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{{{#!latex \section{Logical Implication or Entailment} |
= Logical Implication or Entailment = |
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| \[ X \models y \] 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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| 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 | $$$X \models y$$$ |
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| 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. }}} |
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. |
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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