[FOM] The characteristic S5 axiom and the ontological argument
blumal at mail.biu.ac.il
Thu Apr 9 17:37:13 EDT 2009
Paul Hollander wrote:
>>> For what the skeleton of the argument proves is that: if it is possible
>>> that God exists necessarily then God exists necessarily, which is
>>> but an
>>> instance of the characteristic S5 axiom.
>> I recall that Godel is regarded as drawing this conclusion on the
>> basis of weaker modal axioms than S5, bolstered with his own
>> second-order special axioms and definitions regarding "essence,"
>> "being God-like," "positive property," etc.
> And, Laureano Luna wrote::
>>> I wonder if Godel noticed that the characteristic S5 axiom:
>>> 'If Pos Nec p then Nec p',
>>> will give him straightaway the skeleton of the ontological proof for
>>> the existence of God; and that what remains is the not easy task to
>>> prove the consistency of 'God exists'.
>>> For what the skeleton of the argument proves is that: if it is
>>> possible that God exists necessarily then God exists necessarily,
>>> which is but an instance of the characteristic S5 axiom.
>>> It could very well be that his suspicion of the characteristic S5 axiom
>> held him back.
> I guess Godel did notice that.
> If you think that, having realizing that, Godel could have solved that
> part of the argument with just one stroke, note that simply "if pos
> nec >p, then nec p" is not sufficient for the argument; it requires
> establishing "nec (if p, then nec p)" to derive "if pos p, then pos
> nec >p". Where "p" is, of course, "God exists".
In vol iii of the collected works of Godel, R. M. Adams reproduction
of the skeleton of Godel's proof is :
(i) N [if (Ex)Gx then N(Ey)Gy]
(ii) if M(Ex)Gx then MN(Ey)Gy
(iii)if M(Ex)Gx then N(Ey)Gy (p.390)
In answer to Paul, going from (ii) to (iii) gives us the characteristic
S5 axiom. Regarding Laureano's comment, Godel takes (i) to be an
immediate consequence of his Axiom 4 which is that existence is a
positive property.(p.403) The theorem is arguably no less intuitive than
the axiom. But in any event the skeleton of the argument requires only
the conditional which is the characteristic S5 axiom.
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