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| Deletions are marked like this. | Additions are marked like this. |
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| = Logical Implication = | = Logical Implication or Entailment = |
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| Consider | Consider |
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| X |= y | $$$X \models y$$$ |
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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 logically implies the conclusion. | 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 $$y$$. |
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| To show X |= y, show that X => y is a tautology. | 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. |
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| Therefore a tautology of the form A |= B is called a Logical Implication. | 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) |
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 $$y$$.
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.
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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