Definition:Tableau Proof (Propositional Tableaus)

Definition
Let $\mathbf H$ be a set of WFFs of propositional logic.

Let $\mathbf A$ be a WFF.

A tableau proof of $\mathbf A$ from $\mathbf H$ is a tableau confutation of $\mathbf H \cup \left\{{\neg \mathbf A}\right\}$.

This definition also applies when $\mathbf H = \varnothing$.

Then a tableau proof of $\mathbf A$ is a tableau confutation of $\left\{{\neg \mathbf A}\right\}$.

If there exists a tableau proof of $\mathbf A$ from $\mathbf H$, one can write:
 * $\mathbf H \vdash_{\mathrm{PT}} \mathbf A$

Specifically, the notation:
 * $\vdash_{\mathrm{PT}} \mathbf A$

means that there exists a tableau proof of $\mathbf A$.

Proof System
Tableau proofs form a proof system $\mathrm{PT}$ for the language of propositional logic $\mathcal L_0$.

It consists solely of axioms, in the following way:


 * A WFF $\mathbf A$ is a $\mathrm{PT}$-axiom iff there exists a tableau proof of $\mathbf A$.

Likewise, we can define the notion of provable consequence for $\mathbf{PT}$:


 * A WFF $\mathbf A$ is a $\mathrm{PT}$-provable consequence of a collection of WFFs $\mathbf H$ if there exists a tableau proof of $\mathbf A$ from $\mathbf H$.

Although formally $\mathrm{PT}$ has no rules of inference, the rules for the definition of propositional tableaus can informally be regarded as such.