Generalized Propositional Logics

2021-09-11 128 阅读4分钟

Table of Contents

1. Syntax of L_CON
- 1.1. Abbreviations:
- 1.2. Specification:
2. Extensional Semantics M of L_CON
- 2.1. Abbreviations:
- 2.2. Specification:
3. Variant Semantics M, L, H, K for L_CON

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

1 Syntax of L_CON

1.1 Abbreviations:

L: language;
CON: Set of all Connectives in language L_CON;
CON_n: 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:

L_CON = A × F
A = VAR × CON × PAR
CON = CON₀ × CON₁ × … × CON_n
∇ ∈ CON₁ ; º ∈ CON₂ ;
AF = VAR
F ⊆ A^*
F = AF | ∇ F | F º F

2 Extensional Semantics M of L_CON

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

M_L_CON := 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 L_CON, which are formulars in L_CON 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.
Model_A = {v; v |=_M A}; CounterModel_A = {v; V |\=_M A}.
|=_M A ⇔ ∀ v (v^*(A) = T) ⇔ ∀ v (v |=_M A) ⇔ CounterModel_A = ∅
|\=_M A ⇔ Countermodel ≠ ∅
MT = {A; |=_M A}
M is well-defined ⇔ MT ≠ ∅

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

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.