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I am studying predicate calculus on some lecture notes on my own. I have a question concerning a strange rule of inference called the Thinning Rule which is stated from the writer as the third rule of inference for the the formal system K$(L)$ (after Modus Ponens and the Generalisation Rule):

TR) $ $ if $\Gamma \vdash \phi$ and $\Gamma \subset \Delta$, then $\Delta \vdash \phi$.

Well, it seems to me that TR is not necessary at all since it is easily proven from the very definition of formal proof (without TR, of course). I am not able to see what is the point here.

The Notes are here http://www.maths.ox.ac.uk/system/files/coursematerial/2011/2369/4/logic.pdf (page 14-15)

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    When I studied predicate calculus, such a rule was never mentioned...2012-09-14
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    Maybe the thinning rule should be applied to itself...2012-09-14
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    There is a difference between predicate calculus and propositional calculus, as hinted at by question [1 (e) here](http://www.maths.ox.ac.uk/system/files/attachments/b0109.pdf)2012-09-14
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    @Henry This is exactly the examination paper of the lectures notes I am reading. So what is the answer?2012-09-15
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    How would you demonstrate $\alpha, \beta \vdash \alpha$ without this rule? Certainly $\alpha \vdash \alpha$ can be demonstrated.2012-09-15
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    Sorry for being insistent, @Henry, but I would appreciate a lot if you could give me just the answer. It would be very kind of you.2012-09-16
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    @CarlMummert, It depends on what definition you want to choose for $\vdash$.2012-09-16
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    @user35549: exactly. Once it is out in the open that $\vdash$ might be inductively defined, rather than just meaning "there is a proof", the role of the "thinning rule" becomes much more clear. Peter Smith has commented in more detail in his answer.2012-09-17

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