Generalized Propositional Logics

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A propositional language, denoted as LCON, consists of its Syntax and its Semantics.

1 Syntax of LCON

1.1 Abbreviations:

  1. L: language;
  2. CON: Set of all Connectives in language LCON;
  3. CONn: Set of Connectives with n arguments, n ≥ 0;
  4. A: Alphabet;
  5. F: Set of all Formulars;
  6. AF: Set of all Atomic Formulars;
  7. VAR: Set of all Variables;
  8. PAR: Set of Parenthesis;

1.2 Specification:

  1. LCON = A × F
  2. A = VAR × CON × PAR
  3. CON = CON0 × CON1 × … × CONn
  4. ∇ ∈ CON1 ; º ∈ CON2 ;
  5. AF = VAR
  6. F ⊆ A*
  7. F = AF | ∇ F | F º F

2 Extensional Semantics M of LCON

all connectives in CON are extensional, whose logical value depend on their factors.

MLCON := LV × T × v × v* × MT

  1. Step 1: given the language L we define a set of logical values and its distinguish value T, and define all logical connectives of L
  2. Step 2: we define notions of a truth assignment and its extension;
  3. Step 3: we define notions of satisfaction, model, counter model;
  4. Step 4: we define notions tautology under the semantics M.

2.1 Abbreviations:

  1. LV: Set of Logical Values;
  2. T: designated Logical value for True;
  3. MT: Set of all tautologies in LCON, which are formulars in LCON that are true for all truth assignments.
  4. v: truth assignment function.
  5. v*: truth extension function.

2.2 Specification:

  1. LV; T ∈ LV; Definition for all elements in CON.
  2. v: AF → LV; v*: F → LV.
  3. v |=M A ⇔ v*(A) = T; v |\=M A ⇔ v*(A) ≠ T.
  4. ModelA = {v; v |=M A}; CounterModelA = {v; V |\=M A}.
  5. |=M A ⇔ ∀ v (v*(A) = T) ⇔ ∀ v (v |=M A) ⇔ CounterModelA = ∅
  6. |\=M A ⇔ Countermodel ≠ ∅
  7. MT = {A; |=M A}
  8. M is well-defined ⇔ MT ≠ ∅

3 Variant Semantics M, L, H, K for LCON

3.1 Classical Semantics M.

  1. CON := {¬, ∩, ∪, ⇒, ⇔}.
  2. LV := {F, T}, where F < T, and T is the designated Logical Value.
  3. ¬ : LV → LV = (¬ F = T) × (¬ T = F).
  4. ∩ : LV × LV → LV = min(x, y) for x, y ∈ LV.
  5. ∪ : LV × LV → LV = max(x, y) for x, y ∈ LV.
  6. ⇒ : LV × LV → LV = (¬ x ∪ y) for x, y ∈ LV.
  7. ⇔ : LV × LV → LV = ((x ⇒ y) ∪ (y ⇒ x)) for x, y ∈ LV.
  8. v: AF → LV.
  9. v*(a) = v(a) for a ∈ VAR.
  10. v*(¬ A) = ¬ v*(A).
  11. v*((A ∩ B)) = v*(A) ∩ v*(B).
  12. v*((A ∪ B)) = v*(A) ∪ v*(B).
  13. v*((A ⇒ B)) = v*(A) ⇒ v*(B).
  14. v*((A ⇔ B)) = v*(A) ⇔ v*(B).
  15. v*(A) = T ≡ v |=M A.
  16. v*(A) ≠ T ≡ v |/=M A.
  17. |=M A ≡ ∀v (v |=M A).
  18. =|M A ≡ ∀v (v |/=M A).
  19. T = {A ∈ F: |=M A}.
  20. C = {A ∈ F: =|M A}.

3.2 Lukasiewicz Semantics L.

  1. CON := {¬, ∩, ∪, ⇒}.
  2. LV := {F, ⊥, T}, where F < ⊥ < T, and T is the designated Logical Value.
  3. ¬ : LV → LV = (¬ F = T) × (¬ T = F) × (¬ ⊥ = ⊥).
  4. ∩ : LV × LV → LV = min(x, y); for x, y ∈ LV.
  5. ∪ : LV × LV → LV = max(x, y); for x, y ∈ LV.
  6. ⇒ : LV × LV → LV = (¬ x ∪ y) if x > y else T; for x, y ∈ LV.
  7. v: AF → LV.
  8. v*(a) = v(a) for a ∈ VAR.
  9. v*(¬ A) = ¬ v*(A).
  10. v*((A ∩ B)) = v*(A) ∩ v*(B).
  11. v*((A ∪ B)) = v*(A) ∪ v*(B).
  12. v*((A ⇒ B)) = v*(A) ⇒ v*(B).
  13. v*(A) = T ≡ v |=L A.
  14. v*(A) ≠ T ≡ v |/=L A.
  15. |=L A ≡ ∀v (v |=L A).
  16. =|L A ≡ ∀v (v |/=L A).
  17. LT = {A ∈ F: |=L A}.
  18. LC = {A ∈ F: =|L A}.
  19. LT ⊂ T.

3.2.1 L4 Semantics based on L. Difference as following.

  1. LV := {F, ⊥1, ⊥2, T}, where F < ⊥1 < ⊥2 < T, and T is the designated Logical Value.
  2. ¬ : LV → LV = (¬ F = T) × (¬ T = F) × (¬ ⊥1 = ⊥1) × (¬ ⊥2 = ⊥2).

3.3 Heyting Semantics H.

  1. CON := {¬, ∩, ∪, ⇒}.
  2. LV := {F, ⊥, T}, where F < ⊥ < T, and T is the designated Logical Value.
  3. ¬ : LV → LV = a ⇒ F; for a ∈ LV.
  4. ∩ : LV × LV → LV = min(x, y); for x, y ∈ LV.
  5. ∪ : LV × LV → LV = max(x, y); for x, y ∈ LV.
  6. ⇒ : LV × LV → LV = T if x ≤ y else y; for x, y ∈ LV.
  7. v: AF → LV.
  8. v*(a) = v(a) for a ∈ VAR.
  9. v*(¬ A) = ¬ v*(A).
  10. v*((A ∩ B)) = v*(A) ∩ v*(B).
  11. v*((A ∪ B)) = v*(A) ∪ v*(B).
  12. v*((A ⇒ B)) = v*(A) ⇒ v*(B).
  13. v*(A) = T ≡ v |=H A.
  14. v*(A) ≠ T ≡ v |/=H A.
  15. |=H A ≡ ∀v (v |=H A).
  16. =|H A ≡ ∀v (v |/=H A).
  17. HT = {A ∈ F: |=H A}.
  18. HC = {A ∈ F: =|H A}.
  19. HT ⊂ T.

3.4 Kleene Semantics K.

  1. CON := {¬, ∩, ∪, ⇒}.
  2. LV := {F, ⊥, T}, where F < ⊥ < T, and T is the designated Logical Value.
  3. ¬ : LV → LV = (¬ F = T) × (¬ T = F) × (¬ ⊥ = ⊥).
  4. ∩ : LV × LV → LV = min(x, y); for x, y ∈ LV.
  5. ∪ : LV × LV → LV = max(x, y); for x, y ∈ LV.
  6. ⇒ : LV × LV → LV = (¬ x ∪ y); for x, y ∈ LV.
  7. v: AF → LV.
  8. v*(a) = v(a) for a ∈ VAR.
  9. v*(¬ A) = ¬ v*(A).
  10. v*((A ∩ B)) = v*(A) ∩ v*(B).
  11. v*((A ∪ B)) = v*(A) ∪ v*(B).
  12. v*((A ⇒ B)) = v*(A) ⇒ v*(B).
  13. v*(A) = T ≡ v |=K A.
  14. v*(A) ≠ T ≡ v |/=K A.
  15. |=K A ≡ ∀v (v |=K A).
  16. =|K A ≡ ∀v (v |/=K A).
  17. KT = {A ∈ F: |=K A}.
  18. KC = {A ∈ F: =|K A}.
  19. (KT = ∅) ⊂ T.