1. With respect to a deductive system $\text{L}$, describes any formula $\phi$ for which a semantic presentation of $\text{L}$ has no countermodels to $\phi$, i.e., for which $\varnothing {\models }_{\text{L}}\phi$ with respect to the semantic consequence relation of $\text{L}$. The richness and variety of semantics for non-classical logics permit a number of distinct explications of this notion, including:

1. 1 $\phi$ is valid when $\phi$ is true in every model

2. 2 $\phi$ is valid when every model assigns a designated value to $\phi$

3. 3 $\phi$ is valid when there is no model in which $\phi$ is false

These conditions are classically equivalent, but are distinct in weaker deductive systems. For example, in a paraconsistent logic whose semantics has truth value gluts (i.e, in which models may assign $\phi$ both truth and falsity), a formula $\phi$ may be both true in every model and yet there may still be a model in which it is evaluated as false (although also true). When a deductive system is sound and complete, validity may also be considered the model-theoretic counterpart of theoremhood, so that a sentence is provable from the empty set of assumptions precisely when the sentence is valid.

2. Where ${\models }_{\text{L}}$ is the semantic consequence relation of a logic $\text{L}$, describes any inference $\text{Γ}{\models }_{\text{L}}\phi$ (or $\text{Γ}{\models }_{\text{L}}\Delta$ in the case of multiple-conclusion logic) in which the premisses and conclusion (or conclusions) are related in an appropriate fashion. When $\text{L}$ is classical logic, the relationship corresponding to validity of an inference $\text{Γ}{\models }_{\text{L}}\phi$ is the necessary preservation of truth from $\text{Γ}$ to $\phi$, i.e., the property that for any model $\mathfrak{M}$, the truth of all formulae $\psi \in \text{Γ}$ in $\mathfrak{M}$ guarantees the truth of $\phi$ in $\mathfrak{M}$. Beyond classical logic, there are other ways of characterizing the validity of an inference. In semantics with multiple truth values, the validity of an inference $\text{Γ}{\models }_{\text{L}}\phi$ may be captured as:

1. 1 the preservation of designated values from $\text{Γ}$ to $\phi$

2. 2 the preservation of non-falsity from $\text{Γ}$ to $\phi$

3. 3 that the degree of truth of $\phi$ exceeds the degree of truth of every $\psi \in \text{Γ}$

Some non-classical logics constrain the notion of validity by imposing additional demands on the relationship between premisses and conclusion (or conclusions), e.g., by requiring that the conjunction of the premisses has a model or by requiring that a valid inference preserves subject-matter. Given soundness and completeness of $\text{L}$, validity of the inference $\text{Γ}{\models }_{\text{L}}\phi$ acts as a semantic analogue of the provability of $\phi$ from a set of premisses $\text{Γ}$.