Table of Contents
A propositional language, denoted as LCON, consists of its Syntax and its Semantics.
1 Syntax of LCON
1.1 Abbreviations:
- L: language;
- CON: Set of all Connectives in language LCON;
- CONn: Set of Connectives with n arguments, n ≥ 0;
- A: Alphabet;
- F: Set of all Formulars;
- AF: Set of all Atomic Formulars;
- VAR: Set of all Variables;
- PAR: Set of Parenthesis;
1.2 Specification:
- LCON = A × F
- A = VAR × CON × PAR
- CON = CON0 × CON1 × … × CONn
- ∇ ∈ CON1 ; º ∈ CON2 ;
- AF = VAR
- F ⊆ A*
- 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
- 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
- Step 2: we define notions of a truth assignment and its extension;
- Step 3: we define notions of satisfaction, model, counter model;
- Step 4: we define notions tautology under the semantics M.
2.1 Abbreviations:
- LV: Set of Logical Values;
- T: designated Logical value for True;
- MT: Set of all tautologies in LCON, which are formulars in LCON that are true for all truth assignments.
- v: truth assignment function.
- v*: truth extension function.
2.2 Specification:
- LV; T ∈ LV; Definition for all elements in CON.
- v: AF → LV; v*: F → LV.
- v |=M A ⇔ v*(A) = T; v |\=M A ⇔ v*(A) ≠ T.
- ModelA = {v; v |=M A}; CounterModelA = {v; V |\=M A}.
- |=M A ⇔ ∀ v (v*(A) = T) ⇔ ∀ v (v |=M A) ⇔ CounterModelA = ∅
- |\=M A ⇔ Countermodel ≠ ∅
- MT = {A; |=M A}
- M is well-defined ⇔ MT ≠ ∅
3 Variant Semantics M, L, H, K for LCON
3.1 Classical Semantics M.
- CON := {¬, ∩, ∪, ⇒, ⇔}.
- LV := {F, T}, where F < T, and T is the designated Logical Value.
- ¬ : LV → LV = (¬ F = T) × (¬ T = F).
- ∩ : LV × LV → LV = min(x, y) for x, y ∈ LV.
- ∪ : LV × LV → LV = max(x, y) for x, y ∈ LV.
- ⇒ : LV × LV → LV = (¬ x ∪ y) for x, y ∈ LV.
- ⇔ : LV × LV → LV = ((x ⇒ y) ∪ (y ⇒ x)) for x, y ∈ LV.
- v: AF → LV.
- v*(a) = v(a) for a ∈ VAR.
- v*(¬ A) = ¬ v*(A).
- v*((A ∩ B)) = v*(A) ∩ v*(B).
- v*((A ∪ B)) = v*(A) ∪ v*(B).
- v*((A ⇒ B)) = v*(A) ⇒ v*(B).
- v*((A ⇔ B)) = v*(A) ⇔ v*(B).
- v*(A) = T ≡ v |=M A.
- v*(A) ≠ T ≡ v |/=M A.
- |=M A ≡ ∀v (v |=M A).
- =|M A ≡ ∀v (v |/=M A).
- T = {A ∈ F: |=M A}.
- C = {A ∈ F: =|M A}.
3.2 Lukasiewicz Semantics L.
- CON := {¬, ∩, ∪, ⇒}.
- LV := {F, ⊥, T}, where F < ⊥ < T, and T is the designated Logical Value.
- ¬ : LV → LV = (¬ F = T) × (¬ T = F) × (¬ ⊥ = ⊥).
- ∩ : LV × LV → LV = min(x, y); for x, y ∈ LV.
- ∪ : LV × LV → LV = max(x, y); for x, y ∈ LV.
- ⇒ : LV × LV → LV = (¬ x ∪ y) if x > y else T; for x, y ∈ LV.
- v: AF → LV.
- v*(a) = v(a) for a ∈ VAR.
- v*(¬ A) = ¬ v*(A).
- v*((A ∩ B)) = v*(A) ∩ v*(B).
- v*((A ∪ B)) = v*(A) ∪ v*(B).
- v*((A ⇒ B)) = v*(A) ⇒ v*(B).
- v*(A) = T ≡ v |=L A.
- v*(A) ≠ T ≡ v |/=L A.
- |=L A ≡ ∀v (v |=L A).
- =|L A ≡ ∀v (v |/=L A).
- LT = {A ∈ F: |=L A}.
- LC = {A ∈ F: =|L A}.
- LT ⊂ T.
3.2.1 L4 Semantics based on L. Difference as following.
- LV := {F, ⊥1, ⊥2, T}, where F < ⊥1 < ⊥2 < T, and T is the designated Logical Value.
- ¬ : LV → LV = (¬ F = T) × (¬ T = F) × (¬ ⊥1 = ⊥1) × (¬ ⊥2 = ⊥2).
3.3 Heyting Semantics H.
- CON := {¬, ∩, ∪, ⇒}.
- LV := {F, ⊥, T}, where F < ⊥ < T, and T is the designated Logical Value.
- ¬ : LV → LV = a ⇒ F; for a ∈ LV.
- ∩ : LV × LV → LV = min(x, y); for x, y ∈ LV.
- ∪ : LV × LV → LV = max(x, y); for x, y ∈ LV.
- ⇒ : LV × LV → LV = T if x ≤ y else y; for x, y ∈ LV.
- v: AF → LV.
- v*(a) = v(a) for a ∈ VAR.
- v*(¬ A) = ¬ v*(A).
- v*((A ∩ B)) = v*(A) ∩ v*(B).
- v*((A ∪ B)) = v*(A) ∪ v*(B).
- v*((A ⇒ B)) = v*(A) ⇒ v*(B).
- v*(A) = T ≡ v |=H A.
- v*(A) ≠ T ≡ v |/=H A.
- |=H A ≡ ∀v (v |=H A).
- =|H A ≡ ∀v (v |/=H A).
- HT = {A ∈ F: |=H A}.
- HC = {A ∈ F: =|H A}.
- HT ⊂ T.
3.4 Kleene Semantics K.
- CON := {¬, ∩, ∪, ⇒}.
- LV := {F, ⊥, T}, where F < ⊥ < T, and T is the designated Logical Value.
- ¬ : LV → LV = (¬ F = T) × (¬ T = F) × (¬ ⊥ = ⊥).
- ∩ : LV × LV → LV = min(x, y); for x, y ∈ LV.
- ∪ : LV × LV → LV = max(x, y); for x, y ∈ LV.
- ⇒ : LV × LV → LV = (¬ x ∪ y); for x, y ∈ LV.
- v: AF → LV.
- v*(a) = v(a) for a ∈ VAR.
- v*(¬ A) = ¬ v*(A).
- v*((A ∩ B)) = v*(A) ∩ v*(B).
- v*((A ∪ B)) = v*(A) ∪ v*(B).
- v*((A ⇒ B)) = v*(A) ⇒ v*(B).
- v*(A) = T ≡ v |=K A.
- v*(A) ≠ T ≡ v |/=K A.
- |=K A ≡ ∀v (v |=K A).
- =|K A ≡ ∀v (v |/=K A).
- KT = {A ∈ F: |=K A}.
- KC = {A ∈ F: =|K A}.
- (KT = ∅) ⊂ T.