I’m trying to understand how to justify Universal Generalization of a variable
introduced in an assumed premise. My book seems to indicate that a variable
introduced in an assumed premise stands for a particular but as yet unknown
member of a domain. At this point it is error to universally generalize on this
type of variable. But after the premise is discharged, we are free to consider
the variable as general for any member of the domain and can therefore
universally generalize it. What frees us to consider the variable type
differently? What restricts us to considering it an unknown particular in the
assumed premise?

Thank you for any help you can give.

Hello,

I’m wondering if someone would explain to the extent they care what are
some aspects of anti-foundational logics, where the Axiom of Regularity
or Foundation is relaxed as a requirement.

In an anti-foundational logic, there is no axiom that says a set can not
contain itself as a member element, in the context of sets.  For
example, A = {A} is not a valid set in ZFC for reason of AR.  So with a
different foundation axiom, completely different constructions might be
possible, for example special cases or a class of special cases.

What other of the axioms depend on the regularity axiom to validate
them?

The foundation axiom as it is contains some ambiguity in some
interpretations, basically the discussion here yielded the result that
not the set, but it’s transitive closure (set of all subsets and subsets
of subsets of the set) that is to be tested for regularity against the
actual axiom.

For example, it should be simple enough to list three reasons why the
Axiom of Foundation can be relaxed in a set theory otherwise compatible,
and three why not or that it would be meaningless.  Yet, I do not
specifically do so now.

I’ve been reading about the lambda calculus and other predicate
calculi.  Is there some kind of meta-grammar about the formulaic and
associative stuff like that?  It uses, for example, variable assignation
and other computer programming metaphors and logical constructs.

Have a nice day,

Ross
—
Ross Andrew Finlayson
Finlayson Consulting
Ross at Tiki-Lounge: http://www.tiki-lounge.com/~raf/
"The best mathematician in the world is Maplev in Ontario."  - Pertti L.

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Required to work with Prof. Aaron Sloman on a Leverhulme-funded project
on
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==
====
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School of Computer Science, The University of Birmingham, B15 2TT, UK
EMAIL A.Sloman AT cs.bham.ac.uk   (ReadATas@please !)
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In article <schr5tssordqlabqoi6tll8r0qpqjcu…@4ax.com>,

- Hide quoted text — Show quoted text -

wrja…@kudzucountry.com wrote:
> On Thu, 21 Dec 2000 13:56:55 GMT, S D Rodrian
> <Rodr…@mad.scientist.com> wrote:

> >There ane only "three" dimensions:
> >That is all that’s required to permit
> >existence (and) motion in the universe:
> >Time is but the human habit of timing
> >(most/all) such motions (and so time
> >is a dimension, only a human dimension,
> >as  employed in relativity’s space-time
> >map–i.e. a description/measurement of
> >motions and NOT something which, like the
> >"three" dimensions of reality… permits
> >motions).

> For something to exist, it has to
> exist for a period of time.

The logical error you are committing there
is when you fundamentally believe that
"existence" literally "does not exist" but is
created/destroyed moment to moment and
instant by instant: That is absolutely NOT so!

Existence is all that exists: Right now!

It is not "things" that have some sort of
"absolutely" independent existence, but merely
that bits/portions of existence take on/take off
different shapes/forms: If "it" does not exist
RIGHT NOW, "it" never existed and never
will exist… as existence is sustained NOT by
the creation and/or destruction of its "things"
but by the simpler, more elegant reality that
its "things change."  [i.e. ... bits/portions of
existence are forever becoming other bits
and portions of existence; or, energy cannot
be created/destroyed; or, Newton's laws of
motions; or, "Man's is but a brief and mortal
glimpse of eternity, for him an immortal memory
which he forevermore will hold in his thoughts
until his end, but not until the end;" or, "the
universe is just a jumble of relativistic (independently
recognized) motions, forever some speeding up
while others are slowing down... NOT because
they are beginning/ending, but because those that
are slowing down are doing so as they "conserve"
their energy  (their very selves) "into" those motions
which are speeding up." Et cetera. And this is ever
where we came in.]

What I am describing above is, as I’ve said many
times, the fact that the "forms of matter" are just
that… "forms!" And, in the final analysis, ALL
that exists is MOTION.

There are NO fundamental forms of matter, none
at all [i.e. there are NO indivisible and therefore
"eternal/immortal" "particles" of ANY kind
at ANY level of existence]  as our universe simply
can NOT produce them except it does so by sheer
magic–There is NO possible way to physically
give birth to an Immortal: It can only spring from
the forehead of a god… full-grown & armored, as
the ancient Athenians knew "way back when.".

> Therefore time is as necessary
> a dimension as the other three.

Time has as much to do with existence as the inch:
"Nothing exists just because it is an inch, and
nothing cannot exist because it is not an inch."
As the greatest poet of the Twentieth Century
wrote (recently)… "An hour is but an arbitrary
invention of necessity by man just as a liter also
is but our own arbitrary invention of necessity.
The problem is that whereas no one is claiming
that you can’t have beer without the liter, far , far
too many of those who have chugged down a "few
too many" themselves ARE claiming that you can’t
have MOTION without Time!" But this confusion
will most probably always be with us, since it’s much
more a matter of our "human nature" than anything else.
Just you keep in mind that things are NEVER as
confusing as the confused always MAKE them out to be.

And life will most often be a lot easier/pleasant.

S D Rodrian
web.sdrodrian.com
sdrodrian.com
music.sdrodrian.com

PS All quotations are original with me. And, by the
   way: remember that the description of our reality
   as "three-dimensional" is only a mere conceit of
   convenience: It comes from a mostly arbitrary and
   quite artificial "geometrical object" one seldom
   finds in nature… the box. Our reality should
   more accurately be called NOT "three-dimensional"
   but "spherical" since a perfect sphere describes
   ALL the dimensions possible in our universe (without
   leaving out a single one of them) better than anything
   (else) anyone can think up.

> William R. James

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                         UCLA LOGIC COLLOQUIUM
                        Friday, January 19, 2001
                               4:00 p.m.
                       Mathematical Sciences 6627
                                 UCLA

                             DREW MOSHIER

                           (Chapman College)

       "MULTI-LINGUAL SEQUENT CALCULI: PROOF THEORY AND TOPOLOGY"

We introduce Gentzen-style sequent calculi where the formulas on the
left and right of the turnstile need not come from the same logical
system.  In such calculi, sequents can be seen as consequences between
different domains of reasoning.  We discuss the ingredients of proof
theory needed to set up calculi generalized in this fashion.  The
result is the category MLS (multi-lingual sequent calculus) in which
the morphisms are generalized calculi and objects are calculi that
capture "domain internal" reasoning.

To develop a semantic theory for MLS, we turn to spectral theory,
showing that the MLS objects give rise to spectra (spaces of prime
filters) that are, up to homeomorphism, exactly the locally compact,
strongly sober spaces (aka Nachbin spaces).  Such spaces, studied first
by Nachbin in the 1940′s, are arguably the natural T_0 generalizations
of compact Hausdorff spaces.  The morphisms of MLS correspond to
"continuous relations" between these spaces.  In turn, a continuous
relation from X to Y corresponds to a continuous function from X to the
Smyth powerdomain of Y.  Thus we establish category equivalences: MLS
to N^* (the category of Nachbin spaces and continuous relations) and
MLS to the Kleisli category for the Smyth powerdomain monad on N
(Nachbin spaces and continuous functions).

Several useful topological concepts, such as the Hausdorff condition,
zero-dimensionality, functionality of a relation, and properness of a
function (f^-1 preserves the "well-below" relation on opens), have
interesting characterizations within MLS as proof-theoretic properties
of the corresponding sequent calculi.  As time allows, we will discuss
some of these characterizations and demonstrate how proofs within MLS
of familiar topological results shed light on the topological concepts
involved.

_________________________________________________________________________

                           http://www.math.ucla.edu/~hbe/logic.html

It would be hard to find a better example
than is provided by the post quoted below of
why it’s ALWAYS better to look upon reality
with one’s own eyes… rather than through the
assembled puzzles of formal logic’s painstakingly
deliberately blind "arrangements & arrays" of "all
related minutia/details" (supposedly to "see" the
"big picture" objectively) and/or mathematics [&
every other formal discipline whose function most
properly lies with instituting internal agreements
in their own equations]. In effect, their purpose is
to resolve all "their" internal self-contradictions
rather than to explain anything of the universe
which may appear contradictory to us—That is:
"There are no paradoxes in Nature, only in our
minds." And the true purpose of all such formal
disciplines ought to be to help us understand why
we may not be understanding (interpreting) reality
correctly… rather than the pretense (fallacy) that
they can resolve "the paradoxes inherent to reality"
for our benefit (or, better put, to our gratification
… as if it were the mind that is the true reality and
the universe but merely a puzzled "mirage" which
must inevitably yield (answer) to us once we come
close enough to it). [And you can read in the poster's
appendix the strength of his superstition that he has
eliminated human biases by looking at the matter
NOT with his eyes, but with his formal discipline.]

It is hardly surprising, therefore, that… the Book of
Genesis boasts how language itself ("the word" that
is really "our words") created Heaven and Earth. And
why –to this day– persons who really ought to "know
better" keep proposing physical impossibilities merely
because they are metaphysically possible (and all so
very "real")… in their dreams.

S D Rodrian
http://web.sdrodrian.com
http://www.sdrodrian.com
http://music.sdrodrian.com

re:

In article <93pf3t$hi…@news8.svr.pol.co.uk>,
"Alastair Malcolm" <amalc…@physica.freeserve.co.uk> wrote:

There may be some interest here in the following key points of a
little publicized explanation of our universe, variations of which
several professional researchers are starting to take very seriously.

This post takes as its starting point an objective world, and the
absence of any God/Creator (a justification for this starting point
is in the website given below). The Anthropic Principle can then be
used to reject the case for a single universe: the odds of the
physical laws being precisely tuned by luck to give life as we know
it are extremely remote (for other life-types see web site below,
p102); moreover a similar argument can be used to dismiss the case of
many universes falling under one set of overarching physical laws
(eg many worlds theory of quantum mechanics) as an ultimate
explanation: why should the underlying physical laws happen
to be just right to generate a large number of universes, with a
conveniently wide range of physical conditions? And why this
*particular* set of overarching physical laws?

Which only leaves us with the case of many worlds, *not* constrained
by an overarching set of physical laws: all logically possible
universes exist.  The lesson of the likely infinities of space and
time in *our* universe should have taught us that nature does not
pander to the limitations of our minds, so the slight mind stretching
that is needed to accommodate this case should come as no surprise.

A little further mind stretching is needed to refute the main objection
to the All-Logically-Possible-Universes (ALPU) hypothesis, which is:
why do I happen to be in one of the few possible worlds governed solely
by the relatively simple physical laws – why not in one of the far more
numerous worlds where crazy things happen, like a dragon suddenly
swooping by? To refute this objection, we need a human-bias-free way of
representing our world that automatically encapsulates the physical
laws (rather than the survival-geared image that is created by our
minds), such as some mathematical model based on the (yet to be found)
‘Theory of Everything’, or a minimal length bit string containing all
the information about the universe. Using these, it can soon be shown
(see web pages p105, p112; or the Appendix below for a brief summary)
that (perceptibly) simple universes predominate, so we should not
expect to see dragons or other paranormal events: ALPU is the only
reasonable explanation for our universe.

More details starting at
http://www.physica.freeserve.co.uk/p101.htm

(For anyone interested in fundamental questions in general, please
start at http://www.physica.freeserve.co.uk/p001.htm)

 Alastair Malcolm

APPENDIX: If we break down the human-bias-free representation
of our  universe into logical units (such as the axioms of a
mathematical representation, or the bits of a minimal informational
representation), and let our universe be minimally specifiable by
n such logical units and an alternative universe containing a dragon
by n+d units (more units because dragons add complexity), then we
can say that for all logically possible universes specifiable by
m units (where m is any number one likes n+d), there will be far
more possible combinations of units containing the particular n units
of our universe (with the excess m-n units specifying entities and
space-time regions invisible to us), than those containing the
particular n+d units of dragon-containing universes (which only have
m-n-d extra units for combinations). (Note the number of possible
invisible entities, including those in other space-time regions, will
vastly outweigh possible visible entities like dragons.) Hence we
should be in the simplest possible universe (in terms of visible
entities) consistent with the existence of thinking beings – no dragons
or other paranormal events will occur. (Hence the success of induction,
and Occam’s Razor, as applied to the physical world.)

– 30 —

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From: Russ Schneider <russ…@stargate.net>
To: <SDRodr…@mad.scientist.com>
Sent: Tuesday, January 09, 2001 7:54 AM
Subject: talk.philosphy.misc

> I came across a thread in talk.philosophy.misc
> which you may or may not
> have started concerning time.

Not guilty: Here’s the start/starter
who originally misspelled "dimensions" …

START URL

http://x54.deja.com/[ST_rn=if]/threadmsg_ct.xp?AN=707220356.1&mhitnum=0&
CONTEXT=979451385.1225588741

END URL

> Unfortunately the thread starts in the
> middle and I cannot find the original posting.

Some urls on the threads:

http://www.deja.com/=dnc/[ST_rn=ps]/threadmsg_ct.xp?AN=706597086&fmt=tex
t

Wait! Go to the URL below, where you will
find ALL the relevant different threads
on this & related subjects in one neat bunch
–and remember that you MUST "paste" the entire
URL (below & elsewhere) into your browser, and
NOT just merely click on the http:: link.

START URL

http://x30.deja.com/[ST_rn=if]/topics_if.xp?search=topic&group=alt.sci.p
hysics.relativity&GRPP=945795855.1065549904&title=Related&query=einstein

END URL

You could also go to http://web.sdrodrian.com
where you’ll find a group of links to
practically everything I’ve been posting
for the last years or so on "these subjects."

> I found this quote which I assume is yours,
> "I have explained why there is
> no past/no future" and was wondering if
> you could elaborate on the issue if
> this quote does indeed belong to you.

Guilty: The relevant gist of what you’re after
is probably located somewhere IN
the following single very un-neat post
whose URL is quoted below:

START URL

http://www.deja.com/[ST_rn=if]/getdoc.xp?AN=712813067&fmt=text

END URL

Well, actually, I’ve been arguing the same thing
all over Usenet’s newsgroups for half a dozen years
or so… at almost every level, discipline & approach.

> I used to believe this sort of idea concerning
> time, but a new question has
> raised it’s spectre in my mind and now I’m unsure.

Hope I can either reassure you
on the validity of your former conviction
or, at least, on your ability to eventually
convict yourself on your own!                   [sic]

S D Rodrian
http://web.sdrodrian.com
http://www.sdrodrian.com
http://music.sdrodrian.com

re:

> Any info you can relay me on the topic
> to support your quote would be most
> helpful.
> Thank you.
> Russ Schneider

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I need a reference that points out the fact
that quantified monotone boolean formulas
are decidable in linear time.

A monotone boolean formula is composed of ANDs
ORs and boolean variables, no negations.

They are decided, just by plugging in 0 for xj when
xj is quantified with forall, and 1 when
xj is quantified with exists.  Iff the
assignment evaluates to true, the quantification
is valid.

I remeber discussing this briefly on comp.theory
in early 1998, but I do not know any citation.
Thanks for your help, please send email.
Dan Pehoushek

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In article <29807-3A6276E2…@storefull-264.iap.bryant.webtv.net>,

GravityPhys…@webtv.net (tj Frazir) wrote:
> one last time.

I’ll take you at your word.

> curve in space time

An F-15 over Iraq.

> electrons apear in a patern in time.

Or they crash?

> time …A unit of space

Spaced-out time is meaningless:
Time is simply the timing of
one motion vs another motion(s).

> is ( presure in a piriod of time. )  All space
> is presure maintained.

Well, you have you work cut out
for you… explaining how gravity
and pressure can exist in the same
space at once! Good luck!

> Mater contibutes less presure to space
> than empty space dose.

And water burns, so in order to dowse it
one must pour gasoline over it.

> Electrons
> are part of space not mater.

Everything ultimately reduces to
motion. The problem we have in defining
"energy" ultimately as "motion" is that
our idea of physicality is exclusively based
upon its materiality (so motion without
something material moving is impossible
for our brains to compute).

> HO well….sigh
> Im taking my 1465mpg  88 cadalac
> to detroit this week. !

Wow! Impressed! (Them things’s gotta
weigh a ton!) Are you going to throw it
over your back, or just drag it to Detroit
with a rope?

S D Rodrian
wisdom.findhere.org
sdrodrian.com
web.sdrodrian.com

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   Where can I find the proof of the completeness of
Lukasiewicz-Slupeki’s three-value logic?
   Thanks in advance.

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