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Logical Implication or Entailment

Consider

$X ⊨ y$

where $X$ represents some set of premises and y represents the conclusion. This simply means that the conjuction of all the premises entails the conclusion. We say that $X ⊨ y$ if and only if all the models of $X$ are models of $y$ .

To show $X ⊨ y$ , show that $X \Rightarrow y$ is a tautology. We call a tautology of the form $A ⊨ B$ a Logical Implication.

In predicate calculus, we use $⊢$ to denote deduction

$\nabla ⊢ Q$

where $\nabla$ represents the set of assumptions and $Q$ represents the conclusion. This expression reads " $Q$ is deduced from $\nabla$ ." If $\nabla = \emptyset$ , often denoted $⊢ Q$ , then it is call a proof. That is $Q$ is deduced soley from the axioms.

(FirstOrderMathematicalLogicAngeloMargaris)

See LogicNotes

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-where $X$ represents some set of premises and y represents the conclusion. This simply means that the conjuction of all the premises entails the conclusion. We say that  $X ⊨ y$  if and only if all the models of  $X$  are models of  $y$ .
+where $$X$$ represents some set of premises and y represents the conclusion. This simply means that the conjuction of all the premises entails the conclusion. We say that  $X ⊨ y$  if and only if all the models of  $X$  are models of  $y$ .

Diff for "LogicalImplication"

Logical Implication or Entailment