Freethought & Rationalism Archive

ComestibleVenom · 05-17-2003, 11:36 PM

I think this again goes to prove that modal logic is a toy model of human modal reasoning.

Thomas Ash · 05-19-2003, 01:46 AM

Quote:

Originally posted by 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

Oh, I see. In this case, my only point would be that Argument 1 rests on God being defined as a hypothetically necessary being (which one could argue is not absolutely integral to his definition, though given he is almost always characterised as a necessary, uncaused first cause it probably is.) But in that case, you can't have Axiom 2 without further proof, as the theist would obviously deny that "God could possibly not exist" if they think he necessarily does.

Witt · 05-19-2003, 09:20 AM

Thomas:
Oh, I see. In this case, my only point would be that Argument 1 rests on God being defined as a hypothetically necessary being (which one could argue is not absolutely integral to his definition, though given he is almost always characterised as a necessary, uncaused first cause it probably is.) But in that case, you can't have Axiom 2 without further proof, as the theist would obviously deny that "God could possibly not exist" if they think he necessarily does.

Agreed. The theist needs only to admit that God exists is possibly true to conclude God exists is true, and,
The atheist needs only to admit that God does not exist is possible to conclude that God exists is false.

It seems that we cannot say..<>G & <>~G, is true.

Witt

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05-17-2003, 11:36 PM	#11
ComestibleVenom Veteran Member Join Date: May 2003 Location: Alberta, Canada Posts: 2,320	I think this again goes to prove that modal logic is a toy model of human modal reasoning.

05-19-2003, 09:20 AM	#13
Witt Banned Join Date: May 2003 Location: Toronto Canada Posts: 1,263	Thomas: Oh, I see. In this case, my only point would be that Argument 1 rests on God being defined as a hypothetically necessary being (which one could argue is not absolutely integral to his definition, though given he is almost always characterised as a necessary, uncaused first cause it probably is.) But in that case, you can't have Axiom 2 without further proof, as the theist would obviously deny that "God could possibly not exist" if they think he necessarily does. Agreed. The theist needs only to admit that God exists is possibly true to conclude God exists is true, and, The atheist needs only to admit that God does not exist is possible to conclude that God exists is false. It seems that we cannot say..<>G & <>~G, is true. Witt

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