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| 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 $X \models y$ if and only if all the models of $X$ are models of $y$. | 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 $X \models y$ if and only if all the models of $X$ are models of $y$.\bigskip |
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| 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. | 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 |
\section{Logical Implication or Entailment}
Consider
\[
X \models y
\]
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 $X \models y$ 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 $\nabla$ is the set of assumptions and $Q$ is the conclusion is read ''$Q$ 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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