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= Logical Implication = | [[BR]] |
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Consider | {{{#!latex \section{Logical Implication or Entailment} |
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X |= y | 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 $y$.\bigskip |
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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. | 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 |
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To show X |= y, show that X => y is a tautology. Therefore a tautology of the form A |= B is called a Logical Implication. In predicate calculus, we often times use |- to denote logical implication. In this case we must prove that the predicate calculus expression is valid. |
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) |
\section{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 $y$.\bigskip 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 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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