Diff for "LogicalImplication"

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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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-  ⇤ ← Revision 6 as of 2006-07-18 23:41:16 → 
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+   ← Revision 20 as of 2010-06-03 19:33:36 → ⇥
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-= Logical Implication or Entailment =
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-Consider
+{{{#!latex
\section{Logical Implication or Entailment}
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-[[latex2($$X \models y$$)]]
+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
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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 [[latex2($X \models y$)]] if and only if all the models of [[latex2($X$)]] are models of [[latex2($y$)]].
+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.\bigskip
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-To show [[latex2($$X \models y$$)]], show that [[latex2($$X \Rightarrow y$$)]] is a tautology.

We call a tautology of the form [[latex2($$A \models B$$)]] a Logical Implication.

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