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I have $A$ and $B$ that are contingent formulas. I have to prove that $A \land B $ can not be logically true and $A \lor B$ can not be logically false. How to start unfolding these questions? I know that a formula is logically false iff it is not realizable and that a formula is realizable if it is true in all propositional calculus models.

Let $M$ be a model and if $M \models A$ then $M \models B$ because $M \models A \land B$. If $A = \neg B$ then $¬B∧B$ and then M can not be model because $M \models B$ and $M \models \neg B$ can not be true at the same. So $A \land B$ can not be logically true. Is this ok or am I totally in the wrong path?

mathematicsclass23
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    To say that a formula A is contingent means that there is a model M where A is false. If so, what is the truth value of $A \land B$ in M ? – Mauro ALLEGRANZA Sep 09 '20 at 07:38
  • not sure I understand your second paragraph there...maybe I'm just misinterpreting, it's been a while since I was thinking about such things, but it doesn't look right to me.. – Ettore Sep 09 '20 at 08:20
  • For the second case, consider for contradiction the case when $A \lor B$ is logically false. – Mauro ALLEGRANZA Sep 09 '20 at 09:09
  • Does 'logically true' mean true for all valuations of the variables? And 'logically false' mean false for all valuations of the variables? – Doug Spoonwood Sep 09 '20 at 13:44

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