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In article <1992Apr11.130906.796@gacvx2.gac.edu> 
logic@gacvx2.gac.edu writes:

>In article <1992Apr10.161709.5397@vax.oxford.ac.uk>, 
>loader@vax.oxford.ac.uk (R Loader) writes:

>>In article <1992Apr8.100146.689@gacvx2.gac.edu>, 
>>logic@gacvx2.gac.edu writes:

>>>In article <1992Apr7.124948.5325@vax.oxford.ac.uk>, 
>>>loader@vax.oxford.ac.uk (R Loader) writes:

RL:
>>>> S4 and S5 (and most other common modal logics), are subsystems of classical
>>>> (non-modal) logic extended with the axiom: A iff \box A.

>>>> [...]

>>>> Ralph Loader,
>>>> Mathematical Institute, Oxford.

AH:
>>> S4 and S5 are supersystems, not subsystems, of classical logic..  Neither
>>> contain the axiom "A iff \box A", both contain the axiom "if \box A then A".
>>> 
>>> Al Hart

RL:
>> Sorry, My original posting was ambiguous: I meant ". . . are subsystems
>>   of the logic obtained by adding A iff \box A to classical logic"
>> --
>> Ralph Loader, Oxford.

This is exactly the way I originally interpreted your statement.

AH:
>I see the ambiguity now - missed it during my first post..  however, not having
>extracted your first post, i'm not sure i see the point.  extending classical
>logic with the axiom A iff \box A obviously makes \box completely superfluous.
>
>confused,
>al hart

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.


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