Logic, "and" operator between a set of formulas and a formula
A set $S$ of formulas (propositional logic) is consistent if there is a truth assignment under which all formulas in $S$ are true, in other words if there is a model of $S$. Alternatively, $\{\beta_1, \beta_2, ...,  \beta_n \}$ is consistent, iff $\beta_1 \wedge \beta_2 \wedge ... \wedge \beta_n$ is satisfiable.

You could just find a truth assignment such that $\alpha \wedge \beta_1 \wedge \beta_2 \wedge \beta_3$ is true. If the number of variables is small then you can check it by hand by creating a truth-table, or use Method of analytic tableaux. Also, note that the Satisfiability problem (SAT) is NP complete.

You can also use a resolution procedure to show that a formula is unsatisfiable (inconsistent).