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Clean up NFO page.
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MostAwesomeDude committed Aug 23, 2024
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and
[Shutt-abstractive](https://fexpr.blogspot.com/2013/12/abstractive-power.html).

## Identity
## Examples

As a classic and powerful first example, we can give a redefinition of `{du}`
in NFOL terms.

ro da ro de zo'u:

da du de
========================== (id-def)
ro bu'a zo'u da .o de bu'a

This rule is a second-order *definition*; such rules are not possible within
first-order logic. There are many reasons to justify this sort of explanation
as a definition, but the primary reason is that they could be entered into
Lojban dictionaries. For example, we could define ``{du}`` as:

ro bu'a zo'u x₁ .o x₂ bu'a

And give any tautology, like ``(id-refl)``, as an example:

ro da zo'u da du da

Once so defined, ``{du}`` is technically not in terms of any particular
selbri; rather, it is a global property of *all* selbri!

## Natural numbers

We can give the induction principle for natural numbers with SO quantifiers.

ro bu'a zo'u:

ge li no bu'a gi ro da su'o de zo'u da na.a de bu'a .ije da kacli'e de
---------------------------------------------------------------------- (kacna'u-ind)
ro da zo'u da kacna'u nagi'a bu'a
Metamath statement | Lojban bridi | Description
---|---|---
[df-du](df-du.html) | {go ko'a du ko'e gi ro bu'a zo'u ko'a .o ko'e bu'a} | definition of identity
[ax-nat-ind](ax-nat-ind.html) | {ganai ge li no bo'a gi ro da poi ke'a bo'a ku'o zo'u su'o de zo'u ge da kacli'e de gi de bo'a gi ro da poi ke'a kacna'u ku'o zo'u da bo'a} | induction for natural numbers

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