Differences between revisions 3 and 21 (spanning 18 versions)

Logical Implication or Entailment

Consider

$X ⊨ y$

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 $X ⊨ y$ if and only if all the models of $X$ are models of $y$ .

To show $X ⊨ y$ , show that $X \Rightarrow y$ is a tautology. We call a tautology of the form $A ⊨ B$ a Logical Implication.

In predicate calculus, we use $⊢$ to denote deduction

$\nabla ⊢ Q$

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 $⊢ Q$ , then it is call a proof. That is $Q$ is deduced soley from the axioms.

(FirstOrderMathematicalLogicAngeloMargaris)

See LogicNotes

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-  ⇤ ← Revision 3 as of 2003-10-08 23:43:32 → 
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+   ← Revision 21 as of 2020-01-26 23:04:47 → ⇥
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-Deletions are marked like this.
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-= Logical Implication =
+= Logical Implication or Entailment =
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 Consider
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- X |= y
+ $X ⊨ y$
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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  $X ⊨ y$  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  $X ⊨ y$ , show that  $X \Rightarrow y$  is a tautology. We call a tautology of the form  $A ⊨ B$  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  $⊢$  to denote deduction
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-In predicate calculus, we often times use |- to denote logical implication. In this case we must prove that the predicate calculus expression is valid.
+ $\nabla ⊢ Q$ 

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  $⊢ Q$ , then it is call a proof. That is  $Q$  is deduced soley from the axioms. 

(FirstOrderMathematicalLogicAngeloMargaris)

Diff for "LogicalImplication"

Logical Implication or Entailment