Completeness via correspondence for extensions of first degree - PowerPoint PPT Presentation

Completeness via correspondence for extensions of first degree entailment supplied with classical negation Yaroslav Petrukhin 04/05/2017 yaroslav.petrukhin@mail.ru Lomonosov Moscow State University Completeness via correspondence for extensions of first degree entailment...

Introduction. The purpose of this report is to present a general method of constructing natural deduction systems for all possible unary and binary truth-functional extensions of first degree entailment ( FDE ) supplied with classical negation labeled as BD+ by De and Omori. In so doing I apply Kooi and Tamminga’s technique of correspondence analysis . It was first applied to Priest’s three-valued logic of paradox LP and Kleene’s strong three-valued logic K 3 . My aim is to generalize correspodence analysis to the area of four-valued logics. Completeness via correspondence for extensions of first degree entailment...

To begin with, consider some preliminary. A language L of BD+ is specified by the following grammar: A ∶= p ∣ ¬ A ∣ ∼ A ∣ A ∨ A ∣ A ∧ A , where ¬ is De Morgan negation and ∼ is classical (Boolean) negation. Note that FDE is built in L ’s {¬ , ∧ , ∨} –fragment. Let L ♯ be L ’s extension by unary ⋆ 1 ,..., ⋆ n and binary ○ 1 ,..., ○ m operators, respectively. Let BD ♯ be a logic built in L ♯ . Completeness via correspondence for extensions of first degree entailment...

Truth values. If t is a truth-functional operator then f t is a truth table for t . Let V 4 be a set { 1 ,b,n, 0 } of truth values “true”, “both true and false”, “neither true nor false”, and “false”. The values are ordered as follows: 0 ≼ n , 0 ≼ b , n ≼ 1, b ≼ 1; n and b are incomparable. Let x,y,z ∈ V 4 . Then f ⋆ ( x ) = y ( f ○ ( x,y ) = z ) stands for an entry of a such truth table f ⋆ ( f ○ ) that for each valuation v if v ( A ) = x then v (⋆ A ) = y , for each A ∈ L ♯ (if both v ( A ) = x and v ( B ) = y then v ( A ○ B ) = z , for all A,B ∈ L ♯ ). Completeness via correspondence for extensions of first degree entailment...

Single Entry Correspondence For a four-valued case the following adaptation of Kooi and Tamminga’s definition 2.1 and Tamminga’s definition 1 holds: Definition 1. ( Single Entry Correspondence ) Let Γ ⊆ L ♯ and let A ∈ L ♯ . Let x,y,z ∈ V 4 . Let E be a truth-table entry of the type f ⋆ ( x ) = y or f ○ ( x,y ) = z . Then the truth-table entry E is characterized by an inference scheme Γ / A , if E if and only if Γ ⊧ A . Completeness via correspondence for extensions of first degree entailment...

The purpose of this report is to present such inference schemes of the type Γ / A that each possible truth-table entry E is characterised by some inference scheme. These inference schemes are in fact inference rules. By adding them to a natural deduction system for BD+ , one obtains natural deduction system ND BD ♯ for BD ♯ . Completeness via correspondence for extensions of first degree entailment...

Belnap’s semantics of BD+ Interpretations of BD+ ’s connectives are defined by the following truth tables. A f ¬ f ∼ f ∧ 1 b n 0 f ∨ 1 b n 0 1 0 0 1 1 0 1 1 1 1 1 b n 0 0 1 1 b b n b b b b b b 0 0 1 1 n n b n n n n n n 0 1 1 0 0 0 0 0 0 1 0 b n The entailment relation in BD+ and BD ♯ is defined as follows: Γ ⊧ A iff for each valuation v , if v ( B ) ∈ { 1 ,b } , for each B ∈ Γ , then v ( A ) ∈ { 1 ,b } . Completeness via correspondence for extensions of first degree entailment...

Dunn’s semantics for BD+ . Truth values here are subsets of a set of classical truth values { t , f } , that is { t } , { t , f } , ∅ and { f } which are analogues of values 1, b , n and 0 from Belnap’s semantics. The conditions of truth and falsity for formulas are as follows: t ∈ v (¬ A ) iff f ∈ v ( A ) ; f ∈ v (¬ A ) iff t ∈ v ( A ) ; t ∈ v ( ∼ A ) iff t / ∈ v ( A ) ; f ∈ v ( ∼ A ) iff f / ∈ v ( A ) ; t ∈ v ( A ∧ B ) iff t ∈ v ( A ) and t ∈ v ( B ) ; f ∈ v ( A ∧ B ) iff f ∈ v ( A ) or f ∈ v ( B ) ; t ∈ v ( A ∨ B ) iff t ∈ v ( A ) or t ∈ v ( B ) ; f ∈ v ( A ∨ B ) iff f ∈ v ( A ) and f ∈ v ( B ) . Completeness via correspondence for extensions of first degree entailment...

In terms of J.M. Dunn’s semantics the entailment relation in logics BD+ and BD ♯ is defined as follows: Γ ⊧ A iff for each valuation v , if t ∈ v ( B ) , for each B ∈ Γ , then t ∈ v ( A ) . Completeness via correspondence for extensions of first degree entailment...

Theorem 1. For each L ♯ -formula A : ⎧ ∼ A, ¬ A ⊧ ∼ ⋆ A ∧ ¬ ⋆ A 0 iff ⎪ ⎪ ⎪ ⎪ ⎪ ∼ A, ¬ A ⊧ ∼ ⋆ A ∧ ∼ ¬ ⋆ A n iff f ⋆ ( 0 ) = ⎨ ⎪ ∼ A, ¬ A ⊧ ⋆ A ∧ ¬ ⋆ A b iff ⎪ ⎪ ⎪ ⎪ ∼ A, ¬ A ⊧ ⋆ A ∧ ∼ ¬ ⋆ A ⎩ 1 iff ⎧ ∼ A, ∼ ¬ A ⊧ ∼ ⋆ A ∧ ¬ ⋆ A 0 iff ⎪ ⎪ ⎪ ⎪ ⎪ ∼ A, ∼ ¬ A ⊧ ∼ ⋆ A ∧ ∼ ¬ ⋆ A n iff f ⋆ ( n ) ⎨ = ∼ A, ∼ ¬ A ⊧ ⋆ A ∧ ¬ ⋆ A ⎪ b iff ⎪ ⎪ ⎪ ⎪ ∼ A, ∼ ¬ A ⊧ ⋆ A ∧ ∼ ¬ ⋆ A 1 iff ⎩ Completeness via correspondence for extensions of first degree entailment...

⎧ A, ¬ A ⊧ ∼ ⋆ A ∧ ¬ ⋆ A 0 iff ⎪ ⎪ ⎪ ⎪ ⎪ A, ¬ A ⊧ ∼ ⋆ A ∧ ∼ ¬ ⋆ A n iff f ⋆ ( b ) = ⎨ ⎪ A, ¬ A ⊧ ⋆ A ∧ ¬ ⋆ A b iff ⎪ ⎪ ⎪ ⎪ A, ¬ A ⊧ ⋆ A ∧ ∼ ¬ ⋆ A iff ⎩ 1 ⎧ A, ∼ ¬ A ⊧ ∼ ⋆ A ∧ ¬ ⋆ A 0 iff ⎪ ⎪ ⎪ ⎪ ⎪ n iff A, ∼ ¬ A ⊧ ∼ ⋆ A ∧ ∼ ¬ ⋆ A f ⋆ ( 1 ) ⎨ = ⎪ A, ∼ ¬ A ⊧ ⋆ A ∧ ¬ ⋆ A b iff ⎪ ⎪ ⎪ ⎪ iff A, ∼ ¬ A ⊧ ⋆ A ∧ ∼ ¬ ⋆ A ⎩ 1 Completeness via correspondence for extensions of first degree entailment...

Proof of Theorem 1. As an example, we consider the case f ⋆ ( 0 ) = 1 . (1) f ⋆ ( 0 ) = 1 (assumption). (2) f ⋆ has an entry such that if v ( A ) = 0 then v (⋆ A ) = 1 , for each A and for each v (from (1)). (3) If ( t / ∈ v ( A ) and f ∈ v ( A ) ) then ( t ∈ v (⋆ A ) and f / ∈ v (⋆ A ) ), for each A and for each v (from (2)). (4) t ∈ v ( ∼ A ) and t ∈ v (¬ A ) (assumption). (5) t / ∈ v ( A ) and f ∈ v ( A ) (from (4)). (6) t ∈ v (⋆ A ) and f / ∈ v (⋆ A ) (from (3) and (5)). (7) t ∈ v (⋆ A ) and t ∈ v ( ∼ ¬ ⋆ A ) (from (6)). (8) t ∈ v ( ⋆ A ∧ ∼ ¬ ⋆ A ) (from (7)). (9) If ( t ∈ v ( ∼ A ) and t ∈ v (¬ A ) ) then t ∈ v ( ⋆ A ∧ ∼ ¬ ⋆ A )), for each A and v (from (3)–(8)). Completeness via correspondence for extensions of first degree entailment...

(10) ∼ A, ¬ A ⊧ ⋆ A ∧ ∼¬ ⋆ A , for each A (from (9)). (11) If f ⋆ ( 0 ) = 1 then ∼ A, ¬ A ⊧ ⋆ A ∧ ∼¬ ⋆ A , for each A (from (1)–(10)). (12) ∼ A, ¬ A ⊧ ⋆ A ∧ ∼¬ ⋆ A , for each A (assumption). (13) If ( t ∈ v (∼ A ) and t ∈ v (¬ A ) ) then t ∈ v (⋆ A ∧ ∼¬ ⋆ A ), for each A and v (from(12)). (14) If (( t / ∈ v ( A ) and f ∈ v ( A )) then ( t ∈ v (⋆ A ) and f / ∈ v (⋆ A ))) , for each A and v (from (13)). (15) If v ( A ) = 0 then v (⋆ A ) = 1 , for each A and v (from (14)). (16) f ⋆ ( 0 ) = 1 (from (15)). (17) If ∼ A, ¬ A ⊧ ⋆ A ∧ ∼¬ ⋆ A , for each A , then f ⋆ ( 0 ) = 1 (from (12)–(16)). Completeness via correspondence for extensions of first degree entailment...

Theorem 2. For each L ♯ -formulas A and B : ⎧ ∼ A, ¬ A, ∼ B, ¬ B ⊧∼( A ○ B ) ∧ ¬( A ○ B ) 0 iff ⎪ ⎪ ⎪ ⎪ ⎪ ∼ A, ¬ A, ∼ B, ¬ B ⊧∼( A ○ B )∧ ∼¬( A ○ B ) n iff f ○ ( 0 , 0 ) = ⎨ ⎪ ∼ A, ¬ A, ∼ B, ¬ B ⊧ ( A ○ B ) ∧ ¬( A ○ B ) b iff ⎪ ⎪ ⎪ ⎪ ∼ A, ¬ A, ∼ B, ¬ B ⊧ ( A ○ B )∧ ∼¬( A ○ B ) ⎩ 1 iff ⎧ ∼ A, ¬ A, ∼ B, ∼¬ B ⊧∼( A ○ B ) ∧ ¬( A ○ B ) 0 iff ⎪ ⎪ ⎪ ⎪ ⎪ ∼ A, ¬ A, ∼ B, ∼¬ B ⊧∼( A ○ B )∧ ∼¬( A ○ B ) n iff f ○ ( 0 ,n ) = ⎨ ∼ A, ¬ A, ∼ B, ∼¬ B ⊧ ( A ○ B ) ∧ ¬( A ○ B ) ⎪ b iff ⎪ ⎪ ⎪ ⎪ ∼ A, ¬ A, ∼ B, ∼¬ B ⊧ ( A ○ B )∧ ∼¬( A ○ B ) 1 iff ⎩ Completeness via correspondence for extensions of first degree entailment...

⎧ ∼ A, ¬ A,B, ¬ B ⊧∼( A ○ B ) ∧ ¬( A ○ B ) 0 iff ⎪ ⎪ ⎪ ⎪ ⎪ ∼ A, ¬ A,B, ¬ B ⊧∼( A ○ B )∧ ∼¬( A ○ B ) n iff f ○ ( 0 ,b ) = ⎨ ⎪ ∼ A, ¬ A,B, ¬ B ⊧ ( A ○ B ) ∧ ¬( A ○ B ) b iff ⎪ ⎪ ⎪ ⎪ ∼ A, ¬ A,B, ¬ B ⊧ ( A ○ B )∧ ∼¬( A ○ B ) iff ⎩ 1 ⎧ ∼ A, ¬ A,B, ∼¬ B ⊧∼( A ○ B ) ∧ ¬( A ○ B ) 0 iff ⎪ ⎪ ⎪ ⎪ ⎪ n iff ∼ A, ¬ A,B, ∼¬ B ⊧∼( A ○ B )∧ ∼¬( A ○ B ) f ○ ( 0 , 1 ) = ⎨ ⎪ ∼ A, ¬ A,B, ∼¬ B ⊧ ( A ○ B ) ∧ ¬( A ○ B ) b iff ⎪ ⎪ ⎪ ⎪ iff ∼ A, ¬ A,B, ∼¬ B ⊧ ( A ○ B )∧ ∼¬( A ○ B ) ⎩ 1 Completeness via correspondence for extensions of first degree entailment...