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| = Logical Implication = | = Logical Implication or Entailment = |
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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 logically implies the conclusion. | 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($y$)]]. |
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| Therefore a tautology of the form [[latex2($$A \models B$$)]] is called a Logical Implication. | We call a tautology of the form [[latex2($$A \models B$$)]] a Logical Implication. |
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| 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. | 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!!!) |
Logical Implication or Entailment
Consider
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 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!!!)
See LogicNotes
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