A.1. Basic concepts
| Concept | Negative definition | Positive definition |
|
φ is entailed by Γ:
Γ ⊨ φ |
There is no logically possible world in which φ is false while all members of Γ are true. | φ is true in every logically possible world in which all members of Γ are true. |
|
φ and ψ are (logically) equivalent :
φ ≃ ψ |
There is no logically possible world in which φ and ψ have different truth values. | φ and ψ have the same truth value as each other in every logically possible world. |
|
φ is a tautology:
⊨ φ (or ⊤ ⊨ φ) |
There is no logically possible world in which φ is false. | φ is true in every logically possible world. |
|
φ is inconsistent with Γ:
Γ, φ ⊨ (or Γ, φ ⊨ ⊥) |
There is no logically possible world in which φ is true while all members of Γ are true. | φ is false in every logically possible world in which all members of Γ are true. |
|
Γ is inconsistent:
Γ ⊨ (or Γ ⊨ ⊥) |
There is no logically possible world in which all members of Γ are true. | In every logically possible world, at least one member of Γ is false. |
|
φ is absurd:
φ ⊨ (or φ ⊨ ⊥) |
There is no logically possible world in which φ is true. | φ is false in every logically possible world. |
|
Σ is rendered exhaustive by Γ:
Γ ⊨ Σ |
There is no logically possible world in which all members of Σ are false while all members of Γ are true. | At least one member of Σ is true in each logically possible world in which all members of Γ are true |