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
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
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
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
Comparing the sizes Definition Given a countably infinite set Φ of set labels, the language L is generated by the following grammar: a | t c | ( t ∩ t ) t ::= ϕ ::= | t | ≥ | t | | ¬ ϕ | ( ϕ ∧ ϕ ) , 2
Comparing the sizes Definition Given a countably infinite set Φ of set labels, the language L is generated by the following grammar: a | t c | ( t ∩ t ) t ::= ϕ ::= | t | ≥ | t | | ¬ ϕ | ( ϕ ∧ ϕ ) , • A field of sets model is � X , F , V � where V : Φ → F . 2
Comparing the sizes Definition Given a countably infinite set Φ of set labels, the language L is generated by the following grammar: a | t c | ( t ∩ t ) t ::= ϕ ::= | t | ≥ | t | | ¬ ϕ | ( ϕ ∧ ϕ ) , • A field of sets model is � X , F , V � where V : Φ → F . • Terms are evaluated by � V 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: a | t c | ( t ∩ t ) t ::= ϕ ::= | t | ≥ | t | | ¬ ϕ | ( ϕ ∧ ϕ ) , • A field of sets model is � X , F , V � where V : Φ → F . • Terms are evaluated by � V on F in the obvious way. • | s | ≥ | t | : set s is at least as large as set t : there is an injection from � V ( t ) to � V ( s ). 2
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 , | s | ≥ | t | ↔ | s ∩ t c | ≥ | t ∩ s c | . • For infinite sets s , t , u • | s | ≥ | t | → | s ∩ t c | ≥ | t ∩ s c | is not valid; • ( | s | ≥ | t | ∧ | s | ≥ | u | ) → | s | ≥ | t ∪ u | is valid. 3
More background • The sentences in L valid on finite sets have been axiomatized, with size interpreted as probability, credence, etc.. 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. 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
BasicCompLogic Definition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level. ≥ is a total preorder extending ⊇ . ≥ works well with ∅ . 6
BasicCompLogic Definition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level. ≥ is a total preorder extending ⊇ . ≥ 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 | ∅ | ≥ | t | is a theorem. ≥ is a total preorder extending ⊇ . ≥ 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 | ∅ | ≥ | t | is a theorem. ≥ is a total preorder extending ⊇ . • | s | ≥ | t | ∨ | t | ≥ | s | ; ( | s | ≥ | t | ∧ | t | ≥ | u | ) → | s | ≥ | u | ; • | ∅ | ≥ | s ∩ t c | → | t | ≥ | s | ; ≥ 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 | ∅ | ≥ | t | is a theorem. ≥ is a total preorder extending ⊇ . • | s | ≥ | t | ∨ | t | ≥ | s | ; ( | s | ≥ | t | ∧ | t | ≥ | u | ) → | s | ≥ | u | ; • | ∅ | ≥ | s ∩ t c | → | t | ≥ | s | ; ≥ works well with ∅ . • ¬ | ∅ | ≥ | ∅ c | ; • ( | ∅ | ≥ | s | ∧ | ∅ | ≥ | t | ) → | ∅ | ≥ | s ∪ t | ; 6
From logic to algebra Definition A comparison algebra is a pair � B , �� where B is a Boolean algebra and � is a total preorder on B such that • for all a , b ∈ B , a ≥ B b implies a � b , • ⊥ B �� b for all b ∈ B \ {⊥ B } . 7
From logic to algebra Definition A comparison algebra is a pair � B , �� where B is a Boolean algebra and � is a total preorder on B such that • for all a , b ∈ B , a ≥ B b implies a � b , • ⊥ B �� b for all b ∈ B \ {⊥ B } . Any formula ϕ consistent with BasicCompLogic is satisfiable in a finite comparison algebra, with � interpreting | · | ≥ | · | . ϕ ⇒ Σ ⇒ � B , � , V � ⇒ � V ( T ( var ( ϕ ))) , � , V � � �� � relevant terms, a finite set 7
From logic to algebra Definition A comparison algebra is a pair � B , �� where B is a Boolean algebra and � is a total preorder on B such that • for all a , b ∈ B , a ≥ B b implies a � b , • ⊥ B �� b for all b ∈ B \ {⊥ B } . Any formula ϕ consistent with BasicCompLogic is satisfiable in a finite comparison algebra, with � interpreting | · | ≥ | · | . ϕ ⇒ Σ ⇒ � B , � , V � ⇒ � V ( T ( var ( ϕ ))) , � , V � � �� � relevant terms, a finite set �⇒ � X , F , V � 7
Not enough constraints (111) : 4 (011) : 4 (101) : 4 (110) : 4 (001) : 1 (010) : 2 (100) : 3 (000) : 0 8
Not enough constraints (111) : 4 • (010) and (100) should be (011) : 4 (101) : 4 (110) : 4 finite. • Then all must be finite. • But | (011) | = | (101) | while (001) : 1 (010) : 2 (100) : 3 | (010) | < | (100) | . (000) : 0 8
Message • Any formula ϕ consistent with BasicCompLogic is satisfiable in a finite comparison algebra B , with � interpreting | · | ≥ | · | . 9
Message • Any formula ϕ consistent with BasicCompLogic is satisfiable in a finite comparison algebra B , with � interpreting | · | ≥ | · | . • But the ordering � in this B might not be based on any cardinality comparison. 9
Message • Any formula ϕ consistent with BasicCompLogic is satisfiable in a finite comparison algebra B , with � interpreting | · | ≥ | · | . • But the ordering � in this B might not be based on any cardinality comparison. • We need to know when the ordering arise from cardinality comparison, and add the constraints to the logic. 9
Plan Introduction Laws common to finite and infinite sets A representation theorem Logic with predicates for finite and infinite sets Eliminating extra predicates Further questions 10
Definitions Definition A measure algebra is a pair � B , µ � , where B is a Boolean algebra and µ is a function assigning a cardinal to each element of B such that • if a ∧ b = ⊥ , then µ ( a ∨ b ) = µ ( a ) + µ ( b ), and • µ ( b ) = 0 iff b = ⊥ . 11
Definitions Definition A measure algebra is a pair � B , µ � , where B is a Boolean algebra and µ is a function assigning a cardinal to each element of B such that • if a ∧ b = ⊥ , then µ ( a ∨ b ) = µ ( a ) + µ ( b ), and • µ ( b ) = 0 iff b = ⊥ . Definition A comparison algebra � B , �� is represented by a measure algebra � B , µ � if for all a , b ∈ B , we have a � b iff µ ( a ) ≥ µ ( b ). 11
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