The Logic of Comparative Cardinality Yifeng Ding (voidprove.com) Joint work with Matthew Harrison-Trainer and Wesley Holliday Aug. 7, 2018 @ BLAST 2018 UC Berkeley Group of Logic and the Methodology of Science Introduction • The equational theory of Boolean algebras is also the equational theory of fields of sets, if we only care about Boolean operations. • But more information can be extracted from a field of sets. • We compare their sizes. A field of sets Definition A field of sets (X ; F) is a pair where 1. X is a set and F ⊆ }(X ); 2. F is closed under intersection and complementation. 1 • But more information can be extracted from a field of sets. • We compare their sizes. A field of sets Definition A field of sets (X ; F) is a pair where 1. X is a set and F ⊆ }(X ); 2. F is closed under intersection and complementation. • The equational theory of Boolean algebras is also the equational theory of fields of sets, if we only care about Boolean operations. 1 • We compare their sizes. A field of sets Definition A field of sets (X ; F) is a pair where 1. X is a set and F ⊆ }(X ); 2. F is closed under intersection and complementation. • The equational theory of Boolean algebras is also the equational theory of fields of sets, if we only care about Boolean operations. • But more information can be extracted from a field of sets. 1 A field of sets Definition A field of sets (X ; F) is a pair where 1. X is a set and F ⊆ }(X ); 2. F is closed under intersection and complementation. • The equational theory of Boolean algebras is also the equational theory of fields of sets, if we only care about Boolean operations. • But more information can be extracted from a field of sets. • We compare their sizes. 1 • A field of sets model is hX ; F; V i where V :Φ !F. • Terms are evaluated by Vb on F in the obvious way. •j sj ≥ jtj: set s is at least as large as set t: there is an injection from Vb(t) to Vb(s). Comparing the sizes Definition Given a countably infinite set Φ of set labels, the language L is generated by the following grammar: t ::= a j tc j (t \ t) ' ::= jtj ≥ jtj j :' j (' ^ '); 2 • Terms are evaluated by Vb on F in the obvious way. •j sj ≥ jtj: set s is at least as large as set t: there is an injection from Vb(t) to Vb(s). Comparing the sizes Definition Given a countably infinite set Φ of set labels, the language L is generated by the following grammar: t ::= a j tc j (t \ t) ' ::= jtj ≥ jtj j :' j (' ^ '); • A field of sets model is hX ; F; V i where V :Φ !F. 2 •j sj ≥ jtj: set s is at least as large as set t: there is an injection from Vb(t) to Vb(s). Comparing the sizes Definition Given a countably infinite set Φ of set labels, the language L is generated by the following grammar: t ::= a j tc j (t \ t) ' ::= jtj ≥ jtj j :' j (' ^ '); • A field of sets model is hX ; F; V i where V :Φ !F. • Terms are evaluated by Vb on F in the obvious way. 2 Comparing the sizes Definition Given a countably infinite set Φ of set labels, the language L is generated by the following grammar: t ::= a j tc j (t \ t) ' ::= jtj ≥ jtj j :' j (' ^ '); • A field of sets model is hX ; F; V i where V :Φ !F. • Terms are evaluated by Vb on F in the obvious way. •j sj ≥ jtj: set s is at least as large as set t: there is an injection from Vb(t) to Vb(s). 2 s t • • For finite sets s; t, jsj ≥ jtj $ js \ tc j ≥ jt \ sc j. • For infinite sets s; t; u •j sj ≥ jtj ! js \ tc j ≥ jt \ sc j is not valid; • (jsj ≥ jtj ^ jsj ≥ juj) ! jsj ≥ jt [ uj is valid. Finite sets and infinite sets • Finite sets and infinite sets obey very different laws. 3 Finite sets and infinite sets • Finite sets and infinite sets obey very different laws. s t • • For finite sets s; t, jsj ≥ jtj $ js \ tc j ≥ jt \ sc j. • For infinite sets s; t; u •j sj ≥ jtj ! js \ tc j ≥ jt \ sc j is not valid; • (jsj ≥ jtj ^ jsj ≥ juj) ! jsj ≥ jt [ uj is valid. 3 • The sentences in L valid on infinite sets have been axiomatized, with size interpreted as likelihood or possibilities. • We want to combine them: with no extra constraint on (X ; F), what is the logic? More background • The sentences in L valid on finite sets have been axiomatized, with size interpreted as probability, credence, etc.. 4 • We want to combine them: with no extra constraint on (X ; F), what is the logic? More background • The sentences in L valid on finite sets have been axiomatized, with size interpreted as probability, credence, etc.. • The sentences in L valid on infinite sets have been axiomatized, with size interpreted as likelihood or possibilities. 4 More background • The sentences in L valid on finite sets have been axiomatized, with size interpreted as probability, credence, etc.. • The sentences in L valid on infinite sets have been axiomatized, with size interpreted as likelihood or possibilities. • We want to combine them: with no extra constraint on (X ; F), what is the logic? 4 Outline Introduction Laws common to finite and infinite sets A representation theorem Logic with predicates for finite and infinite sets Eliminating extra predicates Further questions 5 Laws common to finite and infinite sets • if t = 0 is provable in the equational theory of Boolean algebras, then j?j ≥ jtj is a theorem. •j sj ≥ jtj _ jtj ≥ jsj;(jsj ≥ jtj ^ jtj ≥ juj) ! jsj ≥ juj; c •j ?j ≥ js \ t j ! jtj ≥ jsj; c •:j ?j ≥ j? j; • (j?j ≥ jsj ^ j?j ≥ jtj) ! j?j ≥ js [ tj; BasicCompLogic Definition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level. ≥ is a total preorder extending ⊇. ≥ works well with ?. 6 • if t = 0 is provable in the equational theory of Boolean algebras, then j?j ≥ jtj is a theorem. •j sj ≥ jtj _ jtj ≥ jsj;(jsj ≥ jtj ^ jtj ≥ juj) ! jsj ≥ juj; c •j ?j ≥ js \ t j ! jtj ≥ jsj; c •:j ?j ≥ j? j; • (j?j ≥ jsj ^ j?j ≥ jtj) ! j?j ≥ js [ tj; BasicCompLogic Definition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level. ≥ is a total preorder extending ⊇. ≥ works well with ?. 6 •j sj ≥ jtj _ jtj ≥ jsj;(jsj ≥ jtj ^ jtj ≥ juj) ! jsj ≥ juj; c •j ?j ≥ js \ t j ! jtj ≥ jsj; c •:j ?j ≥ j? j; • (j?j ≥ jsj ^ j?j ≥ jtj) ! j?j ≥ js [ tj; BasicCompLogic Definition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level. • if t = 0 is provable in the equational theory of Boolean algebras, then j?j ≥ jtj is a theorem. ≥ is a total preorder extending ⊇. ≥ works well with ?. 6 c •:j ?j ≥ j? j; • (j?j ≥ jsj ^ j?j ≥ jtj) ! j?j ≥ js [ tj; BasicCompLogic Definition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level. • if t = 0 is provable in the equational theory of Boolean algebras, then j?j ≥ jtj is a theorem. ≥ is a total preorder extending ⊇. •j sj ≥ jtj _ jtj ≥ jsj;(jsj ≥ jtj ^ jtj ≥ juj) ! jsj ≥ juj; c •j ?j ≥ js \ t j ! jtj ≥ jsj; ≥ works well with ?. 6 BasicCompLogic Definition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level. • if t = 0 is provable in the equational theory of Boolean algebras, then j?j ≥ jtj is a theorem. ≥ is a total preorder extending ⊇. •j sj ≥ jtj _ jtj ≥ jsj;(jsj ≥ jtj ^ jtj ≥ juj) ! jsj ≥ juj; c •j ?j ≥ js \ t j ! jtj ≥ jsj; ≥ works well with ?. c •:j ?j ≥ j? j; • (j?j ≥ jsj ^ j?j ≥ jtj) ! j?j ≥ js [ tj; 6 Any formula ' consistent with BasicCompLogic is satisfiable in a finite comparison algebra, with interpreting j · j ≥ j · j. ' ) Σ ) hB; ; V i ) h V (T (var('))) ; ; V i | {z } relevant terms, a finite set 6) hX ; F; V i From logic to algebra Definition A comparison algebra is a pair hB; i where B is a Boolean algebra and is a total preorder on B such that • for all a; b 2 B, a ≥B b implies a b, •? B 6 b for all b 2 B n f?B g. 7 6) hX ; F; V i From logic to algebra Definition A comparison algebra is a pair hB; i where B is a Boolean algebra and is a total preorder on B such that • for all a; b 2 B, a ≥B b implies a b, •? B 6 b for all b 2 B n f?B g. Any formula ' consistent with BasicCompLogic is satisfiable in a finite comparison algebra, with interpreting j · j ≥ j · j. ' ) Σ ) hB; ; V i ) h V (T (var('))) ; ; V i | {z } relevant terms, a finite set 7 From logic to algebra Definition A comparison algebra is a pair hB; i where B is a Boolean algebra and is a total preorder on B such that • for all a; b 2 B, a ≥B b implies a b, •? B 6 b for all b 2 B n f?B g. Any formula ' consistent with BasicCompLogic is satisfiable in a finite comparison algebra, with interpreting j · j ≥ j · j. ' ) Σ ) hB; ; V i ) h V (T (var('))) ; ; V i | {z } relevant terms, a finite set 6) hX ; F; V i 7 • (010) and (100) should be finite.