Thanks, man. Do I really have to read all the stuff before Enderton? I want to get to the meat of it right away, but if I really have to...

I'm obviously in too deep here, I'll need to check out some actual mathematical logic books or something. Could you offer a starting point maybe? I have the Book of Proof.

Potentially good example, thanks. But:

I disagree on the argument not making sense deductively (disregarding the hidden premise of the argument)*. While Aristotle might be the only person who is the unique collection of all the traits he possesses, he still belongs to a number of sets according to those traits, unless I'm just in ignorance of the way Socrates is treated by Aristotle. He is, for instance, a man, but I don't think that needs to be assumed up front before the argument is made.

*There needs to be a premise saying that not all men are Socrates, or it doesn't make sense.

However, I think I see what you are saying: Socrates is unique insofar as we cannot treat him as just a man; he is not assumed to be a subset of anything, so, therefore, because of the second premise, we are no longer dealing with sets, or "all" and "some" statements, which goes against the way Aristotle did logic.

So, while this argument might make sense deductively, it doesn't make sense as part of a larger schema.

Formally, an argument is merely an ordered pair <G P> where G is a set of statements and P is a statement. G is the set of premises and P is the conclusion.

An argument is valid if and only if there is no model in which every member of G is true but P is false. So 'validity of an argument' is a semantical notion. — TonesInDeepFreeze

What could be said about an argument, A, whose premises include the entire set of the correct premises of sound argument B, and has the same conclusion as B, but the conclusion is unsound for A, potentially because of added steps or premises? Doesn't that give a model in which every member of G could be true but the conclusion, P, be false?

Or what if there is some sort of recursive step in a valid model, an instance in which correct premises are applied to correct premises in such a way that the conclusion P of argument G becomes false even though all the premises stay true? Would that not be a model that would defy the formal definition?

I'm no expert on proofs, I hope I got that right.

I get that, I'm talking about a way to determine if said theorem can be understood by observing the proof. I do not deny the fact that the proof would be a function of certain foundational axioms if it is true, thus meaning the proof is derivable given those axioms. What I'm talking about is like looking in through a window that may or may not be there on something that we know is there (the proof).

Interesting. What if it applied conversely and allowed us to determine that the proof behind a principle or theorem assumed to be true couldn't be observed or determined?

Thanks. You basically answered my question.

If there was a way to determine if it is possible at all to observe or determine the proof behind a certain unproven but assumedly correct principle or theorem, would that be valuable?

Is it possible to break down a deductively valid mathematical theorem into its constituent parts without knowing the actual argument that is going to be used to prove it? In other words: what if the reasoning cannot be observed at all but we know it's (correct) constituent parts when synthesized add up to a deductively valid argument?

I feel like it exists.

It isn't so much a reference to lexical semantics and whatnot, but that might matter too.

I mean we know the premises, but don't know the format of the argument. Such as whether it is modus tollens or ponens (if you couldn't just guess between those two based on the premises and conclusion).

If all the numbers in your calculations are such that they cancel out and leave you with a nice whole number answer, you're (almost) guaranteed to have solved the problem correctly. — Agent Smith

Assuming a more complex math problem, I think you would mean if the variables cancel out and you are left with an integer? Or maybe a rational number?

If you do mean whole numbers, the set of whole numbers is a countably infinite set, so just coming up with a single whole number doesn't guarantee a whole lot.

Why doesn't this rule apply to real life scenarios? Shouldn't we be going :chin: huh? when after trying to calculate some constants in math and science we find their values to be rather unwieldy/cumbersome/awkward like, for example, the numbers ππ and ee? — Agent Smith

What is unwieldy/cumbersome/awkward about $\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.$? Just because there is no magical logic gate in your calculator than can represent $e$ as a ratio of two integers, or CAS that can compute it algebraically, doesn't mean there is something wrong there.

That thread did little to elucidate anything; the only thing I saw that was relevant was the discussion of what it is "like" to be oneself holistically and whether or not one considers that to be coherent and thus a useful construct.

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Got it.

You just linked me to this page.

Can you expand on what you mean by "incoherent"? And what is it with respect to? Propositions? Arguments? Sentences? Mathematical models? Experimental data?

Can't. We would never be able to tell if a being that came to us was just ultra-powerful, or truly omnipotent. A sufficiently powerful being could just seem omnipotent.

At a 45 degree angle to piss off the motorists, obviously.

Any way I please, I imagine.

Really God could make me have any brain-state, but that doesn't mean that all of those brain-states are the same.

You need to explain why omnipotence implies that God cannot deliberately select a measurable course of action. God could know that if they stimulate my brain the right ways, I'll black out and smack my face against my desk. Or they could stimulate my brain so that I feel intense pleasure. I perceive those things, so, to me at least, the consequences of God's actions are measurable. Or they could do anything to my brain, and I will likely perceive it, and, thus, it is measurable to me.

Why can't God walk with measure?

It was a joke.

Exactly. And having no measures means impotency. — EugeneW

So why wouldn't God's actions be measurable? It's like the walking/circle example I gave. Infinite choices doesn't imply that one cannot deliberately choose a course of action, provided there are parameters or measurable consequences.

God could make my head explode right now, if she wanted. That's pretty measurable.

Its over, Eugene, Bartricks has the high ground.

Eugene, you aren't making sense. Read my reply to you earlier in the thread.

I'm just messing with you.

But we have no idea what logical deductions are valid or not according to your view of God. Any random deduction could be absolutely worthless because its negation could also (secretly) be true. We are just groping in the dark, really.

Yeah, just too bad God is a monster who lets people suffer gratuitously. :joke:

You are pretty clear in your reasoning, I don't get why some of the smart people on this forum don't understand your arguments.

To generate 'explosions' and other such logical dramas one would have to assume the reality of necessity. Yet the reality of necessity is incompatible with the existence of an omnipotent being. Not necessarily incompatible, of course. Just actually incompatible. And thus as an omnipotent being exists, we can safely conclude that there are no necessary truths (including that one). And so if - if - the omnipotent being made a true proposition false at the same time, this not create any explosion, for it remains down to the omnipotent being whether any other propositions are true and false at the same time. — Bartricks

So God could theoretically just choose to make there be no other contradictions, or could choose to make any contradictions they want to be true, true. Got it.

But insisting that an omnipotent being is necessarily omnipotent is to insist that being omnipotent essentially involves an inability - the inability to not be omnipotent. That just seems incoherent to me - indeed, it asserts a contradiction. For how can one say that an omnipotent being is able to do anything if at the same time one insists that there is something that the omnipotent being cannot do, namely divest themselves of their omnipotence? How is that not to assert P and not P? We agree, I take it, that no contradictions are true. — Bartricks

We agree, I take it, that no contradictions are true. — Bartricks

So we just dismiss this contradiction because it goes against our preconceptions? Doesn't it mean god can't be omnipotent? Or something? I mean, surely the principle of explosion or something like that wouldn't follow. But then again God could just make this contradiction not true, or so you claim.

Maybe God exists, has thought about this, and has smoothed it over?

I guess what I'm saying is that if the consequences or parameters of a decision or course of action can be measured, we could theoretically have chosen otherwise; it could have been different.

I don't think the ability to choose between infinite options would render one incapable of choosing. Like you said, one could walk straight in infinite directions starting from a center point, but one would always be walking a measurable distance, and could say at what angle one was walking at if a circle was projected with its center at the center point from which one began walking, with the radius being the line along which one walks.

The existence of infinite options does not mean that one cannot choose a course of action, or could not have chosen a different course of action, or could not have chosen no course of action. After all, you could have chosen a different angle or distance.

But what about the argument that she must be omnipotent in possible worlds too in order to be truly omnipotent? I'm making the argument that if she divests herself of her omnipotence she must necessarily have never been omnipotent - but only in the possibility of her actually taking the route of making herself not omnipotent. It seems to me my argument still stands, unless God violates LNC or chooses to be omnipotent and not omnipotent at the same time.

I also address the contingency in which God violates LNC and makes herself omnipotent again in another thread. I haven't gotten any feedback on it, so I don't know if my reasoning is solid, though.

To put it another way, God is no more bound by the principle of explosion than he is by any other principle. He can make the law of non-contradiction false. So he can make the principle of explosion false too. — Bartricks

But the principle of explosion would be true globally, if not for god, right? How would logical deductions suddenly become valid if LNC doesn't apply for a pair of mutually exclusive propositions? Would God not have to fix the contradiction to make the principle of explosion not true?

Thanks for responding, Bartricks, I appreciate you.

I'm sure I'm not, I just didn't read it anywhere, not trying to plagiarize or anything. I don't know if hardly anything I've written is truly original.

Part of the fun is trying to figure this out on my own, at least partially.

Who knows. I don't really even care.

So my point here is that the ability to break/ignore LNC defeats the OP - i.e. God can simultaneously be be omnipotent over the sum process and divest Herself of Her omnipotence. — EricH

I think it is less so that God is omnipotent and not omnipotent over the sum process, but rather that God can restore her omnipotence at any time, regardless of current status. If God decided to make a contradiction such as: "God is both omnipotent and not omnipotent" true, the principle of explosion would follow and we could then prove any statement or its negation true. Or maybe I'm wrong - I just read about deductive explosion today for the first time.

edit: Whether or not the deductions would be true, I don't know. But it seems to me some logical systems would be all messed up.

second edit: rather, the systems wouldn't be messed up; we would just not be able to rely on logical deductions anymore, I think.

So my point here is that the ability to break/ignore LNC defeats the OP - i.e. God can simultaneously be be omnipotent over the sum process and divest Herself of Her omnipotence. — EricH

I think my argument stands so long as God is bound by LNC, but yes, otherwise it appears to defeat the OP. I'll have to think about this. Thanks for reading and understanding the OP.

Btw, you know what D-kers are? — EugeneW

No, what are they?

What they mean is mental illness, mostly schizophrenia, but often manifests itself with religious symbology and themes. I dont think they meant religiosity is a mental illness, nor saying religious people are mentally ill.
I mention it because “sociopath” seems a pretty drastic take on the comment. — DingoJones

I don't think he is actually a sociopath, I just think that his intense desire to be regarded as a big brain atheist manifests as verging on anti-social behavior.

I regret creating this thread, especially since no one has addressed the original part of my OP.