I have these two statements:
False $\models$ True
Reads as : False logicially entails True if all models that evaluate False to True also evaluate True to True.
True $\models$ False
Reads as : True logically entails False if all models that evaluate to True also evaluate False to True.
If my understanding of the concept of "Entails" is correct, then both of these are incorrect because False can never be evaluated to True. Am I correct in my thinking, or am I missing something? This seems like a trick question, so I'm second guessing myself.
$\endgroup$1 Answer
$\begingroup$You are partly correct.
"True $\models$ False" is indeed false: every model makes "true" true, but no model makes "false" true, so every model provides a counterexample.
However, since no model makes "false" true, "False $\models$ True" is actually true! It's an instance of vacuous implication: think of it as being true for the same reason the statement $$\mbox{"If $0=1$, then I'm the president"}$$ is true.
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