# Consistent Formula

A collection of formulas is consistent if there is a Boolean valuation in which all the formulas are T

To do a consistency check, we can check that:

- The conjunction of the formulas is satisfiable
- there is a Boolean valuation that maps each formula in the set to T
- there is a Boolean valuation that maps the conjunction of the formulas to T
- the conjunction of the formulas is not a contradiction

We use the Semantic Tableaux usually to prove that a set of formulas is inconsistent.

### Proving Consistent/Inconsistent with Semantic Tableaux

- To prove that an argument is inconsistent, just list out the premises and close the branches.

- premise
- premise

- To prove that a Valid Argument:

- premise
- premise
- negation of the conclusion