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Can somone explain the Modal Ontological arg

kp98
Posts: 729
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8/16/2015 12:46:58 PM
Posted: 1 year ago
Well, in this context 'modal' refers to 'Modal logic' which is an extended form of logic that allows for use of more nuanced terms that the hard-edged 'is' and 'is not' of traditional logic. For instance, 'can be', 'must be', 'possibly not be' and so on are terms belonging to modal logic, but are not part of traditional logic.

Once takem seriously, the ontological argument is now generallt considered to be a word-puzzle that purports to prove the existence of God by exploiting the innate ambiguities of ordinary language, expecially when it comes to hard-to-pin-down terms such as 'exists' and 'conceivable'.

So to summarise, the modal ontological argument is the ontological argument expressed in the terms of modal logic. Use google to find any number of examples of the antological argument in modal (and non-modal) form.

Like all good puzzles the problem is not so much that it's wrong (which is rarely disputed) but in the heated discussions about what the mistake is!
RuvDraba
Posts: 6,033
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8/17/2015 5:44:37 PM
Posted: 1 year ago
At 8/15/2015 10:55:46 PM, UtherPenguin wrote:
title explains itself.

arg is short for argument (couldn't fit the whole word in the title)

Uther, Modal logic is a logic of necessity and possibility. It has many uses, among the most common of which is multi-agent reasoning: Agent A reasoning about agent B reasoning about agent A.

We can write necessity and possibility as:

[] X -- 'square X', means X is necessarily true;
<> X - 'diamond X', means X is possibly true.

These are often thought of as interchangable: X is necessarily true means it is not possible that X is false; while X is possibly true means it is not necessarily true that X is false. If we write not as '!' and implication as => then these can be written as

[] X => !<>!X and
<> X => ![]!X (*)

I've starred the second one for convenience, since we'll use it later.

Some other useful laws of modal reasoning:

[]A => A -- if A is necessarily true, then A is true
[]A => [][]A -- if A is necessarily true, then necessarily, A is necessarily true
<> => []<>A -- if A is possible then A is necessarily possible
If A => B, then [] A => [] B (**)

Finally, we can write '&' for 'and' and 'v' for 'or'.

Let 'G' mean 'God exists'. The modal form of Anselm's ontological argument starts with three premises.

(P1) G => []G -- if God exists, then God is necessarily true, since it's not possible that God ever doesn't exist. Another way of saying it is that a 'maximal being', if it exists, exists everywhere and everywhen.
(P2) [] G v ![]G -- either God is necessarily true or not necessarily true. This is the law of excluded middle.
(P3) <>!G => []<>!G -- if God possibly doesn't exist, then necessarily, God possibly doesn't exist. Let's use the possibility equivalence at (*) to rewrite that as:
(P3a) ![]!!G => []![]!!G; or getting rid of double negatives:
(P3b) ![]G => []![]G -- i,e, if God doesn't necessarily exist, then necessarily, God doesn't necessarily exist.

From these we can derive:

(C4) []G v []![]G -- either God is necessarily true, or necessarily, God is not necessarily true. This is just (P2) and (P3) combined
(C5) []![]G => []! G -- if necessarily, God is not necessarily true, then necessarily, there is no God. This comes from refuting (P1) and using (**)
(C6) []G v []!G -- either God is necessarily true, or necessarily, there is no God. This joins (C4) with (C5)

(P4) ![]!G -- Recall that 'God is possible' <>G can be written as ![]!G (see *.) This statement P4 says 'God is possible', or equally 'God is not impossible'.

Therefore:

(C7) []G - God necessarily exists, from (P4) with (C6)
(C8) G - God exists, since whatever is necessarily true is also true.

Feel wiser?

No?

Me either, and reasoning with modal logic used to be an old research area of mine.

There are several criticisms one can level at this 'proof', but rather than level them here I'll leave this post as-is, in case anything needs explaining.

Happy to pull it apart in a later post. :)

Hope this may be useful.