I'm trying to understand propositional logics and the concepts of entailment, but I'm struggling. The concepts don't seem to be difficult in theory, but are very strange-looking when examined. For example, I understand that the formal definition of entailment is that a ⊨ b iff M(a) ⊆ M(b).
However, the first example that my textbook provides is that false ⊨ true but true ⊭ false. I'm struggling to understand how false can entail true.
1 Answer
Natural language: $a$ entails $b$ if, whenever $a$ is true, $b$ is true.
Models: $a$ entails $b$ if every model of $a$ is a model of $b$ - that is, if $M(a)\subseteq M(b)$.
So think about "$\perp$" ("false"). When is $\perp$ true? Or, what is the set of models of $\perp$?
It may be easier to think about the following set-theoretic problem first:
Why is $\emptyset\subseteq A$ for every set $A$?
