896
Comment:
|
900
|
Deletions are marked like this. | Additions are marked like this. |
Line 1: | Line 1: |
= Logical Implication or Entailment = | [[BR]] |
Line 3: | Line 3: |
{{{#!latex \section{Logical Implication or Entailment} |
|
Line 4: | Line 6: |
{{{#!latex2 | |
Line 9: | Line 10: |
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 $y$. | 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 $y$.\bigskip |
Line 11: | Line 12: |
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. | 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 |
Line 13: | Line 14: |
In predicate calculus, we use [[latex2($$\vdash$$)]] to denote deduction | In predicate calculus, we use $\vdash$ to denote deduction |
Line 17: | Line 18: |
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 $\nabla$ 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. |
Line 19: | Line 20: |
(FirstOrderMathematicalLogicAngeloMargaris) |
\section{Logical Implication or Entailment} Consider \[ X \models y \] 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 $y$.\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 $\nabla$ 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)
See LogicNotes
Back to ComputerTerms