Tableau metatheory for propositional and syllogistic logics Part - PowerPoint PPT Presentation

Tableau metatheory for propositional and syllogistic logics Part IV: Abstract tableau notions: rules, branches, tableaux Tomasz Jarmużek Nicolaus Copernicus University in Toruń Poland Logic Summer School, 3th-14th, December 2018, Australian National University

Program of lecture We describe the main part of tableau metatheory: general tableau notions: ◮ all notions are presented as set–theoretical ones (for example: branches are sequences of sets and tableaux are sets of those sequences) ◮ the rest of tableau notions are defined in a similar, formal way: 1. tableau rules 2. branches: open, closed, maximal (aka complete) 3. tableaux: open, closed, complete 4. new notions are also presented — branch consequence relation (as a very special set of branches) and useless variant of branch.

Tableau language – set of expressions We need some language of tableau proofs: set of expressions Ex. Firstly, we list symbols: 1. indexes/labels — set of natural numbers N 2. n–ary functional constants (where n ≥ 1): w 1 1 , w 1 2 , w 1 3 , . . . , w 2 1 , w 2 2 , w 2 3 , . . . , w 3 1 , w 3 2 , w 3 3 , . . . 3. n–ary predicates (where n ≥ 2): r 2 1 , r 2 2 , r 2 3 , . . . , r 3 1 , r 3 2 , r 3 3 , . . . , r 4 1 , r 4 2 , r 4 3 , . . . 4. identity symbol: ≡ 5. semantic negation: ∼ .

Tableau language – set of terms Set of all terms TERM is the least that consists of: w l k ( m 1 , . . . , m l ), where: ◮ k , l , m 1 , . . . , m k ∈ N ◮ l ≥ 1 ◮ w l k is a functional constant. The members of TERM we denote by t 1 , t 2 , t 3 , . . .

Tableau language – set of expressions Definition (Expressions) Ex is the least set that consists of the expressions: ◮ r l ∼ r l k ( m 1 , . . . , m l ) k ( m 1 , . . . , m l ) ◮ i ≡ j ∼ i ≡ j ◮ � A , t 1 , . . . , t n � �∼ A , t 1 , . . . , t n � for all: a. A ∈ For b. i , j , k , l , n , m 1 , . . . , m l ∈ N c. t 1 , . . . , t n ∈ TERM, where n ≥ 1. When the context is clear, we write: ◮ A , t 1 , . . . , t n ◮ ∼ A , t 1 , . . . , t n , removing brackets: � � .

Fundamental tableau notions: function choosing indexes Definition (Function choosing indexes) Function choosing indexes we call a function ◦ : Ex ∪ TERM ∪ P(Ex ∪ TERM) − → P( N ) defined by conditions: ◮ ◦ ( w l k ( m 1 , . . . , m l )) = { m 1 , . . . , m l } ◮ ◦ ( r l k ( m 1 , . . . , m l )) = ◦ ( ∼ r l k ( m 1 , . . . , m l )) = { m 1 , . . . , m l } ◮ ◦ ( i ≡ j ) = ◦ ( ∼ i ≡ j ) = { i , j } ◮ ◦ ( � A , w l 1 k 1 ( x k 1 1 , . . . , x k 1 l 1 ) , . . . , w l n h o ( x h o 1 , . . . , x h o l n ) � ) = ◦ ( �∼ A , w l 1 k 1 ( x k 1 1 , . . . , x k 1 l 1 ) , . . . , w l n h o ( x h o 1 , . . . , x h o l n ) � ) = { x k 1 1 , . . . , x k 1 l 1 , . . . , x h o 1 , . . . , x h o l n } ◮ ◦ ( X ) = � {◦ ( y ) : y ∈ X } , if X ⊆ Ex ∪ TERM, for all A ∈ For and h , i , j , k , l , o , m 1 , . . . , m l , x k 1 1 , . . . , x k 1 l 1 , . . . , x h o 1 , . . . , x h o l n ∈ N .

Fundamental tableau notions: similar sets of expressions Definition (Similar sets of expressions) Let X , Y ⊆ Ex be sets of expressions. Let Z ⊆ N . Set X is similar to Y in respect of Z iff there is a bijection ‡ : ◦ ( X ) − → ◦ ( Y ) (where ◦ ( X ), ◦ ( Y ) are sets of indexes occurring in expressions of X and Y ) such that: (a) for all x ∈ Z , if x ∈ ◦ ( X ), then ‡ ( x ) = x (b) for all kinds of expressions in Ex:

Fundamental tableau notions: similar sets of expressions (a) r l k ( m 1 , . . . , m l ) ∈ X iff r l k ( ‡ ( m 1 ) , . . . , ‡ ( m l )) ∈ Y (b) i ≡ j ∈ X iff ‡ ( i ) ≡ ‡ ( j ) ∈ Y (c) � A , w l 1 k 1 ( x k 1 1 , . . . , x k 1 l 1 ) , . . . , w l n h o ( x h o 1 , . . . , x h o l n ) � ∈ X iff � A , w l 1 k 1 ( ‡ ( x k 1 1 ) , . . . , ‡ ( x k 1 l 1 )) , . . . , w l n h o ( ‡ ( x h o 1 ) , . . . , ‡ ( x h o l n )) � ∈ Y (d) ∼ r l k ( m 1 , . . . , m l ) ∈ X iff ∼ r l k ( ‡ ( m 1 ) , . . . , ‡ ( m l )) ∈ Y (e) ∼ i ≡ j ∈ X iff ∼ ‡ ( i ) ≡ ‡ ( j ) ∈ Y (f) �∼ A , w l 1 k 1 ( x k 1 1 , . . . , x k 1 l 1 ) , . . . , w l n h o ( x h o 1 , . . . , x h o l n ) � ∈ X iff �∼ A , w l 1 k 1 ( ‡ ( x k 1 1 ) , . . . , ‡ ( x k 1 l 1 )) , . . . , w l n h o ( ‡ ( x h o 1 ) , . . . , ‡ ( x h o l n )) � ∈ Y , for all A ∈ For and h , i , j , k , l , o , m 1 , . . . , m l , x k 1 1 , . . . , x k 1 l 1 , . . . , x h o 1 , . . . , x h o l n ∈ N .

Fundamental tableau notions: tableau inconsistency Definition (Tableau inconsistent sets of expressions) Let X ⊆ Ex. We say that X is tableau inconsistent iff it consists one of pairs of the expressions: (a) r l k ( m 1 , . . . , m l ), ∼ r l k ( m 1 , . . . , m l ) (b) i ≡ j , ∼ i ≡ j (c) � A , t 1 , . . . , t n � , �∼ A , t 1 , . . . , t n ) � for all: ◮ A ∈ For and i , j , k , l , m 1 , . . . , m l , n ∈ N ◮ t 1 , . . . , t n ∈ TERM. Otherwise, we call set X tableau consistent . We shortly say that X is t-consistent or respectively t-inconsistent .

Model suitable to a set of expressions Definition (Model suitable to a set of expressions) Let X ∈ Ex. Let M = �{ W i } i ∈ M , { R j } j ∈ N , V � ∈ M i and X ⊆ Ex. Model M is suitable to X iff there are functions: (a) f ′ : N − → M ∪ N (b) f ′′ : N − → � i ∈ M W i such that following conditions are fulfilled:

Model suitable to a set of expressions (a) if � A , w l 1 k 1 ( x k 1 1 , . . . , x k 1 l 1 ) , . . . , w l n h o ( x h o 1 , . . . , x h o l n ) � ∈ X , then: ◮ ( � f ′′ ( x k 1 1 ) , . . . , f ′′ ( x k 1 l 1 ) � ∈ W l 1 f ′ ( k ) 1 , . . . , � f ′′ ( x h o 1 ) , . . . , f ′′ ( x h o l n ) � ∈ W l n f ′ ( h ) o ◮ W l 1 f ′ ( k ) 1 × · · · × W l n f ′ ( h ) o ∈ { W i } i ∈ M ◮ M , � f ′ ( x k 1 1 ) , . . . , f ′ ( x k 1 l 1 ) � , . . . , � f ′ ( x h o 1 ) , . . . , f ′ ( x h o l n ) � | = A (b) if �∼ A , w l 1 k 1 ( x k 1 1 , . . . , x k 1 l 1 ) , . . . , w l n h o ( x h o 1 , . . . , x h o l n ) � ∈ X , then: ◮ ( � f ′′ ( x k 1 1 ) , . . . , f ′′ ( x k 1 l 1 ) � ∈ W l 1 f ′ ( k ) 1 , . . . , � f ′′ ( x h o 1 ) , . . . , f ′′ ( x h o l n ) � ∈ W l n f ′ ( h ) o ◮ W l 1 f ′ ( k ) 1 × · · · × W l n f ′ ( h ) o ∈ { W i } i ∈ M ◮ M , � f ′ ( x k 1 1 ) , . . . , f ′ ( x k 1 l 1 ) � , . . . , � f ′ ( x h o 1 ) , . . . , f ′ ( x h o l n ) � �| = A

Model suitable to a set of expressions (c) if r l k ( m 1 , . . . , m l ) ∈ X , then � f ′′ ( m 1 ) , . . . , f ′′ ( m l ) � ∈ R l f ′ ( k ) (d) if ∼ r l k ( m 1 , . . . , m l ) ∈ X , then � f ′′ ( m 1 ) , . . . , f ′′ ( m l ) � �∈ R l f ′ ( k ) (e) if i ≡ j ∈ X , then f ′′ ( i ) is equal to f ′′ ( j ) (f) if ∼ i ≡ j ∈ X , then f ′′ ( i ) is not equal to f ′′ ( j ) for all A ∈ For and h , i , j , k , l , o , m 1 , . . . , m l , x k 1 1 , . . . , x k 1 l 1 , . . . , x h o 1 , . . . , x h o l n ∈ N .

Complex tableau notions Having a set of expressions Ex we can put very general conditions defining rules. Our rules extend properly a set of expressions and they have also an internal mechanism that blocks extending of t-inconsistent sets. Let us distinguish a set of indexes that plays a role of signs of logical values in our language: LV ⊆ N (for some domain in models W j ). Let Z ⊆ Ex. Z is co–infinite iff N \ ◦ ( Z ) is infinite.

Complex tableau notions Definition (Rule) Assume that P(Ex) is the set of all subsets of the set Ex. Let P(Ex) n be n –ary Cartesian product P(Ex) × · · · × P(Ex) , for some � �� � n n ∈ N P(Ex) n be the union of all such n –ary n ∈ N , and let � Cartesian products that n ≥ 2.

Complex tableau notions Definition (Rule) Assume that P(Ex) is the set of all subsets of the set Ex. Let P(Ex) n be n –ary Cartesian product P(Ex) × · · · × P(Ex) , for some � �� � n n ∈ N P(Ex) n be the union of all such n –ary n ∈ N , and let � Cartesian products that n ≥ 2. n ∈ N P(Ex) n that if ◮ Rule is such a subset R ⊆ � � X 1 , . . . , X n � ∈ R , then the following conditions are satisfied: ◮ X 1 ⊂ X i , for all 1 < i ≤ n ◮ X 1 is t -consistent ◮ if k � = l , then X k � = X l , for all 1 < k , l ≤ n

Rule Definition (Rule cont.) ◮ (Closure under similarity) for any such subset of expression Y 1 that Y 1 is similar to X 1 in respect of LV, there exist such sets of expressions Y 2 , . . . , Y n , that � Y 1 , . . . , Y n � ∈ R and for all 1 < i ≤ n , Y i is similar to X i in respect of LV

Rule Definition (Rule cont.) ◮ (Closure under similarity) for any such subset of expression Y 1 that Y 1 is similar to X 1 in respect of LV, there exist such sets of expressions Y 2 , . . . , Y n , that � Y 1 , . . . , Y n � ∈ R and for all 1 < i ≤ n , Y i is similar to X i in respect of LV ◮ (Existence of a core of rule) for some finite set Y ⊆ X 1 there exists exactly one such n –tuple � Z 1 , . . . , Z n � ∈ R that: 1. Z 1 = Y 2. for any 1 < i ≤ n , Z i = Z 1 ∪ ( X i \ X 1 ) 3. there does not exist a proper subset U 1 ⊂ Y and such n –tuple � U 1 , . . . , U n � ∈ R that for 1 < i ≤ n , U i = U 1 ∪ ( Z i \ Z 1 ) Any such n –tuple � Z 1 , . . . , Z n � is called a core of rule R in � X 1 , . . . , X n �