fom
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Registered:
12/4/12


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


On 2/3/2013 10:19 PM, CharlieBoo wrote: <snip>
>> >>>> In PROLOG we use lowercase words for TERMS >>>> and uppercase words for VARIABLES >> >>>> ATOMIC PREDICATE >> >>> ATOMIC PREDICATE meaning relation? >> >>> CB >> >> 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.
Wellformed formulas are built from the alphabet of a formal language. If the language contains a symbol of negation, then NOT(xex) will be a wellformed 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 firstorder 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 2dimensional subspace of the domain. In a 3dimensional 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 fourfold interrelating POSSIBLE and NECESSARY. Deontic extensions are fourfold interrelating OBLIGATORY and PERMITTED.
For quantificational logic, each variable has this fourfold structure. This corresponds with the indexing of quantifiers found in the cylindrical algebras of Tarski's later work.
The negation symbol masks this fourfold structure in the formation rules for formulas.
Yes. Logic in the absence of NOTjust like in the hardware of your computer systemsis 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 firstorder logicists) because wellformed 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?

