Logical Implication or Entailment

Consider

where X is some set of premises and y is the conclusion. This simply means that the conjuction of all the premises entail the conclusion. We say that latex2($X \models y$) if and only if all the models of latex2($X$) are models of latex2($y$).

To show latex2( $X ⊨ y$ ), show that latex2( $X \Rightarrow y$ ) is a tautology.

We call a tautology of the form latex2( $A ⊨ B$ ) a Logical Implication.

In predicate calculus, we often times use latex2( $⊢$ ) to denote logical implication. In this case we must prove that the predicate calculus expression is valid. (Update this!!!)

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LogicalImplication

Logical Implication or Entailment