Define a proof system for propositional logic that uses truth-tables

Question

I have been struggling with this question, and what I'm being asked to do.

Defining a truth-table is (I think) fine:

For any set of formulas Γ and formula φ, let us call the atoms of Γ, φ the sentence letters that appear either in φ or some formula in Γ.

A truth-table for a set of atoms P1,..., Pn is a table that has one row for each possible interpretation of the atoms P1,..., Pn .

Given a truth-table for the atoms of Γ, φ, and given a formula ψ which is either φ or some formula in Γ, the value of ψ at a given row of the table is the truth-value of ψ on interpretations that corresponds to that row.

But then I am asked to "define the proof system by stipulating when Γ |- φ is the case", and I'm unsure how to go about doing this.

Answer 1

If we have $\Gamma = \{ \gamma_1, \ldots, \gamma_k \}$ and $p_1, \ldots, p_n$ are the atoms occurring into some $\gamma_i$ or $\varphi$, we have to build the truth table with $2^n$ rows : one for each possible interpretation of the atoms $p_1,\ldots, p_n$ and $k+1$ rows : one for each formula.

Then we have to define :

$\Gamma \vdash \varphi$

iff, for each row in the t-t where all the formulae $\gamma_i$ are t, also $\varphi$ is t.