A consequence relation $\models$ (or double turnstile) holding between sets of formulae $\text{Γ}$ and formulae $\phi$ in a language (in multiple-conclusion logic, a relation holding between two sets of formulae $\text{Γ}$ and $\Delta$) that is defined by the semantic properties of $\text{Γ}$ and $\phi$. This is to say that with respect to a deductive system $\text{L}$, whether $\text{Γ}{\models }_{\text{L}}\phi$ obtains is determined by an appropriate semantics for $\text{L}$.

Semantic consequence is typically distinguished from syntactic (or proof-theoretic) consequence on the basis of its appeal to meaning. A proof theory for a system $\text{L}$ generally makes no overt appeal to semantic notions and often can be described in entirely mechanistic terms so that whether $\text{Γ}{⊢}_{\text{L}}\phi$ holds is equivalent to whether a procedure that manipulates the symbols appearing in $\text{Γ}\cup \left\{\phi \right\}$ can be successfully executed (although the method of tableau proof implicitly represents semantic concepts). On the other hand, semantic consequence is governed by the interpretation or meaning that a semantics gives to the formulae of its language.

That $\text{Γ}{\models }_{\text{L}}\phi$ is frequently described as equivalent to validity so that the assertion that whenever each formula in $\text{Γ}$ is assigned some distinguished truth value (or truth values), the formula $\phi$ must also bear some particular truth value (e.g., $\phi$ must be true whenever all formulae in $\text{Γ}$ are true or the degree of truth always exceeds those of the formulae in $\text{Γ}$). In other words, semantic consequence is often defined in terms of truth, a generalization of truth, or a related notion (e.g., non-falsity). However, providing an appeal to truth conditions is not the only way to give an account of meaning, and hence, to define semantic consequence. For example, in the case of proof-theoretic semantics, truth conditions are eschewed in favour of assertability conditions, in which case the meanings of the formulae in $\text{Γ}\cup \left\{\phi \right\}$—and, in turn, whether $\text{Γ}{\models }_{\text{L}}\phi$ holds—can be characterized without the need for truth or similar notions.