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\section{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 $y$.\bigskip

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.\bigskip

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

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LogicalImplication (last edited 2020-02-02 17:44:06 by scot)