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Consider Consider 
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[[latex2($$X \models y$$)]] $$$X \models y$$$
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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 entail the conclusion. We say that [[latex2($X \models y$)]] if and only if all the models of [[latex2($X$)]] are models of [[latex2($y$)]]. 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 [[latex2($$X \models y$$)]], show that [[latex2($$X \Rightarrow y$$)]] 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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We call a tautology of the form [[latex2($$A \models B$$)]] a Logical Implication. In predicate calculus, we use $$\vdash$$ to denote deduction
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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. (Update this!!!) $$$\nabla \vdash Q$$$

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