Definition:Tableau Proof

Definition
There are two kinds of tableau proof in the study of propositional logic.

Natural Deduction
A tableau proof by natural deduction is a technique for presenting a logical argument in a straightforward, standard form.

A tableau proof is a sequence of lines specifying the order of premises, assumptions, inferences and conclusion in support of an argument.

Each line of a tableau proof has a particular format. It consists of five parts:


 * Line: The line number of the proof. This is a simple numbering from 1 upwards.
 * Pool: The list of all the lines containing the pool of assumptions for the formula introduced on this line.
 * Formula: The formula or statement form introduced on this line.
 * Rule: The justification for introducing this line. This should be the appropriate abbreviation of the proof rule being used to derive this line, for example:
 * 1) P (for Premise);
 * 2) A (for Assumption);
 * 3) $$\implies \mathcal{I}\ $$ (for example)
 * Depends on: The lines (if any) upon which this line directly depends. For premises and assumptions, this field will be empty.

At the end of a tableau proof, the only lines upon which the proof depends may be those corresponding to premises.

Length
The length of a tableau proof is the number of lines it has.

Propositional Tableau
Let $$\mathbf H$$ be a set of premises in the form of WFFs of propositional calculus.

Let $$\mathbf A$$ be a proposition in the form of a WFF of propositional calculus.

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 $$\mathbf H$$, we can write:
 * $$\mathbf H \vdash \mathbf A$$

using the same symbology (and meaning) as logical implication.

Similarly, the notation:
 * $$\vdash \mathbf A$$

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

Since, by definition, a tableau confutation is a finite propositional tableau, it follows that all tableau proofs have a finite number of nodes.