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Consider
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[[latex2(X⊨y)]] {{{#!latex2
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 $y$.
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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 [[latex2(A⊨B)]] a Logical Implication.
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To show [[latex2(X⊨y)]], show that [[latex2(X⇒y)]] is a tautology.

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

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

Logical Implication or Entailment

LogicalImplication (last edited 2020-02-02 17:44:06 by 68)