Q:

Propositional Logic: Models/Counter-Models

I conducted an extensive search on Google, Math. StackE and ProofWiki before posting.  
Given the following task: (Given a single specification)
Use truth tables to check if the specification is consistent, and if it is - provide a model of the specification.  If not, provide a counter model.  
I know that a specification is consistent if there is at least one line in the truth table which has all true values for all the propositions.  And I know that a model of a specification is essentially the set of interpretations for the prop.symbols, which give out those 'all-true' rows.  
However, what is a counter model?  How can I construct a counter model, if I have an inconsistent formula?  What purpose do they serve? 
Thank you

A:

If the specification (say $\Gamma$) is consistent, then, as you rightly said, there will be a row of the truth table that assigns 'true' to all $\phi \in \Gamma$. If, however, $\Gamma$ is not consistent, then some formulas $\phi, \psi \in \Gamma$ are such that $\phi \not\equiv \psi$. This means that no row in the truth-table can satisfy both $\phi$ and $\psi$. But since $\phi$ and $\psi$ are members of $\Gamma$, any row of the truth-table will then be a counter-model of $\Gamma$.
For example, suppose $\Gamma = \{P, \lnot P\}$. Since $\Gamma$ contains an inconsistent pair, it's not satisfiable.  Therefore, both assignments of truth-values to P will be counter-models for $\Gamma$, so you could just give the following assignment as an answer: 'true' $\mapsto$ 'P'.