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In article <1992Apr13.114824.1745@nic.csu.net> 
suchenek@nic.csu.net writes:

>In article <1992Apr12.193331.11012@husc3.harvard.edu>,
>zeleny@zariski.harvard.edu (Mikhail Zeleny) writes: 

MZ:
>> Observe that Ralph says that S4 and S5 are *subsystems* of classical logic
>> extended with the axiom: A iff \box A. In that, he is absolutely correct: a
>> subsystem needn't contain the axiom in question, but only a deductively
>> closed proper subset of its consequences, e.g. those generated by \box A
>> \supset A in the presence of classical laws.  However, I am not sure
>> whether he could do without an additional rule of inference, viz. the Rule
>> of Necessitation:
>> 
>> 		A
>> 	       ___
>> 
>> 	      \box A
>> 
>> which prima facie seems to be required for adequately characterizing S4 and
>> S5, even if all of their theorems can probably be obtained by the above
>> process.

MS:
>How, without the necessitation rule (and scheme K), can you get 
>
>     p \supset \box (\box (p \supset q) \supset (\box p \supset \box q))
>
>(which is a theorem of S4 and S5), without proving too much, e.g.
>
>      p \supset \box p
>
>(which is not a theorem of S4 and S5), where p is a propositional variable?

Easy.  We select a subsystem generated by all axioms of S5 (or S4).  Do you
doubt that these axioms could be obtained from A \equiv \box A ?  In fact,
on the purely syntactic level of abstraction we are considering, we needn't
even concern ourselves with RNec and other such matters.

>Greetings,
>
>Marek Suchenek

regards,
mz

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