fom
Posts:
1,098
Registered:
12/4/12
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Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Posted:
Feb 4, 2013 12:25 AM
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On 2/3/2013 10:19 PM, Charlie-Boo wrote:
<snip>
>>
>>>> In PROLOG we use lowercase words for TERMS
>>>> and uppercase words for VARIABLES
>>
>>>> ATOMIC PREDICATE
>>
>>> ATOMIC PREDICATE meaning relation?
>>
>>> C-B
>>
>> RELATION
>> p(a, b, e)
>
> If wffs are built on relations then { x | x ~e x } is not a wff
> because ~e is not a relation.
Well-formed formulas are built from the alphabet
of a formal language. If the language contains
a symbol of negation, then NOT(xex) will be a
well-formed formula.
>
> We don?t need ZF - at all. All we need is Naïve Set Theory, a
> complete formal definition of wff and recognition that x ~e x is not a
> relation due to diagonalization on sets.
The reason for ZF and other inquiries into the
foundations of mathematics has to do with a coherent
explanation for the utility of an otherwise incoherent
collection of mathematical techniques. If such an
explanatory role is not forthcoming, such theories
at least organize mathematical techniques into a
science (in the sense of Aristotle) connected by
the argumentation of proofs.
>
> Logic = Set Theory
If this is true, it is not the logic of which
you are thinking.
What you are taking for granted is the structure
of logic without a negation symbol. The negation
you use in your programming has no reality in the
underlying computer architectures.
More formally, what you are taking for granted
is that only 14 of the 16 basic Boolean functions
are linearly separable switching functions. The
two that are not are logical equivalence (LEQ) and
exclusive disjunction (XOR).
These particular connectives become problematic
when considered in the context of classical quantificational
logic because of the relation of identity, definition,
and description.
The standard account of identity (for example as
discussed under "relative identity" at
plato.stanford.edu) addresses trivial identity,
that is, x=x, and substitutivity.
What is not addressed is informative identity,
that is, x=y.
In classical model theory, however one has
determined an object in a model and a name
for that object has consequences for the
satisfaction map. That is how the classical
model theory interprets x=y.
In 1971 Tarski directed his attention to the
representation of first-order logic in the context
of algebraic logic. In those deliberations, he
introduced the axiom
AxAy(x=y <-> Ez(x=z /\ z=y))
In the formulation of these "cylindrical algebras"
the formula
x=y
corresponds geometrically to a 2-dimensional subspace
of the domain. In a 3-dimensional domain, this is
a hyperspace separating the domain into two regions.
This suggests that there is a fundamental geometric
reason for LEQ and XOR to not be represented in the
underlying propositional logic by linearly separable
switching functions.
>
> Logic = NOT AND OR EXISTS simple, easy
>
What happens if I take NOT away?
There is no real way to post this picture to a newsgroup.
It is the ortholattice which is an atomic amalgam of a
Boolean lattice with 4 atoms (the usual 16 element lattice
associated with basic Boolean functionality) and a
Boolean lattice with 3 atoms.
....................................TRU....................................
............................./.../..//\...\.................................
......................../..../.../../....\...\.............................
.................../...../..../..../.........\.....\.......................
............../....../...../....../...............\......\.................
........./......./....../......../.....................\.......\...........
....../......./......./........./...........................\........\.....
.....IF......NAND.......IMP.....OR.........................ALL........NO...
..../.\.\..../.\.\..../..|.\..././\.\\..................../...\.....././...
.../...\./\......\./\....|./..\./..\...\...\...................../.........
../../..\...\.../...\./.\|...../.\..\....\............/....../...\../......
.//......\.../\.../....\.|.\../....\.\......\...\......../.................
LET.......XOR..FLIP....FIX..LEQ.....DENY........./.../............/\.......
.\\....../...\/...\..../.|./..\...././......./...\.......\.................
..\..\../.../...\.../.\./|.....\./../...../...../.............../....\.....
...\.../.\/....../.\/....|.\../.\../.../.../...........\.........\.........
....\././....\././....\..|./...\.\/.//.....................\./.......\.\...
.....NIF......AND......NIMP.....NOR........................OTHER......SOME.
......\.......\.......\.........\.........................../......../.....
.........\.......\......\........\...................../......./...........
..............\......\.....\......\.............../....../.................
...................\.....\....\....\........./...../.......................
........................\....\...\..\..../.../.............................
.............................\...\..\\/../.................................
...................................NTRU....................................
The two lattices share TRU, NTRU, OR, and NOR. The structure of
the unfamiliar lattice has
SOME=EXISTSWITH=Ex
OTHER=EXISTSWITHOUT=Ex-
ALL=Ax
NO=Ax-
with
ALL=NOT(OTHER)
NO=NOT(SOME)
on the basis of the order relation alone.
This construction, while described specifically
for quantificational logic here, actually characterizes
the geometric (in the sense of an atomic lattice) structure
of any extension to propositional logic with negation.
Modal extensions are four-fold interrelating POSSIBLE and
NECESSARY. Deontic extensions are four-fold interrelating
OBLIGATORY and PERMITTED.
For quantificational logic, each variable has this
four-fold structure. This corresponds with the indexing
of quantifiers found in the cylindrical algebras of
Tarski's later work.
The negation symbol masks this four-fold structure in
the formation rules for formulas.
Yes. Logic in the absence of NOT--just like in the
hardware of your computer systems--is not easy.
> ZF Set Theory = a dozen messy axioms for which people can?t even agree
> on the specifics ??
>
There are actually an infinity of axioms (damn those
first-order logicists) because well-formed formulas are
separately generated and present in the axioms of separation.
> There are a dozen set theories and a dozen interpretations of the most
> popular set theory, and 2 or 3 versions of it (with or without Choice,
> etc.) none of which decide any of the important questions of set
> theory due to exhaustive work (a waste!) by Godel and Cohen.
That is an odd thing to say. While I find forcing to be nonsense
in set theory (but I reject the axiom of extension as foundational)
it is extremely important to recursion theory, is it not? And that
has consequences for the practical application in computational
contexts, does it not?
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