On Wed, May 23, 2012 at 04:41:56PM +0200, Bruno Marchal wrote:
> 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".

Actually, thinking about your thesis, I don't recall you ever once
using the symbol =>. Instead, you tend to write


I do appreciate the distinction, though!

> 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.

Why is a => Ba true in Kripke models? Surely, it is possible for a to
be true, yet false in some successor world?


Prof Russell Standish                  Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics      hpco...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au

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