Therefore

The word therefore and its associated symbol ( ∴ ) are used in logic to indicate a relationship of logical consequence, whether that be a syntactic provability, semantic implication or merely intended entailment.

To say Γ ∴ φ is to say that a set of formulae, Γ, logically entails some other formula, φ. It may mean that φ is provable from Γ, that φ follows from Γ, or both. It may also mean that φ is the expected consequence of Γ, or that φ follows from Γ non-deductively.

Provability, or syntactic conseqence

The relationship of provability, Γ ⊢ φ refers to situations where a proof can be constructed in a logical system to demonstrate that, for reasons of syntax, Γ ∴ φ.

Implication, or semantic consequence

The relationship of implication, Γ  φ refers to situations where we can semantically demonstrate Γ ∴ φ because φ actually follows from the premises of Γ.

Intended entailment

Unlike the fairly strict relations of provability and implication, the symbol ∴ is sometimes used by logicians to present arguments or argument forms that have not yet been proven either syntactically or semantically.

Similarly, the symbol ∴ may be used to note conclusions of non-deductive reasoning, where neither implication nor provability appropriately describe the relationship between the premises and the conclusion.