Witt

G = God exists, <> means 'it is possible that', [] means 'it is necessary that'

Argument 1.

Axiom 1. <>~G
It's possible that God does not exist.

Axiom 2. <>G->G
If, God exists is possible, then it is true.

1. [](<>G->G)

by, necessitation on Axiom 2
(If P is an axiom or theorem then []P is a theorem)

2. [](~G->~<>G)

by: 1, p->q.<->.~q->~p

3. [](~G->[]~G)

by: 2, ~<>p<->[]~p

4. <>~G-><>[]~G

by: 3, [](p->q).->.<>p-><>q

5. <>~G->[]~G

by: 4, <>[]p<->[]p

6. []~G

by: Axiom 1, 5, Modus Ponens

7. ~G

by: 6, []p->p

Q.E.D.


Argument 2.

Axiom 1. <>~G
It's possible that, God does not exist.

Axiom 2. G->[]G
If God exists, then necessarily God exists.

1. [](G->[]G)

by, necessitation on Axiom 2

2. [](~[]G->~G)

by: 1, p->q.<->.~q->~p

3. [](<>~G->~G)

by: 2, ~[]p<-><>~p

4. []<>~G->[]~G

by: 3, [](p->q).->.[]p->[]q

5. <>~G->[]~G

by: 4, []<>p<-><>p

6. []~G

by: 5, Axiom1, Modus Ponens

7. ~G

by: 6, []p->p

Q.E.D.


Argument 3.

Axiom 1. <>~G
It's possible that, God does not exist.

Axiom 2. <>G->[]G
If it's possible that God exists then it's necessary.

1. ~[]G->~<>G

by: Axiom1, p->q.<->.~q->~p

2. <>~G->[]~G

by: 1, ~[]p<-><>~p, ~<>p<->[]~p

3. []~G

by: 2, Axiom1, Modus Ponens

4. ~G

by: 3, []p->p

Q.E.D.

What do you think?

Witt

Wiploc

Why would you say this? You could start with the axiom that everything that exists is necessary, and reason from there; but you didn't call it an axiom. You treated it as self-evident. It doesn't seem self evident to me. I think I could have ordered the pork chop for dinner. I think the South could have won the civil war. I think lots of things happen that are iffy or optional, that are unnecessary.
crc

Witt

Originally posted by Witt

(If P is an axiom or theorem then []P is a theorem)
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wiploc: Why would you say this? You could start with the axiom that everything that exists is necessary, and reason from there;

(If P is an axiom or theorem then []P is a theorem), is a standard rule of inference in Modal Logic.

The axioms of logic are not factual truths, rather they are asumed to be analytic truths.

Even if we claim the axiom 'necessarily everything exists', we can still describe objects that do not exist. For example: the present king of France, that which is not equal to itself, the Russell class, the whole number between 1 and 2, etc.
God is a described object. If 'God' is defined by a contradictory predicate then God does not exist.
How do we know that a particular definition, of God is not contradictory?

wiploc: but you didn't call it an axiom. You treated it as self-evident. It doesn't seem self evident to me.

x exists for all x, is an implicit axiom of classical first order logic, imo.

x exists, iff, x=x.
and
x=x for all x, is an axiom of this logic.

wiploc: I think I could have ordered the pork chop for dinner. I think the South could have won the civil war. I think lots of things happen that are iffy or optional, that are unnecessary.
crc

I agree, things that happen are contingent situations which could have been otherwise.
Of course, all contingent truths..those dependent on empirical situations are non-necessary.
God exists is analytic, ie. non-synthetic, if true at all.

x exists, is analytic for all x.
But, God is not a value of x. God is a described object which may or may not exist.

Witt

Wiploc

I thought modal logic dealt with necessary beings and contingent beings; but if P necessarily implies []P, then there can be no contingent beings.
crc

Witt

wiploc:
I thought modal logic dealt with necessary beings and contingent beings; but if P necessarily implies []P, then there can be no contingent beings.
crc

P represents a proposition..a statement that is either true or false.

What is: necessary, possible, contingent, factual, etc., is its truth not a being.

"..but if P necessarily implies []P, then there can be no contingent beings."

I would rather say, if P necessarily implies []P, then P is not a contingent truth. That is to say, P must be necessarily true or necessarily false.

Witt

Thomas Ash

Could you please justify Stage 5 in argument 1:

You seem to be saying that as it's possible that God doesn't exist (Axiom 1, which sounds reasonable enough), God necessarily doesn't exist.
Well, by that logic, as it's possible that there's buried treasure down the back of my sofa, there is buried treasure down the back of my sofa.
Wait, I'm bunking off work early to head home to my shovel...

Witt

Thomas Ash: A problem
Could you please justify Stage 5 in argument 1:

quote:
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5. <>~G->[]~G
--------------------------------------------------------------------------------

Thomas: You seem to be saying that as it's possible that God doesn't exist (Axiom 1, which sounds reasonable enough), God necessarily doesn't exist.

That's correct, if we also accept axiom 2. G->[]G, if God exists then necessarily God exists.

Thomas: Well, by that logic, as it's possible that there's buried treasure down the back of my sofa, there is buried treasure down the back of my sofa.
Wait, I'm bunking off work early to head home to my shovel...

Before you get that shovel, you need to admit as an additional axiom: 'there's buried treasure down the back of my sofa' implies necessarily 'there's buried treasure down the back of my sofa'.
This is false even if its is true that there's treasures there.
I doubt that you will accept that axiom.

The two axioms might make sense for God but they certainly do not make sense for your treasure.

Witt

Witt

Re: A problem
Thomas Ash: A problem
Could you please justify Stage 5 in argument 1:

quote:
--------------------------------------------------------------------------------
5. <>~G->[]~G
--------------------------------------------------------------------------------

Thomas: You seem to be saying that as it's possible that God doesn't exist (Axiom 1, which sounds reasonable enough), God necessarily doesn't exist.

That's correct, if we also accept axiom 2. G->[]G, if God exists then necessarily God exists.
-------------------------------------

Oops! I forgot to justify stage 5 in argument 1.

4. <>~G -> <>[]~G

5. <>~G -> []~G

by: 4, <>[]p <-> []p


<>[]p <-> []p is a theorem of standard modal logic eg. (S5).

If we replace <>[]~G with []~G, in 4. <>~G -> <>[]~G we get 5. <>~G -> []~G.

Witt

Thomas Ash

That's what I don't get. Could you please explain, to someone not fully versed in modal logic, why this axiom is accepted. It seems to say that if something could perhaps be necessary, it is necessary. And the point of my sofa example was that anything could perhaps be a necessary truth, but just happens not to be (regrettably ...)

Witt

Witt:
<>[]p <-> []p is a theorem of standard modal logic eg. (S5).

Thomas:
That's what I don't get. Could you please explain, to someone not fully versed in modal logic, why this axiom is accepted. It seems to say that if something could perhaps be necessary, it is necessary.

Not quite, it says that: it is possible that (p is necessary) iff p is necessary.

No factual truths are necessary. There are no contingent necessities.
No tautologies are factual. There are no factual theorems.

Necessity is true only for: deductive truths, analytic truths, apriori truths, tautologies. Otherwise it is contardictory.
Possibility is false only for contradictions otherwise it is true.

<>([]p) is read, it is possible that p is necessarily true.

If p is factually true, []p is contradictory and <>[]p is contradictory.
If p is factually false, []p is contradictory and <>[]p is contradictory.
If p is tautologous []p is tautologous and <>[]p is tautologous.
If p is contradictory then []p is contradictory and <>[]p is contradictory.

In every case <>[]p is equivalent to []p.

Thomas:
And the point of my sofa example was that anything could perhaps be a necessary truth, but just happens not to be (regrettably ...)

[](there is buried treasure down the back of my sofa) is contradictory, even if 'there is buried treasure down the back of my sofa', is factually true.

Whatever just happens to be true is not necessarily true.

Witt