Calling out the Logical Lion (Dr.S.)

Originally posted by vistesd

Am looking at Goedel’s incompleteness theorems. Can find no on-line reference to the “Comparison Devil.” Can you expand on that just a bit?

That's my own term, meant to draw attention to the fact that theoretically, it is not necessarily easy or even possible to deductively compare some propositions to the set of axioms serving as the standard of truth in a manner sufficient to decide the truth of those propositions. It is for this reason that I left "compare" as an abstract idea rather than giving an explicit algorithm for executing a comparison.

Just like a train needs to be well-behaved (e.g., not moving relative to you) in order for you to measure its length with a yardstick, propositions need to be well-behaved in order to satisfactorily compare them to axioms. One way that propositions can misbehave in this sense is by refering to themselves or to the universe of discourse, as does Goedel's cleverly constructed G.

Originally posted by vistesd

(1) If God’s existence is taken as axiomatic for any (logical) system purporting itself to demonstrate such existence, then (a) that is a logical error in terms of the direction of proof; and (b) that error can only be rectified by identifying an alternative axiom, independent of the existence axiom, that ends up replacing it.

(2) Not all true propos ...[text shortened]... statement of its own consistency without itself being inconsistent (2nd incompleteness theorem).

Yes, those are correct, although (1) is the only one with any practical relevance.

Originally posted by DoctorScribbles
Yes, those are correct, although (1) is the only one with any practical relevance.

Thank you.

Neither Antony Flew's How To Think Straight, nor my Logic: A Very Short Introduction addressed axioms. Can you recommend a good logic book for me to step up the ladder a bit more? (I like books that are tough for me, and sometimes force me to go back to the more introductory accounts, but are not insurmountable.)

Originally posted by DoctorScribbles
Plenty of rhymes rhyme wit axiom
And I like tha shape of yo' mama's rack, see I'm
Always hittin' it from behind in the back seat o' my
Benz and tha whole BWA's got my back, so, um,
Better step off cuz I gotz tha maximum,
No, make that the blaximum,
Rhymin' skillz, you just got tha whaximum.

Now, you tryin' to show off? You tryin' to get fancy?
To get in ma' face is a little bit chancy.
You ain't got the groove to step to my dance, see?
Even Ivanhoe wishes yo' mom ended her pregnancy!

To rhyme with that word, you needed neologism,
Dim cognates, homophones and weak symbolism.
Fo' you, I lay down my sage aphorism:
You a foreigner here, and I'm an autochthonism.

So let me proposition you with ma' own axiom,
Watch how my question indubitably taxes him!
Vexin' him, tauntin' him, baitin' him, waxin' him.
He offers the minimum, I offer the maximum.

'All niggas is equal.' That's the law of the land.
Whether evolved from a blob or the work of God's hand,
or the wave of Spaghetti Monster's Noodly strand,
'We is all equal.' That's a standard command.

Well, I've come to question whether this would be true,
Whether God, Fate or chance this equality did imbue,
And this is the negation I offer for review,
Orwell had it right: I'm mo' equal than you!

That's right, son, you gots it. I shine where you flounder.
My melody's tighter; my rhythm, it's sounder.
Yo' thoughts is so trite, and mine are profounder.
They call you: Confused One. Me? The Astounder.

I gots the wisdom of Solomon, his harem twice again.
I call down my rhymes like fire down from heaven.
Letters of blood come out o' ma' pen,
When I finish rhymin', the folk say 'Amen!.'

Rappin' at you is throwin' pearls to swine.
You lucky that you can even follow ma' line.
I gots followers, a temple, a church, and a shrine.
You gots a card trick. Me? Water to wine.

Hearin' my word is to drink livin' water.
Piss me off, son, and you gonna get slaughtered.
Yo' girl is so cheap, five dolla's I bought her.
I just had yo' mom, next up is yo' daughter.

In the hospital of rhyme, I heal at a glance.
My revelatory speech puts all in a trance.
My word makes the blind see, deaf hear, and lame dance.
I'm the docta, not you. Now go clean bed pans.

You come from Decapolis? It seems that you do.
The impediment of speech was a really big clue.
Ephphatha, I say! Start yo' rhymin' anew.
Yo' beat is confused, yo' words are askew.

You just been healed, but still you a mute?
The little you say in my presence do pollute.
Touch the hem of my rhyme? That's still yo' pursuit?
I healed yo' fat lips, but yo' brain's too minute.

Mess with ma' logos and you'll have atone.
You're just an ass-zephyr, I'm a rhymin' cyclone.
I'm prolific, wise, and powerful and you're an unknown.
You just a sinner and I done cast the first stone!

Even Bartemeus can see that yo' the inferior type.
Like Legion, you foamin' and raisin' up hype.
You flip jive like you been hittin' the pipe.
My sandals you can't carry, but my ass you can wipe.

So how is we equal, huh? You a punk. I'm a king.
Yo' well has run dry, mine's like a new spring.
You fall when I fly, you stammer when I sing.
I'm the Pontifex Maximus. Now come kiss my ring.

Nemesio

I'm comin at ya fast, like da runs, bitch
cos I'm e-coli and yo ring gots da itch
see ma lyrics is spreadng deadly like viral,
running round yo gut and yo mind in a spiral

cos ma rhymes is perfection
gonna need an injection
to deliver yo brain from infection
i'll kill ya stone dead wit ma lyrical inflection

like an arrow I go straight for yo sista
cos she know I'm a sexual zapatista
startin revolution in her innocent mind
while she watchin me doin yo mama from behind

ya think I's gots time to listen to yo rap
but dog, it ain't nothin but an old man's pap
pipe and slippers and an afternoon nap
yo rhymes got to minds out for da generation gap.

so just know that when yo walkin wit a frame
too passed it ta play the freestylin game
that yo old, dog! Ain't got long ta feel da shame
and yo lil sista gonna be choking down ma name

Originally posted by vistesd
[b]Or perhaps the question intended was "Can all true propositions in a universe of discourse be deduced from its axioms?" Again, No. This is one aspect of the Comparison Devil. See Goedel for more details.

Yes, that was it. I remembered it backwards. I’m assuming that an axiom might be drawn from inductive or abductive inference.

Am looking at ...[text shortened]... atement of its own consistency without itself being inconsistent (2nd incompleteness theorem).[/b]

Goedel's theorem is well understood by a vanishingly small fraction of the population (sadly, I can't count myself within that fraction). I think it may not be appropriate to extend Goedel's result about mathematics to arbitrary logical systems.

Wikipedia gives this summary of the result:

"For any consistent formal theory that proves basic arithmetical truths, an arithmetical statement that is true but not provable in the theory can be constructed. That is, any theory capable of expressing elementary arithmetic cannot be both consistent and complete."

and goes on to say that an analagous proof is possible for any logical system "of a certain expressiveness."

Clearly, there are dangers in trying to apply the theorem to just any system; as a trivial example I doubt the logical system whose only axiom is "frogs are blue" satisfies the expressiveness requirement. It is easy to see that there are no true statements besides the axiom in this system, so that there are no unprovable but true statements either. I believe to satisfy the requirements of Goedel's proof a logical system must have an "expressiveness" at least on the order of arithmetic (because Goedel's proof uses numbers).

Similarly, if you're trying to be rigorous here you can only say that (assuming the supposed theistic system is sufficiently "expressive" and is consistent) that not all true statements in the system can be proved--not necessarily that all true statements about God cannot be proved. Another trivial example: if I tack on the axiom "frogs are orange" to mathematics I can still prove all true statements about frogs in my system (i.e., none).

Perhaps an interesting excercise would be "what is a minimal set of axioms required to deductively prove the existence of God?"--just as mathematicians judge the elegance of a logical system by the parsimony of its axiomatic foundations.

Originally posted by vistesd
Thank you.

Neither Antony Flew's How To Think Straight, nor my Logic: A Very Short Introduction addressed axioms. Can you recommend a good logic book for me to step up the ladder a bit more? (I like books that are tough for me, and sometimes force me to go back to the more introductory accounts, but are not insurmountable.)

Part I of Godel, Escher, Bach is my recommendation. I think you'll find that it meets your criteria while being a very enjoyable read.

Originally posted by Nemesio
Now, you tryin' to show off? You tryin' to get fancy?
To get in ma' face is a little bit chancy.
You ain't got the groove to step to my dance, see?
Even Ivanhoe wishes yo' mom ended her pregnancy!

To rhyme with that word, you needed neologism,
Dim cognates, homophones and weak symbolism.
Fo' you, I lay down my sage aphorism:
You a foreigner here, a ...[text shortened]... come kiss my ring.

Nemesio

I'm all outta recs, and that's a damn shame
Cuz you be bustin' mad verses, not a single one lame
Each crafted so fine, and no two the same
Not bad for a guy with a Mexican name

I ain't gonna lie, a full rec you deserve
But Russ be all stingy, I ain't got none in reserve
That niggardly bastard, he sure has some nerve
If I see him out walkin', just watch my Benz swerve

10 recs a month? What's that, some joke?
Every time I post, niggaz oughtta go broke
Mad as hell, I might have a stoke
Cuz I can't give a brother his proper toke

Originally posted by DoctorScribbles
I'm all outta recs, and that's a damn shame
Cuz you be bustin' mad verses, not a single one lame
Each crafted so fine, and no two the same
Not bad for a guy with a Mexican name

I ain't gonna lie, a full rec you deserve
But Russ be all stingy, I ain't got none in reserve
That niggardly bastard, he sure has some nerve
If I see him out walkin', ...[text shortened]... go broke
Mad as hell, I might have a stoke
Cuz I can't give a brother his proper toke

I gots to give credit where credit is due.
Ma' healin's done worked a damn miracle in you.
Why they did what they did -- if I only knew...
In tossin' out the water, they tossed the baby out, too.

They's ain't no doubt that yo' claim is correct.
I'd be a fool to deny it respect.
I gots a bit to say on this particular subject:
They ain't be no doubt that the reccing is wrecked.

Yo' rhymin's all tight, concise, even funny.
Yo' assessment is right, precise, on the money.
You have good insight, representin' it cleva'.
And in tribute to you, here's my shortest rap eva'.

Peace out,
Nemesio

Originally posted by Nemesio
I gots to give credit where credit is due.
Ma' healin's done worked a damn miracle in you.
Why they did what they did -- if I only knew...
In tossin' out the water, they tossed the baby out, too.

They's ain't no doubt that yo' claim is correct.
I'd be a fool to deny it respect.
I gots a bit to say on this particular subject:
They ain't be no doubt t eva'.
And in tribute to you, here's my shortest rap eva'.

Peace out,
Nemesio

Yo, better study up on Doubtin' Thomas
It would do you some good, that's an MF promise
Cuz you so gullible, or maybe just vain
That you didn't realize that I was just playin'

I ain't meant a word I said
You let my compliments go to yo' head
You really think I was respectin' yo' rhymes?
Bitch please, mine be Benjamins while yo's be dimes

You was all like, "Damn, Cribs thinks I can really bust!"
You so weak, when God formed you from dust
And blew life all right up in yo' nose
You collapsed to the ground like some wrinkled clothes

And when he wanted yo' rib to make you a wife
You was all like, "No, please, no, I'm scared of that knife"
And when you ate from the tree of knowledge and belief
You saw you were naked, but didn't need a leaf

And when the Devil offered to save your soul
He offered salvation, if you could meet one goal
You had to bust just one verse that wasn't weak
But you mumbled and stumbled and couldn't even speak

So the Devil spoke up and said "God Damn!
You one no-rhymin' sorry excuse fo' a man!
You ain't earned salvation, but I'll give it just as well,
Cuz I sure ain't want yo' whack ass comin' down to hell!"

Thoughts of heaven and hell and evolution
Here be my rhyme givin' you a contusion
A lyrical thread of religious thought
Wisdom from Rev. K, Nem and Doc can only be bought

Simpletons all round lookin fo cheap grace
Disparagin Darwin who descended from apes
Comin at ya in this here season of Lent
Doc Scribs wit his syllogistic axiomatic bent

Yo I'm preachin a new gospel of sure-fire doom
Yo, Mokko , get ova here and wash my feet wit perfume
Fill my cup full of vintage wine
As I lie back and play chess sublime.

Judas Iscariots all around me surroundin
Plottin against BWA they be houndin
Makin accusations denyin my defense
Yellin, "Hey, nigga, get off tha fence."

Condemnin my black ass to hell by way of the cross
Crucify him! Crucify him! RHP albatross
But like Jesus before me who defeated tha grave
Rev. K wit tha white womens still all the rave

Hrothgar's Prayer:

I'm here to repent of erroneous wayz:
"Father, fo'give me, lest I live out my dayz
aware that my rhyming just fails to amaze.
No brotha flows, who his Saviour betrays!

The sin I confess: my earlier rhyme
was not tight like those of an earlier time.
As RC I spat lyricz in metre sublime;
my rhymin's a mortal sin now, and a crime."

"You is white," you may say, "a regular hick,
it's cool you can't master this fly rhymin' trick."
"My son," you may say "you's so white you look sick
while Cribs is a brotha and N-Dawg a spic."

To you, Lawd, here is my angry reply:
"What about Slug and that Eminem guy?
The Bard, Poe and I all spun rhymes that were fly
and we're white, Lawd, yo' answer does not satisfy!"

Yo', the Saviour of Rap is so whack I bet he is not even going to answer my question properly...

Originally posted by ChronicLeaky
Hrothgar's Prayer:

I'm here to repent of erroneous wayz:
"Father, fo'give me, lest I live out my dayz
aware that my rhyming just fails to amaze.
No brotha flows, who his Saviour betrays!

The sin I confess: my earlier rhyme
was not tight like those of an earlier time.
As RC I spat lyricz in metre sublime;
my rhymin's a mortal sin now, and a c ...[text shortened]... is so whack I bet he is not even going to answer my question properly...

Yo, CL, yo complaint is reminiscent of Job
I rememba when young RC was seekin out ho's
Rhymes so tight wit analytical might
Now you whinin worse than Elie Wiesel in "Night"

What's wrong wit you , young RC
Come on out and be who you was meant to be
Smokin, tokin and a pokin
Humblin Ivanhoe in Hoboken

Forget all dat forgiveness and last dayz crap
It be time for a reunion of caninus lupus aka wolfpack
Gatherin by the fire relivin the glory
Of a night far off yo the Ivanhoe story.

Nerdcore:😞

Bumped to read...There's some good stuff in here--nerdcore aside.