> Let n equal 3 if God exists, or 5 if God does not exist. Is n prime?
Sure, I'll happily quibble! You're using excluded middle to assert that n is either 3 or 5, but you haven't justified that excluded middle holds for the proposition "God exists".
If you’re going to quibble over whether LEM is justifiable in this case, then you need to justify why you’re only concerned about LEM; why not drop ex falso too (Kolmogorov had serious issues with this axiom, and initially considered it to be incompatible with a constructive logic) and work in a paraconsistent logic?
Also, depending on the precise formulation of this proposition, it doesn’t necessarily need LEM.
Why does it rely on the axiom of choice? It’s a consequence of the obvious fact that for every integer n, there exists a program that prints n. This doesn’t require choice.
(I agree that there are formulations which don't require LEM, but it's important to be precise, especially when writing to dismiss a common misconception among people who haven't got the concepts crisply in their minds. "Is this quantity computable?" is very close in spirit to "can I compute the value of this quantity?", and LEM is exactly the kind of axiom which shows the difference.)
> Ex-falso has trivial computational content, and we use it all the time: it's `panic!()`.
I have major issues with the computational content of a bottom type; my interpretation of the BHK semantics would not admit ex falso as constructive.
Of course this all of this really stems from one’s notion of semantic truth: any non-trivial semantic theory of truth can be argued as the “one true logic”, and there’s not much anyone can say otherwise! As I see it, it really comes down to how useful it turns out to be in modelling the particular domain of discourse.
As it happens, I’m more than happy to accept classical logic: I believe the principle of bivalence is how the world works, and as a result I’m forced to admit LEM if I want my proof calculus to be complete.
You are right that pointing out the connection to LEM is important and worthwhile, regardless of formulation.
Also whether n is prime is up to the will of God, if God exists.
A God is not necessarily bound to the "laws of physics" or even basic logical necessities (a God that does comes from a specific line of theological reasoning, not the general case).
He could even make 6 odd if He so wished - altering all math and logical consistency and the whole universe, or just make just 77 even and every other number odd, making so every mathematician finds the new arrangement perfectly consistent and consider it to always have been the valid one!
So the answer does kind of "depend on your religious beliefs".
The idea of "new doesn't affect old" is an idea based on logical and temporal consistency. God is not bound to those.
In fact the current system of odd/even - as well as any other system and law, including the "excluded middle" - can be understood to be God's arbitrary creation. You consider it logically necessary and absolute just because God willed it so, you'd be considering a different one with any arbitrary change as just as logically necessary and absolute if/when God willed it :)
People create a lesser conception of God, with specific boundaries and limitations, and then gloat how he is limited. Well, let's instead start with the more common conception of God as limitless and with total arbitrary power.
That entirely depends on your specific god belief. Suppose someone believes that god is the eternal principle of consistency?
>Well, let's instead start with the more common conception of God as limitless and with total arbitrary power.
If god is capable of anything can god decide to make it such that there is no god? Can god decide to self-limit god's own powers? If god intrinsically encompasses all possibilities does that include the possibility of godlessness?
You quickly get to then point where the concept ceases having any meaning.
My sentiments exactly.
The whole point of the comment was that the part saying "The answer does not depend on your religious beliefs" was wrong.
>If god is capable of anything can god decide to make it such that there is no god? Can god decide to self-limit god's own powers? If god intrinsically encompasses all possibilities does that include the possibility of godlessness?
Yes. Trivially so. And He can even make it so godlessness involves the presense of God too, without there being any logical inconsistency even (since he's so powerful he shapes logic, not the other way around).
>The idea of "new doesn't affect old" is an idea based on logical and temporal consistency. God is not bound to those.
I'm not a theologist, but I don't think god recreating the world such that 3 is not prime would affect the discussion if 3 is prime right now.
In fact, I don't think god is even capable of making 3 not prime. Prime numbers are defined such that 3 is prime, it's not possible and a logical contradiction to make 3 composite without changing the definition. And the discussion is about the current definition.
>In fact the current system of odd/even - as well as any other system and law, including the "excluded middle" - can be understood to be God's arbitrary creation
But it's a human creation, god didn't invent parity or formal logic.
God is bound to His Word and as the saying goes 'it is written':
"Never will you find a change in the way of ALLAH. [Qur'an 33:62]
> Arbitrary creation
"HE created the heavens and the earth in accordance with the requirements of wisdom." [Qur'an 39:5]
p.s. Computing (al-Hesaab) is also addressed. Apparently God and Gauss are both rather fond of harmonic systems:
"HE It Is Who made the sun radiate a brilliant light and the moon reflect a lustre, and ordained for it proper stages, that you might know harmonic measure and mathematics. ALLAH Has not created this system but in accordance with the requirements of truth. HE details the signs for a people who possess knowledge. " [Qur'an 10:5]
Only if you choose to misinterpret the point.
Except if they wanted to say "The answer, iff your religious beliefs are conveniently tame and constrained, so that your God has limited powers and is bound to respecting excluded middle, the laws of physics and other such constraints, doesn't depend on your religious beliefs".
Any other movie quotes to put forward in lieu of counter-arguments?
Logical consistency only cares about propositions and their relationship, it doesn't care whether an entity involved in one is fictitious or not.
In contrast laws of physics are a real world thing, and any omnipotent being can meddle with it, basically by definition.
You can't, under your logical constraints.
God has no such constraints, not just in the physical but also in the logical realm. He can make it so that assuming the law of excluded middle and not assuming it at the same time, is compatible and consistent.
Yep. Possible, and endless fun for a certain sort of person!
>Like, is there a point in asking/answering anything if anything goes?
Sure, since at any given time anything God wills goes!
Have you noticed that God being involved in the topic of a conversation seems to cut everyone's IQ by like 50%? And there are other topics that have this effect as well.
If you're allowed to invoke God ironically in the question, you're allowed to take that seriously (meta-ironically?) in the rebuttal!