Eliminating Undecidability and Incompleteness in Formal Systems

[wtf]

The trivial counter-example is the married bachelor of sciences.

LOL. Good one!

But then again I don't believe natural language can be axiomatized. PeteOlcott does, if I understand him correctly.

ps -- How about this guy?

http://johnbatchelorshow.com/

How critical is the exact spelling of a word? This guy is a Batchelor if not a bachelor.

[Logik]

ps -- How about this guy?

http://johnbatchelorshow.com/

How critical is the exact spelling of a word? This guy is a Batchelor if not a bachelor.

So now you are demonstrating that my is_bachelor() function is incomplete? Nice!

There. Now it checks surname also.

Code: Select all

def is_bachelor(person):
  if person.sex.lower() == 'male' and not person.marital_status:
    return True
  elif person.bachelor_degree:
    return True
  elif person.surname.lower() == "bachelor":
    return True
  else:
    return False

Of course, it is still incomplete because....

bachelor, noun, historical.
a young knight serving under another's banner

[wtf]

elif person.surname.lower() == "bachelor":

You have to check all possible alternate spellings too, like "batchelor".

a young knight serving under another's banner

Which illustrates my point that you can't axiomatize natural language. And when Carl Sandburg wrote that the fog creeps in on little cat feet, how would a computer parse that? Poets stretch the bounds of natural language every day. And language changes over time.

[Skepdick]

But then again I don't believe natural language can be axiomatized.

Which illustrates my point that you can't axiomatize natural language. And when Carl Sandburg wrote that the fog creeps in on little cat feet, how would a computer parse that? Poets stretch the bounds of natural language every day.

Have you heard of Pāṇini and Aṣṭādhyāyī and how it relates to modern-day natural language processing? Pāṇini's formalised grammar is equivalent to a Type 0 formal grammar in the Chomsky's hierarchy, only it predates it by 2500 years.

Sanskrit is basically a formalised (axiomatised?) natural language by virtue of Sanskrit speakers voluntarily adhering to Pāṇini's grammar rules as best as possible. Or would that make Sanskrit a naturalised axiomatic language?

Some still consider Sanskrit to be sufficiently concise and unambiguous to suffice as a scientific language. And you can write poetry in it.

And language changes over time.

Precisely the problem all formal notations attempt to solve, no? Linguistic (and therefore semantic) non-determinism.

[wtf]

Sanskrit is basically a formalised (axiomatised?) natural language by virtue of Sanskrit speakers voluntarily adhering to Pāṇini's grammar rules as best as possible. Or would that make Sanskrit a naturalised axiomatic language?

I read the Wikipedia article on Sanskrit and I did not discern any connection to formal axiomatic systems. I confess to not understanding the intent of your remark. It may well be true (I have no way of knowing) that Sanskrit operates according to specific rules that can be called axiomatic. If so, I'm wrong in saying that we can't axiomatize natural language.

In fact I may be entirely wrong in my own premise. ZFC is a rigid formal mathematical system, yet within that structure, amazing feats of creative thinking can take place. Perhaps it depends on what level the formal system is. What I mean is, if the axioms declare a specific meaning for each word, or lists all the allowable words, that's too rigid for a language to grow. In natural language, the meaning of words changes all the time, and new words get added to the language.

On the other hand Chomsky talks about the deep structures common to all natural languages.

So perhaps I was being too literal or simplistic when I say you can't axiomatize natural language. If the axiomatic structure allows for creativity and change, then a language can be axiomatized while still allowing for creativity.

[Skepdick]

Sanskrit is basically a formalised (axiomatised?) natural language by virtue of Sanskrit speakers voluntarily adhering to Pāṇini's grammar rules as best as possible. Or would that make Sanskrit a naturalised axiomatic language?

I read the Wikipedia article on Sanskrit and I did not discern any connection to formal axiomatic systems. I confess to not understanding the intent of your remark. It may well be true (I have no way of knowing) that Sanskrit operates according to specific rules that can be called axiomatic. If so, I'm wrong in saying that we can't axiomatize natural language.

The Sanskrit page doesn't cover the technical aspect. Starti with Panini grammar.

https://en.wikipedia.org/wiki/P%C4%81%E ... %81y%C4%AB
https://en.wikipedia.org/wiki/P%C4%81%E ... al_systems

[wtf]

The Sanskrit page doesn't cover the technical aspect. Starti with Panini grammar.

I believe I already acknowledged your general point; but I can't understand the specific reference to such a narrow technical area, what some ancient writer said about an ancient language. I am afraid I can't dive into the arcana with you on this particular topic. But I've already conceded your general point. Perhaps there's something about your posts that still eludes me. You asked if I am familiar with this particular 5th century BCE Sanskrit philologist.

Well no, as it happens. Would I be expected to be?

[Skepdick]

I believe I already acknowledged your general point; but I can't understand the specific reference to such a narrow technical area, what some ancient writer said about an ancient language. I am afraid I can't dive into the arcana with you on this particular topic. But I've already conceded your general point. Perhaps there's something about your posts that still eludes me. You asked if I am familiar with this particular 5th century BCE Sanskrit philologist.

Well no, as it happens. Would I be expected to be?

You made the comment that natural languages are not axiomatizable.
I offered you Sanskrit as the counter-example as an FYI. No deep analysis or engagement or experitse is expected.

2500 years ago, in India a Turing-complete grammar was invented (Panini grammar). It's used as a natural language to this day.

I don't care if you are 'right' or 'wrong' - I am not in this to win arguments. At most I expect an 'Oh! Cool!"

[wtf]

I don't care if you are 'right' or 'wrong' - I am not in this to win arguments. At most I expect an 'Oh! Cool!"

Oh! Cool.

[Speakpigeon]

The trivial counter-example is the married bachelor of sciences.

LOL. Good one!

But then again I don't believe natural language can be axiomatized. PeteOlcott does, if I understand him correctly.

ps -- How about this guy?

http://johnbatchelorshow.com/

How critical is the exact spelling of a word? This guy is a Batchelor if not a bachelor.

It's essentially irrelevant. "Informal" arguments are assessed as we assess what other people say, by interpreting what they say in light of what we understand the context is. Sure, we may make mistakes in doing so but anything we do is subject to mistakes and error and failures. No big deal.

There is as you know the "principle of charity", whereby you try your best to interpret what people say in the most favourable light. People stop doing it whenever they have some reason to dislike the speaker. Disagreements are more usually proof of bad tempter and malicious intent.

Informal arguments are not rigorous because they don't need to be. Trying to make hay out of this simple fact of life is just pathetic.

If you want to claim you don't understand an argument, just imagine it's a big guy making it in your face and see if you would keep pretending you don't understand it.

I guess the Internet makes words even cheaper than ever before.
EB

[Skepdick]

There is as you know the "principle of charity", whereby you try your best to interpret what people say in the most favourable light.
People stop doing it whenever they have some reason to dislike the speaker.

I haven't seen an ounce of charity from you on this forum.

Does that mean you dislike everybody (but yourself)?

Grumpy Frenchie...

[Eodnhoj7]

Undecidability and incompleteness are assumptions that if defined would require the same logical systems they serve to define them.

The end.