# Re: The limit of all computations

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On 23 May 2012, at 07:21, Russell Standish wrote:```
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```On Tue, May 22, 2012 at 09:56:24AM -0500, Joseph Knight wrote:
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On Tue, May 22, 2012 at 7:36 AM, Stephen P. King <stephe...@charter.net >wrote:
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```On 5/21/2012 6:26 PM, Russell Standish wrote:

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Yes, that is the usual meaning. It can also be written (DP or not COMP).
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"=>" = "or not"]

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Actually "a implies b" is defined as "not a or b".

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Whoops! (#>.<#)
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To be sure I usually use "->" for the material implication, that is "a -> b" is indeed "not a or b" (or "not(a and not b)").
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The IF ... THEN used in math is generally of that type.

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I use a => b for "from a I can derive b, in the theory I am currently considering".
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For any theory having the modus ponens rule, we have that "a -> b" entails (yet at another meta-level) "a => b". This should be trivial. For many quite standard logics, the reciprocal is correct too, that is: "a = > b" entails "a -> b". This is usually rather hard to prove (Herbrand or deduction theorem). It is typically false in modal logic or in many weak logics. For example the normal modal logics (those having Kripke semantics, like G, S4, ...) are all close for the rule a => Ba, but virtually none can prove the formula a -> Ba. This is a source of many errors.
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Simple Exercises (for those remembering Kripke semantics):
1) find a Kripke model falsifying "a -> Ba".
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2) explain to yourself why "a => Ba" is always the case in all Kripke models.
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I recall that a Kripke model is a set (of "worlds") with a binary relation (accessibility relation). The key is that Ba is true in a world Alpha is a is true in all worlds Beta such that (Alpha, Beta) is in the accessibility relation.
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A beginners course in logic consists in six month of explanation of the difference between "a -> b" and "a => b", and then six month of proving them equivalent (in classical logic).
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"a => b" is often written:

a
_

b

Like in the modus ponens rule

a   a -> b
________

b

Bruno

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Visiting Professor of Mathematics      hpco...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au
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