Pleese help me prove the following:

Let S be a structure, while S* is a substructure of S, that’s definable
in S by a first order formula. Prove that if the theory of S* is
undecidable, then the theory of S is undecidable as well.

I think the MAD Fleet (Mutually Assured Destruction Fleet) would be
created from both NATO and the UN. It would be a NATO-UN Fleet and
would be permanent once formed. It would have contributions from any
country willing to contribute in manpower and in equipment and arms and
in money. Once formed and solving a problem like Lebanon will not
disband the MAD Fleet. Once formed it will be a permanent fleet because
nuclear weapons are a permanent fixture in human society.

To have solved the Pakistan versus India crisis of circa 2000 when
those two countries threatened a localized nuclear war over Kashmir,
the MAD Fleet would have parked off the coast of Pakistan and India and
told both countries that if either one launched a nuclear weapon the
MAD Fleet would annihilate the country. Luckily the skirmish defused
peacefully and a MAD Fleet was not needed in the early years of 2000.
But Pakistan and India could flair up again and so we need a MAD Fleet
should it recurr.

To have solved North Korea the MAD Fleet would surround North Korea,
and if the Fleet is fired upon or impeded by the North Koreans mandates
a regime change in North Korea.

Ditto for the situation in Iran.

But let me outline what the MAD Fleet would do for the situation in
Lebanon right now with the Hezbollah missile fire into Israel.

The MAD Fleet first principles would be that Peace must be first
priority. That we can change our countries for the better via peaceful
means. And that the world no longer needs violence to change our
political ambitions. So a cease fire would be demanded by the MAD
Fleet.

The MAD Fleet would have a police force CSI type unit. A unit that can
identify the maker of those Hezbollah missiles. Are they Iranian
missiles? This unit will provide physical evidence as to whether they
are Iranian or Syrian missiles.

The MAD Fleet would say that Lebanon is two countries at this moment in
time and call them Lebanon and Hezbollah-land. So Lebanon is two
countries really and the MAD Fleet treat it as such because the Fleet
is not into games-by-children. The MAD Fleet is into peace and making
the world safe from nuclear war.

The MAD Fleet would look at this situation of Hezbollah firing missiles
into Israel and fast forward into the future. The outlook is bleak
because those missiles would be nuclear tipped in the future. The MAD
Fleet takes that worst case scenario and responds to the present day
situation as what will come in the future.

Thus, it is easy to see that the Hezbollah of July 2006 is a projected
future of a Iran in 2008 that fires several nuclear tipped missiles
from southern Lebanon and thereby leveling Israel. The MAD Fleet
projects the present problem into the future and if it sees what it
does not like to see, then it preempts that future from turning into
reality.

So, the MAD Fleet today would be parked in the Middle East. It would
divide Lebanon into two countries, call them Lebanon-one and
Hezbollah-land. The MAD Fleet would demand Israel to stop its offensive
campaign and to retreat. The MAD Fleet would demand Hezbollah to stop
its missile firing, if not, ground troops of the MAD Fleet would go
into Hezbollah land and fight them. By the way, Israel can contribute
troops to the MAD Fleet.

The MAD Fleet, if Hezbollah does not stop with its missiles would send
in the CSI unit of the MAD Fleet to identify the maker. If the physical
evidence points to Iran or Syria, the MAD Fleet would have to blockade
Iran and issue an ultimatum, that the next launch of a Iranian missile
into Israel would mean a mandatory regime change of Iran.

In other words, our small world is no longer amenable to lousy
governments that will create a nuclear war or use nuclear weapons.

The clear message out of this will be– if you are a government in our
modern world and you build nuclear weapons or threaten the world peace
with missiles then you will be regime changed.

The reason the world sailed through the Cold War without nuclear war
(although we came close to it in the Cuban Missile Crisis 1960s) is
because of MAD concept and the Cuban blockade was the world’s first MAD
Fleet. We are presently naked without a MAD Fleet. And unless we deploy
a MAD Fleet soon, the world will be ignited and fall into a nuclear
war, or countries leveled by nuclear weapons with a world wide
radiation pollution.

The MAD Fleet is our best sanity insurance policy for diplomacy, for
nuclear weapons dismantling, and for countries to never want to build
nuclear weapons, and to prevent a nuclear war.

The trouble with the USA after the Berlin Wall came down, is that the
USA failed to realize that it was the concept of MAD that won the Cold
War, and it was the image of the Cuban Missile Crisis of the USA fleet
out there that was burned into the memory of the Cold War.

Today, the world is naked and unsafe, because it has missing the MAD
Fleet and what it does to nuclear weapons.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Hello everyone. There are some things that are unclear to me about the
fascinating Busy Beaver sequence.

I will explain my reasoning, and please correct me if I made any
mistake.
Let’s take the Turing machine that searches for a proof of Godel’s G.
Let’s say it has 2000 states. We can neither prove that this machine
halts, since that would prove that G has a proof, and we can’t prove
that this machine would not halt, since that would prove that G doesn’t
have a proof, which is G. So it seems like we can’t prove whether this
machine halts. But that also means that we can’t prove what is
BB(2000), even if we had all the other machines completely analyzed.

Can anyone resolve the matter?

http://web.lemoyne.edu/~giunta/Clausius.html
http://www.mdpi.org/lin/clausius/clausius.htm
Rudolf Clausius, Ueber die bewegende Kraft der Wärme, Annalen der
Physik und Chemie, 79, 368-97, 500-24 (1850):
"The ONLY change will occur in the distribution of the heat, since more
heat will be transferred from B to A than from A to B, and so on the
whole heat will be transferred from B to A. By repeating these two
processes alternately it would be possible, without any expenditure of
force OR ANY OTHER CHANGE, to transfer as much heat as we please from a
cold to a hot body, and this is not in accord with the other relations
of heat, since it always shows a tendency to equalize temperature
differences and therefore to pass from hotter to colder bodies."

http://www.fourmilab.ch/etexts/einstein/specrel/www/
On the Electrodynamics of Moving Bodies, By A. Einstein, June 30, 1905:
"We will raise this conjecture (the purport of which will hereafter be
called the "Principle of Relativity”) to the status of a postulate,
and also introduce another postulate, which is only apparently
irreconcilable with the former, namely, that light is always propagated
in empty space with a definite velocity c which is INDEPENDENT OF THE
STATE OF MOTION OF THE EMITTING BODY."

Ilya Prigogine, From Being to Becoming, Freeman and Co, 1980, p. 5:
"We must distinguish TWO TERMS IN THE ENTROPY CHANGE, dS: the first,
deS, is the transfer of entropy across the boundaries of the system;
the second, diS, is the entropy produced within the system."

Pentcho Valev

If God discovered laws of physics and math, then it means that these
laws were already in existence and hence created by someone else. For
God to be omnipotent, laws or logic must be created by God. And it can
be done only through WISH.

For example, he must take control of logic of human being just by wish
and make us *believe* that 1 + 1 can be equal to 3.

I think WISH is answer to logic.

Hello,

Does Science give us knowledge about Reality? Is Scientific knowledge
universal, necessary and certain? Or does it simply explain and
describe human experience? Is there such a thing as reality which is
independent from human experience? How can we know?  Is this the right
newgroup for these questions? Should I head to talk.philosophy.misc
instead? Should I have cross-posted there? ..?/?/ ..??/

Cheers

Ankmin

  — The person who cuts the Gordian knot instead of unravelling
     it is pretty sure to be entangled by it himself before long —

                                        Stephen F. Barker
                                   "Philosophy of Mathematics"

                   ____________________________

    This post concerns the nature of implication, with particular
reference to the relation of implication to material implication, and
the relation of validity to truth.

    Various notions of implication have existed since antiquity.  A
brief summary of some ancient Greek views on the subject is given by
William and Martha Kneale ["The Development of Logic" (1962), pp. 128-
29]:

               "Sextus Empiricus refers three times to debate on
  this topic between Diodorus and his pupil Philo. Other logicians
  added their suggestions and the dispute became so well known
  in antiquity that Callimachus wrote an epigram saying ‘Even
  the crows on the roofs caw about the nature of conditionals’.
    "In the first place where he mentions the dispute Sextus lists
  four views which had been held about the nature of conditionals.
  In the translation which follows they are distinguished by
  numbers.
                     ’[1] Philo says that a sound conditional is one
that does not begin with a truth and end with a falsehood, e.g. when
it is day and I am conversing, the statement "If it is day, I am con-
versing". [2] But Diodorus says it is one that neither could nor can
begin with a truth and end with a falsehood. According to him the
conditional statement just quoted seems to be false, since when it is
day and I have become silent it will begin with a truth and end with
a falsehood. But the following statement seems to be true: "If atomic
elements of things do not exist, then atomic elements of things do
exist." For he maintains it will always begin with the false antecedent
"Atomic elements of things do not exist" and end with the true
consequent "Atomic elements of things do exist". [3] And those who
introduce the notion of connexion say that a conditional is sound when
the contradictory of its consequent is incompatible with its
antecedent. According to them the conditionals mentioned above are
unsound, but the following is true: " If it is day, it is day." [4]
And those who judge by implication say that a true conditional is one
whose consequent is contained potentially in its antecedent. According
to them, the statement "If it is day, it is day" and similarly every
conditional which is repetitive will apparently be false; for it is
impossible for a thing to be contained in itself.’

    " [...] A passage in Cicero suggests that the third view may be
  that of Chrysippus. For it is argued that, if Chrysippus allows the
  truth of the conditional statement ‘If Fabius was born at the
  rising of the dog-star, then Fabius will not die at sea’, he must
  also allow that ‘Fabius was born at the rising of the dog-star’
  and ‘Fabius will die at sea’ are incompatible (‘pugnant’). This view
  is also attributed to the Stoics by Diogenes Laertius. The fourth
  view, on the other hand, is probably Peripatetic. This is suggested
  both by use of the word ‘potentially’ and by the rejection of the
  form ‘If P then P’, which the Peripatetics regarded as a piece of
  useless Stoic verbalism."

    Various notions of implication have been propounded in modern
times.  A standard definition is given at Wikipedia:

    "The set A entails the set B if and only if, in every model
    in which all sentences in A are true, all sentences in B are
    also true (A |= B)."

            ( http://en.wikipedia.org/wiki/Entailment )

    Dissatisfaction with standard accounts of implication has led to
the formulation of various alternative notions.  Among those I am aware
of are Lewis and Langford’s Strict Implication (¬<>( p & ¬q)), which
is a family of implications depending on which of their axiom sets you
fit it with; Lukasiewicz’s implication for his 3-valued logic; and a
host of other modal and many-valued implications.  I have seen many
references to Anderson and Belnap’s entailment relation; I would sup-
pose that in Heyting’s intuitionist calculus, implication is, in ef-
fect, somewhat different than is standard.

                            * * * * *

    Some time ago, never mind how long precisely, I took an introducto-
ry course in logic at the extension school of a large university.  On
the first day the instructor happened to mention, among other things,
we would be studying the relation of validity to truth.

    It turned out that, for sentences generally, the relationship is
simple: every valid sentence is true, because a valid sentence is
necessarily true by definition.  For implications in particular, we
learned that no valid implication has a true premise and a false
conclusion.

    I will take a closer look at this last result.  How does a valid
implication screen out false conclusions from true premises?  What, in
general, are the logical relationships among true propositions, false
propositions, and valid implications?

    I employ a visual aid, the following Figure 1, to rehearse the
basic facts of the matter, which are well known:

        premise  |  conclusion  |  implication
     |———–|————–|—————||———–
     |   true    |     true     |      valid    ||   possible
     |———–|————–|—————||———–
     |   true    |     true     |    invalid    ||   possible
     |———–|————–|—————||———–
     |   true    |    false     |      valid    || impossible
     |———–|————–|—————||———–
     |   true    |    false     |    invalid    ||   possible
     |———–|————–|—————||———–
     |  false    |     true     |      valid    ||   possible
     |———–|————–|—————||———–
     |  false    |     true     |    invalid    ||   possible
     |———–|————–|—————||———–
     |  false    |    false     |      valid    ||   possible
     |———–|————–|—————||———–
     |  false    |    false     |    invalid    ||   possible
     |———–|————–|—————||———–

                            Figure 1

    Of the eight a priori possible combinations there is only one
combination that is not possible in actuality: there are no valid
implications with true premises and false conclusions.  There is
otherwise no limitation whatsoever on the relation between truth
and validity in an implication.  Knowledge of the truth values of
the premise and conclusion will in no case allow us to establish
the validity of an implication, and will be of only occasional use
in establishing the invalidity of an implication.

    Implication is therefore not truth-functional.  Its truth value,
its validity, is not a function of the truth values of its arguments.
That the premise be true and the conclusion false is a sufficient con-
dition, but not a necessary one, for an implication to be false. There
is otherwise no logical dependency between truth and validity in an
implication.

    So why is it that this one combination is not allowed?  What is the
nature of this rather tangential relation between validity and truth?
Granted, nearly the whole utility of implication rests on this slight
relationship, but could any sensitive soul fail to be moved by that
lone irregularity in the fourth column?  Does this solitary asymmetry
exist merely by some definitional fiat?  Does the fact that there are
are no valid implications with true premises and false conclusions
correspond to any objective reality, or is it simply that we refuse to
countenance the existence of such a thing?

    I am reminded of a story from the Mumonkan, a collection of Zen
stories:  A buffalo tries to leave a field in which he is grazing by
leaving through the gate. His horns go through, his head goes through,
his four legs go through, but his tail won’t go through.  Why won’t his
tail go through?

    Why won’t that little tail go through?  Answer, or you lose your
Buddha-nature!

                            * * * * *

    One of the chief reasons for dissatisfaction with the standard
account of implication, as mentioned above, is confusion regarding the
relation of material implication to implication proper.  A clear expo-
sition of material implication and its relation to implication has
proven problematical [see Matthew C. Clarke "A Comparison of Techniques
for Introducing Material Implication"
( http://www.cs.cornell.edu/Info/People/gries/symposium/clarke.htm )].

    Material implication famously gives rise to the so-called
"paradoxical" formulae such as p->(q->p) and p->(¬p->q). There are
also
the annoying trivial implications p->V and C->p, where "V" is any valid
formula, and "C" is any contradiction.

    One source of difficulty may be the very name "material
implication", which suggests that material implication is a sort of
implication, as horned sheep are a sort of sheep, or a fast car is a
sort of car.  The situation is, actually, that implication is a sort
of material implication: no valid implication has a true premise and a
false conclusion — therefore every valid implication is a true materi-
al implication.  The modifier "material" in this context does not, as
is usual, restrict the reference of the modified common noun; it does
not elect out a sub-class of the indicated class. Like "shoe tree",
"knuckle sandwich", or "iron horse" the modifying adjective acts ir-
regularly to change the reference of the noun other than by election,
while changing itself and the noun into a compound noun.

    Here the modifying adjective expands, rather than contracts, the
reference of the modified noun: implications are a (proper) subset of
the material implications — an implication is a material
…

read more »

I was playing around with an axiomatic system, when I noticed a sort of
break in symmetry that occured between two identical systems (in that
they say the same thing, that P and Q are true).

The first system consists of:
1)P
2)P -> Q

And the second system consists of:
1)P
2)Q

Now if I perform the same operation on both, namely negating each
axiom, watch!

First:
1)~P
2)P^~Q

Second:
1)~P
2)~Q

The first system is contradictory, but the second is not necesarily
contradictory.  Thoughts?

I once heard from a professor that Lob’s Theorem in mathematical logic,
which is ‘O(O A -> A) -> OA’, was used by a philosopher of Ethics before
  it had been shown in Godel-Lob Logic (it was probably used in
connection to deontic logic).

Can anyone verify this?  I would love to read the original paper

Matthew P. Wampler-Doty

This is a webpage that basically gives all answers that people are
looking for. If you are interested in looking at it please check it
out. For the German vrsion please click on the German flag.
http://www.theomonistik.net/ee/374.htm