Semantically

Logic

Formal logic aims to determine whether arguments are deductively valid, that is, whether the premises entail the conclusion. Entailment can be defined in terms of syntax or in terms of semantics. Syntactic entailment, expressed with the symbol ${\displaystyle \vdash }$, relies on rules of inference, which can be understood as procedures to transform premises and arrive at a conclusion. These procedures only take the logical form of the premises on the level of syntax into account and ignore what meaning they express. Semantic entailment, expressed with the symbol ${\displaystyle \vDash }$, looks at the meaning of the premises, in particular, at their truth value. A conclusion follows semantically from a set of premises if the truth of the premises ensures the truth of the conclusion, that is, if any semantic interpretation function that assigns the premises the value true also assigns the conclusion the value true.