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Version 4 |
| In logic, an \emph{inference rule} is a rule whereby one may correctly |
In logic, an \emph{inference rule} is a rule whereby one may correctly |
| draw a conclusion from one or more premises. For example, the law of |
draw a conclusion from one or more premises. For example, the law of |
| the contrapositive allows one to conclude a statement of the form |
the contrapositive allows one to conclude a statement of the form |
| \[ \neg Q \Rightarrow \neg P \] |
\[ \neg Q \Rightarrow \neg P \] |
| from a premise of the form |
from a premise of the form |
| \[ P \Rightarrow Q. \] |
\[ P \Rightarrow Q. \] |
| Here, `$P$' and `$Q$' are propositional variables, which can stand for |
Here, `$P$' and `$Q$' are propositional variables, which can stand for |
| arbitrary propositions. A popular way to indicate applications of rules |
arbitrary propositions. A typical application of the law of |
| of inference is to list the premises above a line and write the |
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| conclusions below the line. For instance, we might indicate the law |
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| of the contrapositive thus: |
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| \[ |
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| {P \Rightarrow Q \over \neg Q \Rightarrow \neg P} |
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| \] |
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| A typical application of the law of |
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| contrapositive would be to conclude "If my clothes are dry, then it is not |
contrapositive would be to conclude "If my clothes are dry, then it is not |
| raining", from "If it rains, then my clothes will be wet." which could be |
raining", from "If it rains, then my clothes will be wet." (In this |
| expressed as follows using the notation described above: |
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| \[ |
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| {\hbox{If it rains, then my clothes will be wet.} \over |
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| \hbox{If my clothes are dry, then it is not raining.}} |
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| \] |
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| (In this |
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| instance, $P$ is ``It is raining'' and $Q$ is ``My clothes are dry''. |
instance, $P$ is ``It is raining'' and $Q$ is ``My clothes are dry''. |
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| An important feature of rules of inference is that they are purely formal, |
An important feature of rules of inference is that they are purely formal, |
| which means that all that matters is the form of the expression; |
which means that all that matters is the form of the expression; |
| meaning is not a consideration in applying a rule of inference. |
meaning is not a consideration in applying a rule of inference. |
| Thus, the following are equally valid applications of the rule of |
Thus, the following are equally valid applications of the rule of |
| the contrapositive: |
the contrapositive: |
| \[ |
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| {\hbox{If the jabberwocky is mimsy, then the toves blithe.} \over |
From "If the jabberwocky is mimsy, then the toves blithe", conclude |
| \hbox{If the toves are not blithing, then the jabberwocky is not mimsy.}} |
if "The toves are not blithing, then the jabberwocky is not mimsy". |
| \] |
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| \medskip |
From "If my cat has a tail, then my cat is a dog", conclude |
| \[ |
"If my cat is not a dog, then my cat does not have a tail". |
| {\hbox{If my cat has a tail, then my cat is a dog.} \over |
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| \hbox{If my cat is not a dog, then my cat does not have a tail.}} |
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| \] |
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| In the first example, the statements are nonsense and in the second |
In the first example, the statements are nonsense and in the second |
| example, the statements are false, but this doesn't matter --- both |
example, the statements are false, but this doesn't matter --- both |
| examples constitute valid apllications of the rule of the contrapositive. |
examples constitute valid apllications of the rule of the contrapositive. |
| Of course, in order to draw valid conclusions, we need to start with |
Of course, in order to draw valid conclusions, we need to start with |
| valid premises, but the point of these examples is clarify the |
valid premises, but the point of these examples is clarify the |
| distinction between valid statements and valid applications of |
distinction between valid statements and valid applications of |
| rules of inference. |
rules of inference. |
