Logical Foundations of Categorization Theory 4th SYSMICS Workshop: - PowerPoint PPT Presentation

Logical Foundations of Categorization Theory 4th SYSMICS Workshop: Duality in algebra and logic Alessandra Palmigiano joint ongoing work with Willem Conradie, Peter Jipsen, Krishna Manoorkar, Sajad Nazari, Nachoem Wijnberg... 16 September 2018 1 1/17

What is categorization? From Wikipedia: Categorization is the process in which ideas and objects are recognized, differentiated, and understood. Ideally, a category illuminates a relationship between the subjects and objects of knowledge. Categorization is fundamental in language, prediction, inference, decision-making and in all kinds of environmental interaction. 2 2/17

Overview and General Motivation ◮ Truly interdisciplinary: philosophy, cognition, social/management science, linguistics, AI. ◮ rapid development, different approaches ; ◮ emerging unifying perspective: categories are dynamic in their essence; they shape and are shaped by processes of social interaction. ◮ Data-driven developments, both empirical and theoretical. ◮ However, what is lacking : ◮ a common ground for the various approaches; ◮ formal models addressing dynamics and connections with the processes of social interaction . ◮ Research program: logic as common ground; dynamics as starting point rather than outcome; systematic connection between dynamics and processes of social interaction. 3 3/17

Contrasting Views on Categorization Classical (Aristotle) ◮ membership in a category defined by satisfaction of features. ◮ categorization: deductive process of reasoning with necessary and sufficient conditions; ◮ categories have sharp boundaries; no unclear cases. ◮ categories are represented equally well by each of its members. Prototype (Rosch) ◮ some category-members more central than others (prototypes). ◮ categorization: inductive process of establishing similarity to prototype; ◮ categories have fuzzy boundaries; membership is graded. 4 4/17

Meanwhile, in logic... Mathematical theory of LE-logics (LE: lattice expansions) the integrated SYSMICS approach: ◮ algebraic and Kripke-style semantics; ◮ generalized Sahlqvist theory; ◮ semantic cut elimination, FMP; ◮ Goldblatt-Thomason theorem. Can we make intuitive sense of LE-logics? 5 5/17

Basic lattice logic & main ideas Language: L ∋ φ ::= p ∈ Prop | ⊤ | ⊥ | φ ∧ φ | φ ∨ φ Lattice Logic: Set of L -sequents φ ⊢ ψ ◮ containing: p ⊢ p ⊥ ⊢ p p ⊢ ⊤ p ⊢ p ∨ q q ⊢ p ∨ q p ∧ q ⊢ p p ∧ q ⊢ q ◮ closed under: φ ⊢ χ χ ⊢ ψ φ ⊢ ψ χ ⊢ φ χ ⊢ ψ φ ⊢ χ ψ ⊢ χ φ ⊢ ψ φ ( χ/ p ) ⊢ ψ ( χ/ p ) χ ⊢ φ ∧ ψ φ ∨ ψ ⊢ χ 6 6/17

Basic lattice logic & main ideas Language: L ∋ φ ::= p ∈ Prop | ⊤ | ⊥ | φ ∧ φ | φ ∨ φ Lattice Logic: Set of L -sequents φ ⊢ ψ ◮ containing: p ⊢ p ⊥ ⊢ p p ⊢ ⊤ p ⊢ p ∨ q q ⊢ p ∨ q p ∧ q ⊢ p p ∧ q ⊢ q ◮ closed under: φ ⊢ χ χ ⊢ ψ φ ⊢ ψ χ ⊢ φ χ ⊢ ψ φ ⊢ χ ψ ⊢ χ φ ⊢ ψ φ ( χ/ p ) ⊢ ψ ( χ/ p ) χ ⊢ φ ∧ ψ φ ∨ ψ ⊢ χ Challenge: Interpreting ∨ as ‘or’ and ∧ as ‘and’ does not work, since ‘and’ and ‘or’ distribute over each other, while ∧ and ∨ don’t. Proposal: Interpreting φ ∈ L as other entities than sentences? Examples: categories, concepts, theories, interrogative agendas. The interpretation of ∨ and ∧ in all these contexts is ok with failure of distributivity Approach: ◮ Understand LE-logics as the logics of these entities ; ◮ integrate LE-logics into more expressive logics capturing how these entities interact (e.g. with sentences, actions etc.). 6 6/17

Polarity-based semantics of LE-logics Formal contexts ( A , X , I ) are abstract representations of databases: X I A A : set of Objects X : set of Features I ⊆ A × X . Intuitively, aIx reads: object a has feature x 7 7/17

Polarity-based semantics of LE-logics Formal contexts ( A , X , I ) are abstract representations of databases: X I A A : set of Objects X : set of Features I ⊆ A × X . Intuitively, aIx reads: object a has feature x 7 7/17

Polarity-based semantics of LE-logics Formal contexts ( A , X , I ) are abstract representations of databases: X I A A : set of Objects X : set of Features I ⊆ A × X . Intuitively, aIx reads: object a has feature x 7 7/17

Polarity-based semantics of LE-logics Formal contexts ( A , X , I ) are abstract representations of databases: X I A A : set of Objects X : set of Features I ⊆ A × X . Intuitively, aIx reads: object a has feature x Formal concepts: “rectangles” maximally contained in I 7 7/17

Complex algebras ( abcd , ∅ ) y x z ( ab , x ) ( cd , z ) X � I ( bc , y ) A a c ( b , xy ) ( c , yz ) b d ( ∅ , xyz ) Language: L ∋ φ ::= p ∈ Prop | ⊤ | ⊥ | φ ∧ φ | φ ∨ φ Lattice Logic: Set of L -sequents φ ⊢ ψ ◮ containing: p ⊢ p ⊥ ⊢ p p ⊢ ⊤ p ⊢ p ∨ q q ⊢ p ∨ q p ∧ q ⊢ p p ∧ q ⊢ q ◮ closed under: φ ⊢ χ χ ⊢ ψ φ ⊢ ψ χ ⊢ φ χ ⊢ ψ φ ⊢ χ ψ ⊢ χ φ ⊢ ψ φ ( χ/ p ) ⊢ ψ ( χ/ p ) χ ⊢ φ ∧ ψ φ ∨ ψ ⊢ χ 8 8/17

Formal contexts as L -models ( abcd , ∅ ) p q y x z ( ab , x ) ( cd , z ) V ( q ) X V ( p ) � I ( bc , y ) A a c ( b , xy ) ( c , yz ) b d p pq q ( ∅ , xyz ) Let P = ( A , X , I ) and P + be the complex algebra of P . Models: M := ( P , V ) with V : Prop → P + V ( p ) := ([ [ p ] ] , ( [ p ] )) membership: M , a � p iff a ∈ [ [ p ] ] M description: M , x ≻ p iff x ∈ ( [ p ] ) M 9 9/17

Formal contexts as L -models ( abcd , ∅ ) p q y x z ( ab , x ) ( cd , z ) V ( q ) X V ( p ) � I ( bc , y ) A a c ( b , xy ) ( c , yz ) b d p pq q ( ∅ , xyz ) M , a � ⊥ iff ∀ x ( aIx ) M , x ≻ ⊥ always M , a � ⊤ always M , x ≻ ⊤ iff ∀ a ( aIx ) M , a � φ ∧ ψ iff M , a � φ and M , a � ψ M , x ≻ φ ∧ ψ iff for all a ∈ A , if M , a � φ ∧ ψ , then aIx M , a � φ ∨ ψ iff for all x ∈ X , if M , x ≻ φ ∨ ψ , then aIx M , x ≻ φ ∨ ψ iff M , x ≻ φ and M , x ≻ ψ M | = φ ⊢ ψ iff [ [ φ ] ] ⊆ [ [ ψ ] ] iff ( [ ψ ] ) ⊆ ( [ φ ] ) 10 10/17

Expanding the language with modal operators Enriched formal contexts: F = ( A , X , I , { R i | i ∈ Agents } ) R i ⊆ A × X and ∀ a (( R ↑ [ a ]) ↓↑ = R ↑ [ a ]) and ∀ x (( R ↓ [ x ]) ↑↓ = R ↓ [ x ]) ⊤ y a = x x z d = z X � y I A c b a c b d ⊥ Language: L ′ ∋ φ ::= p ∈ Prop | ⊤ | ⊥ | φ ∧ φ | φ ∨ φ | � i φ � i φ : concept φ according to agent i Logic: ◮ Additional axioms: ⊤ ⊢ � i ⊤ � i φ ∧ � i ψ ⊢ � i ( φ ∧ ψ ) φ ⊢ ψ ◮ Additional rule: 11 � i φ ⊢ � i ψ 11/17

Interpretation of � i -formulas on enriched formal contexts ⊤ y x z a = x d = z X � y I A c b a c b d ⊥ V ( � i φ ) = � i V ( φ ) = ( R ↓ )] , ( R ↓ )]) ↑ ) i [( [ φ ] i [( [ φ ] M , a � � i φ iff for all x ∈ X , if M , x ≻ φ , then aR i x M , x ≻ � i φ iff for all a ∈ A , if M , a � � i φ , then aIx 12 12/17

Epistemic interpretation ‘Factivity’ � i p ⊢ p corresponds to R i ⊆ I If agent i is aware that object a has feature x , then a has x ‘objectively’ (i.e. according to the database). ‘Positive introspection’ � i p ⊢ � i � i p corresponds to ∀ x ( R ↓ [ x ] ⊆ R ↓ [ I ↑ [ R ↓ [ x ]]]), i.e. R i ⊆ R i ; R i , i.e. if agent i is aware that object a has feature x , then i must also be aware that a has all the features shared by all the objects which i is aware have feature x . 13 13/17

Core concept: Typicality ◮ in conceptual spaces, the prototype of a formal concept is defined as the geometric center of that concept ; ◮ the closer (i.e. more similar) an object is to the prototype, the stronger its typicality. ◮ Advantage: visually appealing ; ◮ Disadvantage: does not explain the role of agents in establishing the typicality of an object relative to a category. 14 14/17

Logical formalization of typicality i ∈ Agents ; let S ∋ s = i 1 · · · i n finite sequence of agents. Let L C ∋ φ ::= p ∈ Prop | ⊤ | ⊥ | φ ∧ φ | φ ∨ φ | � i φ | C ( φ ) � C ( φ ) stands for � s φ s ∈ S where for any s ∈ S , � s φ := � i 1 · · · � i n φ. Hence [ [ C ( φ )] ] can be understood as the set of prototypes of φ . Interpretation of C -formulas on models M , a � C ( ϕ ) iff for all x ∈ X , if M , x ≻ ϕ , then aR C x M , x ≻ C ( ϕ ) iff for all a ∈ A , if M , a � C ( ϕ ), then aIx , R C := � s ∈ S R s , and R s ⊆ A × X defined by induction on s ∈ S ◮ if s = i then R s := R i ; ◮ if s = ti , then R ↑ s [ a ] := R ↑ t [ I ↓ [ R ↑ i [ a ]]] 15 15/17

Gradedness of non-typicality if a / ∈ [ [ C ( φ )] ] then � � R ↓ a / ∈ [ [ � s ] ] φ = s [( [ φ ] )] . s ∈ S s ∈ S So a must fail the typicality test for some s ∈ S , and this failure can be more or less ‘severe’: Definition: a is at least as typical as a member of φ than b is if { s ∈ S | b ∈ R ↓ )] } ⊆ { s ∈ S | a ∈ R ↓ s [( [ φ ] s [( [ φ ] )] } . 16 16/17