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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 K3. 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 ft is a truth table for t. Let V4 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 ∈ V4. Then f⋆(x) = y (f○(x,y) = z) stands for an entry

• f 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 ∈ V4. 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

• f 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 NDBD ♯ 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∼ 1 b b n n n b 1 1 f∧ 1 b n 1 1 b n b b b n n n f∨ 1 b n 1 1 1 1 1 b 1 b 1 b n 1 1 n n 1 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: f⋆(0) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff ∼A,¬A ⊧∼⋆A ∧ ¬ ⋆ A n iff ∼A,¬A ⊧∼⋆A∧ ∼¬ ⋆ A b iff ∼A,¬A ⊧ ⋆A ∧ ¬ ⋆ A 1 iff ∼A,¬A ⊧ ⋆A∧ ∼¬ ⋆ A f⋆(n) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff ∼A,∼¬A ⊧∼⋆A ∧ ¬ ⋆ A n iff ∼A,∼¬A ⊧∼⋆A∧ ∼¬ ⋆ A b iff ∼A,∼¬A ⊧ ⋆A ∧ ¬ ⋆ A 1 iff ∼A,∼¬A ⊧ ⋆A∧ ∼¬ ⋆ A

Completeness via correspondence for extensions of first degree entailment...

f⋆(b) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff A,¬A ⊧∼⋆A ∧ ¬ ⋆ A n iff A,¬A ⊧∼⋆A∧ ∼¬ ⋆ A b iff A,¬A ⊧ ⋆A ∧ ¬ ⋆ A 1 iff A,¬A ⊧ ⋆A∧ ∼¬ ⋆ A f⋆(1) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff A,∼¬A ⊧∼⋆A ∧ ¬ ⋆ A n iff A,∼¬A ⊧∼⋆A∧ ∼¬ ⋆ A b iff A,∼¬A ⊧ ⋆A ∧ ¬ ⋆ A 1 iff A,∼¬A ⊧ ⋆A∧ ∼¬ ⋆ A

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: f○(0,0) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff ∼A,¬A,∼B,¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff ∼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) f○(0,n) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff ∼A,¬A,∼B,∼¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff ∼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)

Completeness via correspondence for extensions of first degree entailment...

f○(0,b) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff ∼A,¬A,B,¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff ∼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) f○(0,1) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff ∼A,¬A,B,∼¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff ∼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)

Completeness via correspondence for extensions of first degree entailment...

f○(n,0) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff ∼A,∼¬A,∼B,¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff ∼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) f○(n,n) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff ∼A,∼¬A,∼B,∼¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff ∼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)

Completeness via correspondence for extensions of first degree entailment...

f○(n,b) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff ∼A,∼¬A,B,¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff ∼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) f○(n,1) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff ∼A,∼¬A,B,∼¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff ∼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)

Completeness via correspondence for extensions of first degree entailment...

f○(b,0) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff A,¬A,∼B,¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff 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) f○(b,n) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff A,¬A,∼B,∼¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff 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)

Completeness via correspondence for extensions of first degree entailment...

f○(b,b) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff A,¬A,B,¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff 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) f○(b,1) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff A,¬A,B,∼¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff 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)

Completeness via correspondence for extensions of first degree entailment...

f○(1,0) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff A,∼¬A,∼B,¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff 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) f○(1,n) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff A,∼¬A,∼B,∼¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff 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)

Completeness via correspondence for extensions of first degree entailment...

f○(1,b) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff A,∼¬A,B,¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff 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) f○(1,1) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ iff A,∼¬A,B,∼¬B ⊧∼(A ○ B) ∧ ¬(A ○ B) n iff 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)

Completeness via correspondence for extensions of first degree entailment...

A natural deduction system for FDE is as follows (this system was first introduced by Priest): (¬¬I) A ¬¬A (¬¬E) ¬¬A A (∨I1) A A ∨ B (∨I2) B A ∨ B (∨E) [A] [B] A ∨ B C C C (∧I) A B A ∧ B (∧E1) A ∧ B A (∧E2) A ∧ B B (¬ ∨ I) ¬A ∧ ¬B ¬(A ∨ B) (¬ ∨ E) ¬(A ∨ B) ¬A ∧ ¬B (¬ ∧ I) ¬A ∨ ¬B ¬(A ∧ B) (¬ ∧ E) ¬(A ∧ B) ¬A ∨ ¬B

Completeness via correspondence for extensions of first degree entailment...

Rules for Boolean negation are as follows: (EFQ) A ∼A B (EM) A∨ ∼A (∼¬E) ∼¬A ¬ ∼A (¬ ∼E) ¬ ∼A ∼¬A Thus, now we have inference rules of a natural deduction system NDBD+ for BD+. A rule of inference of the form R⋆(x,y) A1,... ,An B corresponds to an entry f⋆(x) = y of a truth table f⋆ and a rules of the form R○(x,y,z) A1,... ,Am B corresponds to an entry f○(x,y) = z of a truth table f○.

Completeness via correspondence for extensions of first degree entailment...

Each connective ⋆ needs 4 rules of the form R⋆(x,y) and each connective ○ needs 16 rules of the form R○(x,y,z). These rules are abovementioned inference schemes. Here is an example. According to Theorem 1, the rule R⋆(0,0) corresponds to the entry f⋆(0) = 0 of the truth table f⋆: R⋆(0,0) ∼A ¬A ∼⋆A ∧ ¬ ⋆ A A derivation of A from Γ in NDBD+ and NDBD ♯ is defined in a standard way in linear “Fitch-style” format.

Completeness via correspondence for extensions of first degree entailment...

Theorem 3. Let Γ ⊆ L♯ and let A ∈ L♯. Then Γ ⊢ A in NDBD ♯ iff Γ ⊧ A in BD♯. This theorem is proved, using Kooi and Tamminga’s method which is a kind of Henkin-style technique.

Completeness via correspondence for extensions of first degree entailment...

Correspondence analysis and implicational extensions of BD+. BN4.

One can extend BD+ by an implication from Brady’s logic BN4. f↦ 1 b n 1 1 n b 1 b n n 1 n 1 n 1 1 1 1

Completeness via correspondence for extensions of first degree entailment...

E4

Besides, one can consider an implication of Robles’ logic E4. f↣ 1 b n 1 1 b 1 b n 1 b 1 1 1 1

Completeness via correspondence for extensions of first degree entailment...

Par

Moreover, BD+ can be extended by an implication of Popov’s logic Par. f⇒ 1 b n 1 1 b n b 1 b n n 1 1 1 1 1 1 1 1

Completeness via correspondence for extensions of first degree entailment...

FDEP

One more interesting implication can be defined via BD+’s connectives as follows: A → B ∶=∼A ∨ B. This implication was first studied by Dmitry Zaitsev. f→ 1 b n 1 1 b n b 1 1 n n n 1 b 1 b 1 1 1 1

Completeness via correspondence for extensions of first degree entailment...

Concluding remarks.

In summary, the result presented in this report allows to get immediately adequate natural deduction systems for all possible truth-table expansions of BD+. Consequently, a problem for future research arises: to formulate theorems 1 and 2 without the use of Boolean negation, in other words, to apply the technique of correspondence analysis to FDE directly, without recourse to BD+. There is a full paper on correspondence analysis for BD+: Petrukhin, Yaroslav, “Correspondence analysis for first degree entailment”, Logical Investigations, 22(1) (2016): 108-124.

Completeness via correspondence for extensions of first degree entailment...

Thank you for attention!

Completeness via correspondence for extensions of first degree entailment...