: propositions in this logic are either true or false. In the first case, the truth value 1 is attributed to the proposition, while in the second case the truth value is 0. Given the truth values of two propositions p and q, it is possible to derive the truth values of the negation ‘not p’ (and ‘not q’), the conjunction ‘p and q’, the disjunction ‘p or q’ and the implication ‘p implies q’. These formulas are denoted as p, p&q, p q and p q. The truth values are calculated using the operations1 , , and :: _ ! : ∗ t ) Table 1.1: Truth tables of the operations in classical logic. x y x x y x y x y : ∗ t ) 0 0 1 0 0 1 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 For more complicated formulas the truth values can be computed in the same way. For exam- ple, if p is true and q is false, then the truth value of (p q) ((p q) q) is calculated as follows: the truth value of p q is 1 0 = 0, so the truth_ value! of !(p !q) q is 0 0 = 1. The truth value of p q is 1 0!= 1. So we) conclude that the truth value of!(p q)! ((p )q) q) is 1 1 = 1. Interestingly,_ t the truth value of this formula is always 1, even_ if other! truth! values! are attributed) to p and q. Such formulas are called tautologies. If a formula ' is a tautology, this is denoted as = '. More generally, for a set of formulas Γ, Γ = ' means “no matter what truth values are attributedj to the propositions, if the truth values ofj the formulas in Γ are 1, then the truth value of ' is 1". The two values 0 and 1, together with the defined operations, form a Boolean algebra [47]. Therefore we say that this Boolean algebra is the semantics of classical logic. Saying that (p q) ((p q) q) is a tautology in classical logic, is the same as saying that (x y) _ ((x! y) !y) =!1 is an identity in this Boolean algebra (meaning “whatever value of thet Boolean) algebra) ) we give to x and y, the calculation of (x y) ((x y) y) yields 1”). t ) ) ) 1Note that we use different symbols, to distinguish the logical connectives from the corresponding operations. Only for the negation we employ the same symbol. 1 2 CHAPTER 1. INTRODUCTION Now, identities in this Boolean algebra are also identities in every other Boolean algebra2 (we say that this Boolean algebra generates all Boolean algebras). Therefore classical logic does not only have the Boolean algebra with two elements as semantics, but also the whole variety of Boolean algebras: the general semantics of classical logic consists of all Boolean algebras. Interestingly it is also possible to describe classical logic without using semantics. This is done with axioms and deduction rules, which allow to prove a formula from a set of formulas. When a formula ' is provable from a theory Γ, this is denoted as Γ '. Two important results in classical logic are soundness (if Γ ', then Γ = ') and completeness` (if Γ = ', then Γ '). We write 3 this shortly as Γ ' iff Γ`= '. j j ` ` j Now, for the truth values of several propositions one might prefer more than the two options 0 (false) and 1 (true). Indeed, for vague propositions like ‘it is raining hard’, it would be useful if one could attribute an intermediate truth value, somewhere between ‘false’ and ‘true’.