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In logic, a logical matrix is a set of truth values, some of which are distinguished, along with (at least) two operations on the truth values. It provides semantics for a propositional logic.

Definition
A logical matrix is a system $$\mathfrak{M} = \langle M, D \rangle$$ where $M$, typically but not necessarily an algebra, is a set of elements called truth values, $$D \subseteq M$$ is a subset of the truth values which are called designated. $$\mathfrak{M}$$ may also include a collection of operations on $M$, especially if $M$ is not an algebra.

Value Assignments and Validity
A value assignment $v$ of $$\mathfrak{M} = (M, D, f, g)$$ is a function $v$ from the formulas of a propositional logic $L$ to $M$ such that for any formulas $$\varphi$$ and $$\psi$$, $$v(\varphi \to \psi) = f(\varphi, \psi)$$ and $$v(\neg \varphi) = g(\varphi)$$.

An argument consisting of a set of formulas $$\Gamma$$ (called premises) and a formula $$\varphi$$ (called the conclusion) is said to be valid in $$\mathfrak{M}$$, written $$\Gamma \models_\mathfrak{M} \varphi$$, if any value assignment which assigns a distinguished truth value to each formula of $$\Gamma$$ also assigns a distinguished value to $$\varphi$$.

A formula $$\varphi$$ is valid in $$\mathfrak{M}$$, written $$\models_\mathfrak{M} \varphi$$, if for any any value assignment of $$\mathfrak{M}$$, the value assigned to $$\varphi$$ is distinguished.

If a formula is a valid in a logical matrix $$\mathfrak{M}$$ if and only if it is a theorem of a propositional logic $L$, i.e., $$\vdash_L \varphi \iff \models_\mathfrak{M} \varphi$$ for all $$\varphi$$, then $$\mathfrak{M}$$ is said to be characteristic of $L$. If a matrix $$\mathfrak{M}$$ is characteristic of some logic, then it is said to be completely axiomatisable.

Classical Propositional Logic
A characteristic matrix of classical propositional logic is $\left( \{0, 1\},\ \{1\},\ f(a, b) = \begin{cases} 0,\ a = 1 \text{ and } b = 0 \\ 1,\ \text{ otherwise} \end{cases},\ g(a) = \begin{cases} 0,\ a = 1 \\ 1,\ a = 0 \end{cases} \right)$. Note that this matrix provides a semantics for classical propositional logic which does not rely on the presence of any additional structure associated with the set of truth values such as a partial order.