Differences between revisions 12 and 22 (spanning 10 versions)
Revision 12 as of 2007-02-16 02:10:04
Size: 911
Editor: yakko
Comment:
Revision 22 as of 2020-02-02 17:44:06
Size: 925
Editor: scot
Comment:
Deletions are marked like this. Additions are marked like this.
Line 1: Line 1:
[[BR]]

{{{#!latex2
\section{
Logical Implication or Entailment}
= Logical Implication or Entailment =
Line 7: Line 4:
\[
    X \models y
\]
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 $X \models y$ if and only if all the models of $X$ are models of $y$.
Line 12: Line 5:
To show $X \models y$, show that $X \Rightarrow y$ is a tautology. We call a tautology of the form [[latex2($$A \models B$$)]] a Logical Implication. $$$X \models y$$$
Line 14: Line 7:
In predicate calculus, we use [[latex2($$\vdash$$)]] to denote deduction
\[
     \nabla \vdash Q
\]
where $\nabla$ is the set of assumptions and $Q$ is the conclusion is read ''$Q$ 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)
}}}
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)

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

Back to ComputerTerms

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