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:On Tue, May 22, 2012 at 7:36 AM, Stephen P. King <stephe...@charter.net>wrote:On 5/21/2012 6:26 PM, Russell Standish wrote:Yes, that is the usual meaning. It can also be written (DP or notCOMP)."=>" = "or not"]Actually "a implies b" is defined as "not a or b".Whoops! (#>.<#)

`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)").`

The IF ... THEN used in math is generally of that type.

`I use a => b for "from a I can derive b, in the theory I am currently`

`considering".`

`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.`

Simple Exercises (for those remembering Kripke semantics): 1) find a Kripke model falsifying "a -> Ba".

`2) explain to yourself why "a => Ba" is always the case in all Kripke`

`models.`

`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.`

`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).`

"a => b" is often written: a _ b Like in the modus ponens rule a a -> b ________ b Bruno

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