= Logical Implication or Entailment =

Consider 

$$$X \models y$$$

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 $$X \models y$$ if and only if all the models of $$X$$ are models of $$y$$.

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

In predicate calculus, we use $$\vdash$$ to denote deduction

$$$\nabla \vdash Q$$$

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 $$\vdash Q$$, then it is call a proof. That is $$Q$$ is deduced soley from the axioms. 

(FirstOrderMathematicalLogicAngeloMargaris)

See LogicNotes

Back to ComputerTerms