+-'1^1h Closed Set Logic in Categories William James Department of Philosophy at The Universitv of Adelaicle August, 1996 Table of Contents Introduction 1 PART I: Preliminaries Chapter 1. Basic Category Theory 25 1. Categories and Morphisms 25 2. Yoneda 42 3. Adjoints . 44 Chapter 2. Basic Topos Theory 49 1. Toposes 49 2. Topos Logic 53 3. Image Factorisation 56 Chapter 3. The HA Dual 58 1. Languages, Logics and Dual Algebras 60 2. Paraconsistent Algebras 69 3. Intuitionism's Dual t.l 4. Individual Logics and Natural Duals 76 PART II: Categorial Semantics for Paraconsistent Logic Chapter 4. The Complement Classifier 80 1. The Classifier 83 2. Complement Classifler vs. Subobject Classifier 85 Chapter 5. The Quotient Object Classifier 88 1. Quotient Object Lattices 91 2. The Functor QUO 95 Chapter 6. A Functor Category r02 1. Component Algebras 106 2. Operator Arrows 110 PART III: Sheaf Concepts Chapter 7. Sheaves: a brief history of the structure 116 Chapter 8. Closed Set Sheaves I27 1. Presheaves on Categories 130 2. Pretopologies and Topologies for Categories 133 3. Subobject Classifiers in Sheaf Categories 740 4. Closed Set Sheaves r47 Chapter 9. Brouwerian Algebras in Closed Set Sheaves 150 1. Component Algebras of the Classifi.er Object 152 2. Component Algebras and Natural Transformations 155 Chapter 10. Grothendieck Toposes 159 1. Pretopologies and Sheaves revisited 160 2. Subobject Classifiers in Grothendieck Toposes 162 3. Brouwerian Algebras in the Classifier Object 777 Chapter 11. Covariant Logic Objects 176 1. A Paraconsistent Logic Object in a Covariant Functor Category 177 Chapter 12. Covariant Sheaves 185 1. Co-topologies 186 2. Categorial Co-topoiogies on Closed Set Topologies 190 3. Sheaves on Co-topologies 1.94 Chapter 13. Sheaf Spaces on Finite Closed Sets 206 1. Sheaves and Sheaf Spaces 27L 2. From Presheaves to Sheaf Spaces 275 3. From Sheaf Spaces to Sheaves 220 4. Equivalence of Categories 224 PART IV: Theories and Relevance Chapter 14. Inconsistent Theories in Categories 234 1. Many Sorted Languages 238 2. Geometric Logic, Sites and Language Aigebras 243 Chapter 15. The Omega Monoid 256 1. De Morgan Monoids 257 Chapter 16. Conclusions 26r Bibliographies 263 u1 Abstract In this work we investigate two related aspects of a clualisation program for the usual intuitionist logic in categories. The dualisation prograrr h.as a,s its encl the presentation of closed set, or paraconsistent, logic in placr: of the usual open set, or intuitionist, logic found in association with toposes. \Äie address ourselves pa,rticrr- larly to Brouwerian algebras in categories as the duals of the usual Heyting algebras. The first aspect of the program is that of external or ex-ca,tegorial dualisation of logic structures b¡' interpretation of order. This appears in the u'ork as an exarnirration of the notion of a complement classifier. We also use ex-categorial dualisation as a, tool to prompt the development of a categorial proof and rnodel theory adequa,te to the task of modelling theories generated by inconsistency tolerant logics. We rnake an initial attempt to develop dual logic structures by considering quotient object classifiers in place of subobject classifiers. Ex-categorial dualisation of structure was always meant to act as an indication of the existence of categorial entities that di- rectly satisfy dual descriptions, so the bulk of the work is concerned with the second aspect of the dualisation program: the discovery of logic objects within categories that exhibit paraconsistent algebras in their own right. Our investigation focuses on sheaves for their algebraic properties in relation to base space topologies. We define the notion of a sheaf over the closed sets of a topological space. We find essentially two things. First, logic objects in contravariant sheaf categories contain component Brouwerian algebras but are not generally themselves Brouwerian algebras within their categories. A corollary is that subobject lattices in Grothendieck toposes are Brouwerian algebras (but not naturally so). Second, paraconsistent logic objects do exist. We describe one such within a category of covariant sheaves. As a corolì.ary we find that the original ex-categorial dualisation idea represented by the notion of a complement classifier has an instantiation in categories. Our paraconsistent logic object proves to be the object of a genuine complement classifier. IV Staternent of Originality and Consent This work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or writ- ten by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being available for loan and photocopying. 1n 1 6 Introduction When a set of sentences is closed under some consequence relation and und.er uniform substitrrtion of sentences for atomic sentences we have a sentential logic. A paraconsistent logic is one which allows that sets of sentences contain a sentence and its negation and be closed with respect to the logic's consequence relation without containing every other sentence. The logic is said to tolerate inconsistency. It is rarely remarked that closed set topologies form algebras for exactly this sort of logic while it is well known that their duals, the open set topologies, form the algebras for the logics of Intuitionism. In as much as it is exactly the Intuitionism algebras that are known to occur in and around topos theor¡, it is perhaps surprisin¡ç that category theory, with its awareness of duality, should have so little to note on the topic of paraconsistency. It is at least true that inconsistency toleration is exactly the right sort of notion to use in the development of any type of machine that requires input so with the emphasis of category theory tending toward useful applications, particularly those computational, it is appropriate that we investigate the underdeveloped area of closed set logic within categories. The project of this thesis began with the study of closed set topologies as algebras for paraconsistent logics. These were to be developed as tire duals of the Intuitionist logics. The background assumption for the usual formalisation of Intuitionism is that any sentence is interpreted on some open set of a topological space. For a sentence ,S, then, the negated sentence -,9 is the largest open set for which 5 íì -S is empty. The dual position assumes that any sentence is interpreted on some closed set. We can then interpret a set .,9 in relation to a sentence ,S by allowing .S to be the smallest closed set for which S U.S contains everJ/ other closed set. The operators - and r are then formaily dual; among other theorems we will have that ,S lì -,S need not be empty. This is to say that sentences ^9 and -.9 are such that they cannot both be false but that (given a big enough set of designated values) they may both be true. The background assumption that any sentence is valued on a closed set allows us to avoid the suggestion that - is a subcontrary operator rather than negation operator. It follows that the logics that arise as the duals of intuitionist logics are genuinely paraconsistent. With the introduction of Mortensen and Lavers' complement classifiers and complement toposes the idea of dualisirrg logics was linked formally with category and topos theory. The structures that most obviously linhed Heyting algebras (and so Intuitionism) and topos theory were the sheaves definecl over open set topologies. And so arose the idea of investigating the effect on categorial logic of defining sheaves over topologies of closed sets. The overall aim was and remains one of dualising the logics built into the structures of toposes and categories. The importance of paraconsistent versions of categorial logic is in terms of categories as semantic objects for assignment functions that determine inconsistent theories: the logic of the category provides the deduction relation under which the modelled theory is closed; inconsistent theories require (and are generated by) " paraconsistent deduction relation. Now clearly, categories are not the only seman- tic objects that we might use in inconsistency model theory. Equally clearly there has been little or no work on this type of model theory done for categories. Fur- thermore, category theory is an important mathematical discipline; if we regard paraconsistent logic as significant, then investigation of its place in category the- ory is mandated. Some note of it has already been made. In the introduction to Lawvere and Schanuel's Categories in Continuum Pltysics,1986, Lawvere notes in connection with sheaves and categories that a property of complements in algebras of closed sets is that the intersection of a closed set and its compiement will not invariably be the least element of the algebra; in other words. using closed set lat- tices as logical algebras produces logics in which a formula and its negation have, 2 as a rule, truth values with non-zero intersections. Just above I claimed that paraconsistent logic allows for the existence of in- consistent theories. The idea is this: a theory is a set of sentences closed under a deduction relation; if the deduction relation is paraconsistent, then the presence of inconsistent sentences within the theory need not mean that the theory contain all other sentences and be rendered trivial. We may think of paraconsistent deduction as one that limits the (deductive) impact of contradiction. This is clifferent from being huppy to have one's theories loaded with inconsistencies. The paraconsistent logics are, in this light, a way of dealing with problems that arise fïom otherwise good mathematical and philosophical ideas.