| 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 |