The system RCA0 is central to "Reverse Mathematics". It can be proven
consistent in first order Peano arithmetic PA. But what happens if we
add the PA induction schema to RCA0? I was wondering what you would
get.

…The claim that there is a real connection between experience and
brain is taken from the idea that the correlation is ‘proved’ from a
correlation of their activities. But this can’t be taken as a ‘proof’
that brain causes experience, because the brain activity is defined or
pointed out by experience. Can we pull a logical rule out of this?

1) Only a ‘correlation’, association, or mapping, can be made between
elements or objects from incommensurable frameworks. In our example,
brain and experience are incommensurable objects because the
frameworks of mind and matter have different rules for the behaviours
of their objects. So only a correlation between the events of each can
be made. We can say only that there is a correlation between mind
events and material events. Mind and matter do not partake in
reciprocal causality.

2) A ‘proof’ that there is a connection between objects requires that
the objects are not incommensurable but are taken from the same
framework.*  So we cannot prove that there is a real connection
between mind and matter. Proofs are always carried out in the
framework of the objects being proved. We cannot then, prove from a
correlation of brain activity and experience a causal relationship
between them.

The logical rule I want to pull out of this is really about fine-
tuning the concept of relationship. Framework disjunctive objects in
‘relationship’ are mappings or correlations, as is the case for common
framework objects in ‘relationship’. However, only relationship of the
latter kind can be ‘proven’. A proof establishes a mapping between
objects as their necessary condition. No such condition exists between
objects from incommensurable frameworks, between the objects of brain
and experience, for example.

* ( Note here that a correlation requires no proof. A set also
requires no proof for its element membership, but unlike a correlation
a set, if it is empirically sound, is constructed of commensurable
objects)

Hellow all,

I. AIM

  A theory that can proof the existence of universal sets like the set
of all sets, the set of all sets bigger than a set, the set of all
sets that are not a specific set, …etc, and that has theorems that
enable us to make consistent logical analysis of these sets without
being involved in the known main paradoxes of Russell’s , Burali-Forti
and Cantor’s.

 And at the same time a theory that we can construct ordinals in it
in a simple manner, and that has a lot of the structure that standard
set theories like ZF and MK has.

So this theory differe from ZOT ( see topic: christmass  theory in
this usenet ) in that is has a subtheory of it resembling the standard
theories, so it is not as distant from standard set theories as ZOT
is.

II. General idea about this theory:

 DST is named after the most essential feature of this theory that is
it has two primitive binary relations_internal membership ‘e’  and
external membership ‘@’.
So yex is read as: y is an internal member of x
and y@x is read as: y is an external member of x.
In this theory every object is an external member of itself.
Two objects are identical iff they have the same internal
and external members , and for the sake of simplicity all sets have
only one external member that is themselfs. Comprehension requires
external and internal membership to satisfy a certain predicate P or
be the set itself, the following is the exposition of this theory.

III. EXPOSITION:

DST is the collection of all sentences entailed (from first order
logic with identity and the primitive binary relations e and @) by the
following non logical axioms:

1) External self inclusion: Ax x@x

2) Extensionality: Az ( ((zex or z@x) <-> (zey or z@y)) -> x=y ).

3) Comprehension schema: if P is a formula in whcih x is not free,
then all closures of
E!xAy( (yex or y@x) <-> (P(y) or y=x) )
are axioms

Definition:

x is P_defined <-> Ay( (yex or y@x) <-> (P(y) or y=x) )

4) Simplification: Ax Ay (y@x -> y=x)

5) Internal membership: Ax ( x is P_defined -> (P(x) -> xex) ).

Definition:

x is transitive <-> Ay Az ( (zey & yex) -> zex )

x is regular <-> ( Ey yex -> Ey( yex & y disjoint x) ).

y subset x <-> Az( zey -> zex )

x is ordinal <-> ( x is transitive & Ay(yex->y is transitive) &
Ay( y subset x -> y is regular ) ).

7) Ordinal existence: Ex x is ordinal

7) Ordinal internal membership schema:

For every formula P, the sentence

(Ax(P -> x is ordinal) & ~Ax( x is ordinal -> P ) ) ->

Ax ( x is P_defined -> (~P(x) -> ~xex) ) .

is an axiom.

Ordinal successor schema:

For every formula P , the sentence

(Ax(P -> x is ordinal) & ~Ax( x is ordinal -> P ) )  ->

Ax( x is P_defined -> Ey( xey & y is an ordinal ) )

is an axiom.

/ Theory definition finished.

Some Theorems:

E!x Ay ~yex

Proof: let P<-> ~y=y
from comprehension: E!xAy( (yex or y@x) <-> (~y=y or y=x) )
we have ~P(x)
Now we have
( Ay(~y=y -> y is ordinal) & ~Ay( y is ordinal -> ~y=y ) )
because Ey y is ordinal ( axiom 6)
So from ordinal membership we have ~xex.
Now every y that is not x will not be internal nor external member of
x. So x has no internal member at all, and it only has itself as
external member.

Theorem proved.

Definition: x=0 <-> Ay ~yex

Theorem: Ey ( y is ordinal & 0ey )
Proof Ordinal successor schema with P<->~y=y

Theorem: E!x Ay ( yex <-> y=0 )

Proof: let P<-> y=0
from comprehension we have:

E!xAy( (yex or y@x) <-> (y=0 or y=x) )

Suppose that x=0
then from axiom of internal membership we
will have 0e0, a contradiction!
 thus ~x=0
Now from comprehension 0 is either internal
or external member of x,But from axiom
of simiplification we only have x@x
and since ~x=0 , then 0ex.
all other y that do not satify y=0 or y=x
are neither internal nor external members of x.
Now is xex or ~xex?

Now P<->y=0 do satisfy the condition

(Ax(y=0 -> x is ordinal) & ~Ax( x is ordinal -> y=0 ) )
because of the axiom of ordinal successor since 0 is
~y=y_defined so there exist y that has 0 as a member
and y is ordinal, thus not every ordinal is 0 ( see theorem
above).

then since ~P(x) then we have ~xex. ( schema of ordinal internal
membership ).

 Theorem proved.

In this way we can build all the ordinals we need in this theory,
therefore bypassing the difficulty of constructing ordinals in ZOT
 ( see thread: Christmass theory in this usenet ).

IV. The position of this theory from known Paradoxes:

This theory avoids Russell’s paradox.

Let P<-> ( ~yey & y@y )

Then we will simply have the set x of all these sets that are not
internal members of themselfs OR the set x itself ,Now if ~xex then x
is fulfilling P above, then xex a contradiction!
On the other hand if xex then from axiom of internal membership we
either have xex or ~xex, but we have xex thus the result is xex
 No contradiction!

Thus x is an internal member of itself.

 This theory avoids Burali-Forti paradox.

In this theory the segment of all ordinals smaller than the set of all
ordinas is the set of all ordinals itself ( due to the uniqueness
condition in comprehension ) thus Burali-Forti paradox avioded.

NOTE: the conditions for Ordinal internal membership is not fulfilled
for the formula x is ordinal since we have
Ax( x is ordinal -> x is ordinal ).

So the set of all ordinals or itself would be not an ordinal that is
in itself.

This theory avoids Cantor’s paradox:

Cantor’s argument for every set being smaller than its power set is
not prooved in this theory. Here the set of all sets will be its power
set resolving the paradox.

So this theory aviods the major paradoxes, and at the same time
we can construct ordinals in it in a much simpler way than ZOT.

V FINAL word. This theory seems complicated with a lot of fixes to
allow for the construction of ordinals ( Axiom 6, and schemas 7 and
, thus it might be inconsistent.

Zuhair

WHY עשוהי, I.E. JESUS, IS NOT THE JEWISH MESSIAH IF הוהי is the LORD

Many Christians claim that הוהי is the LORD and Elohim, while his son
עשוהי is the Messiah.

All Jews reject this and rightfully so.  If it were true, don’t you
think there would be at least one passage in the old testament to that
effect, saying, oh, by the way, הוהי has a son named עשוהי and he will
come as the Messiah?

But there is no such passage, not even the hint of such a passage, and
so the theory of these “Christians” is rightfully rejected by almost
all Jews as unfounded according to the Jewish religion.

Now, my hypothesis, however distasteful it may be to Jews, is a
possibility they cannot altogether deny if they wish to be thought of
as rational human beings.

I claim that Moshe did not know the true identity of the LORD, and
this is signified by the fact that the LORD would not allow Moshe to
see HIS face, according to the account of Moshe’s enlightenment
experience that is given in Exodus itself.

I claim that the true name of the LORD is עשוהי or JHVSA or Jehoshua
Moshiach, aka Jesus Christ.  I claim that JHVH is the Divine Name of
the angel Gabriel, even as I AM THAT I AM = EHYEH ASHER EHYEH = AHJH
ASR AHJH is the Divine Name of the angel Michael, who is the angel of
the LORD seen by Moshe in the burning bush.  When it says that the
LORD knew Moshe face to face, it means that HE knew him face to face
through HIS angel, namely Michael as I AM THAT I AM.  It is from
Michael that Moshe received his prophetic utterances, known
collectively as the Torah, even as Mohammed received his prophetic
utterances, known as the Koran, from the angel Gabriel.

The LORD HIMSELF was very clear that Moshe could not see HIS face,
i.e. know who and what HE truly is, and live.  That is why the
teaching of Jehoshua Moshiach emphasized the willingness to die to see
G-D.  Even Jesus Himself had to die in order to return to His own
rightful estate as the LORD G-D.  The angel of the LORD, Michael =
Mahavatar Babaji = ABBA = the Heavenly Father of Jehoshua Moshiach,
while He was incarnated as Jesus Christ.  Hence the statement of
Jesus:  “Believe me that I am in the Father, and the Father in me: or
else believe me for the very works’ sake.”  (John 14:11)

My conjecture, if true, points to the reconciliation of the three
great monotheistic religions.  The alternative is endless fighting,
for none of the three will relinquish their claims.  The Jews suffer
more than anyone from the sin of Moshe in fabricating the name of the
LORD.  The penalty for that sin, multiplied by the ongoing
participation of all Jews, was the destruction of the Temple and
diaspora, i.e. sangsara, that will last forever, unless the Jews
somehow discover the Truth and repent.

As a half Jew myself, I am not just proselytizing or evangelizing.  I
am simply sharing what seems to work for me in finding a Truth of the
Bible that is not monstrous, but is instead both believable and
intelligible.  Moreover, as a physicist interested in the foundations
of quantum theory, I am interested in finding a Truth that will answer
our deepest questions about reality.  I believe I have found it.

THE SIMPLE ABBREVIATED TRUTH ABOUT G-D

The simple abbreviated truth is that Moses could only see the backside
of the LORD he saw in the burning bush, and so he could not discern
His true identity.  You can’t identify someone from behind, especially
if you don’t already know who they are.  The true name of the LORD was
and is JEHOSHUA, or JHVSA, not JHVH.  The entire Torah and all
secondary literature is in a sense a great big fraud, and you can see
that if you go back to its origin in the actual experience of Moses as
described within the Torah itself.  Dogma is only as good as the depth
of enlightenment of the one or ones fashioning the dogma.

BTW, Moses did see the angel of the LORD in the burning bush, and it
was the LORD through that same angel who knew him face to face. Never,
according to the Torah, was there face to face contact between the
LORD Himself and Moses.  The LORD absolutely forbade that.

SCRIPTURAL PROOF OF MY CONJECTURE

From:  http://www.answering-christianity.com/godtitle.htm :

“Mighty God” in Isaiah 9:6 is “El Gibor”.  This is not exactly “Mighty
God,” but close.  “Strong" is more correct (but it is different from
strong of “hazak”).  Here both El and Gibor are nouns – this is short
full spell is “El Hu Gibor.”

Anyway “El Gibor” and “Gabriel” are same thing.  They both mean
“Strong God.” “Gabriel” is an angel’s name in the Bible.

The word “Gibor” in Isaiah 9:6 and the word “Gibor” of Gabriel have
exactly the same root, and they are both the same word.  The word
itself can also be translated as “Man of God.”  [END QUOTE]

So, the very passage used by Christians to prove that Yeshua is the
Jewish Messiah, instead proves that Jews and Christians alike worship
the Antichrist, the Abomination of Desolation, namely the angel
Gabriel in the holy place of the LORD, where he ought not to sit.

Now the Gabriel that responds to the invocations of Jews and
Christians alike is NOT the holy angel Gabriel, for he would not
imposter the LORD.  Rather, it is an astral plane imposter of the
angel Gabriel, who is himself a fallen angel of the worst sort.

All of the monstrosities of the old testament, as well as those same
monstrosities carried over into the new testament by the
misunderstandings of the disciples of Jesus, are the work of this
fallen angel imposter of the LORD.

Now, this is not to say that the formula in Isaiah is entirely false.
In Judaism the usual designation for the one that is worshipped is
“LORD God.”  God, or El, aka Eloah, aka Elohim, is the suffix found in
the name of each Archangel:  Micha-el, Gibor-el, etc.  Furthermore, it
is well-known that ALLAH comes from AL ILAH, where ILAH is equated to
ELOAH.

So, Islam is that religion wherein the angel of the LORD can actually
be the LORD, but that is not the case with Judaism, which is a
Messianic religion geared toward the advent of Jehoshua Moshiach, the
LORD, designated by the letters JHVSA.  One can only conclude that the
child referred to in Isaiah 9:6 is Mohammad.  He is called a child in
the same sense that Moshe is sometimes referred to as a child.  There
may be a reference by implication to Moshiach, yet another child, but
the passage literally refers to Mohammad, the messenger of ALLAH
through the angel Gibor-el, whose Divine Name is JHVH.

Moshe was confused about this and all subsequent Jewish prophets and
rabbinical commentators were also confused.  Moshe’s confusion began
with his enlightenment experience, wherein he saw the LORD in the
burning bush, but only HIS back parts could he see.  From that limited
experience, he could not properly identify the LORD.

HOW THE DELUSION THAT JHVH IS THE LORD BEGAN

I don’t see a contradiction between the fact that Eve said she had
gotten a man from JHVH and the fact that the Patriarchs did not know
the name JHVH.  Eve’s delusion was that the LORD was JHVH.  She got
that from the serpent.  Moshe ben Adam inherited that delusion from
his line of Jewish mothers.  What could be more natural.  The delusion
had gone into remission at the time of Abraham, Isaac, and Jacob, but
resurfaced again with Moshe.

Eve’s belief that JHVH is the LORD is like an Oedipal complex.  Its
remission at the time of the Patriarchs is like the period of
latency.  Its re-emergence at the time of Moshe is like puberty, where
the Oedipal complex resurfaces, but this time resolved and integrated
into the social aspect of life.  The question is whether the
integration into social life is adequate or psychotic.  That is a
matter of personal judgment.  I believe that the monstrosities of the
old testament tip the balance scale toward the psychotic.  It would be
far better from the spiritual point of view to just uproot the Oedipal
complex altogether and establish a new foundation for the structure of
the psyche.  That is the idea of a new testament and covenant.

THE SECRET DOCTRINE OF MONOTHEISM UNVEILED

Exodus
33:11 And the LORD spake unto Moses face to face, as a man speaketh
unto his friend.
33:18 And he said, I beseech thee, show me thy glory.
33:19 And he said, I will make all my goodness pass before thee, and I
will proclaim the name of the LORD before thee; and will be gracious
to whom I will be gracious, and will show mercy on whom I will show
mercy.
33:20 And he said, Thou canst not see my face: for there shall no man
see me, and live.
33:21 And the LORD said, Behold, there is a place by me, and thou
shalt stand upon a rock:
33:22 And it shall come to pass, while my glory passeth by, that I
will put thee in a cleft of the rock, and will cover thee with my hand
while I pass by:
33:23 And I will take away mine hand, and thou shalt see my back
parts: but my face shall not be seen.

Deuteronomy
34:10 And there arose not a prophet since in Israel like unto Moses,
whom the LORD knew face to face…

Now, how are we to understand this apparent contradiction?  It is
essential, for the understanding of who and what G-D is depends upon
it.  On the one hand, the LORD speaks to Moses face to face, and yet
Moses cannot see His face at all, but only His backside.  Before we
make an attempt at an exegesis, we must consider also the following:

Exodus
3:2 And the angel of the LORD appeared unto him in a flame of fire out
of the midst of a bush: and he looked, and, behold, the bush burned
with fire, and the bush was not consumed.
3:3 And Moses said, I will now turn aside, and see this great sight,
why the bush is not burnt.
3:4 And when the LORD saw that he turned aside to see, God called unto
him out of the midst of the bush, and said, Moses, Moses. And he said,
Here am I.
3:5 And he said, Draw not nigh hither: put off thy shoes from off thy
feet, for the place whereon thou standest is holy ground.
3:6 Moreover
…

read more »

To Marshall and other dullwits:

According to you:  only systems with symbols that you like constitute
logical systems.  Poor logic.

Hi all,

It is often told that ZF(C) avoids Russell’s paradox, because of its
limited comprehension schema (Separation). This is a falsy!

The correct statment is that Separation prevents the exitence of the
set of all sets that are not in themselfs USING the negation Russell’s
paradox (which is already a theorem of FOL), and that is not
prevention of Russell’s paradox due to separation per se.

What prevents the existence of Russell’s paradox without using the
negation of the paradox itself is actually Regularity and Pairing.

Because the set of all sets V will be prevented since it would be in
itself by definition and from pairing we can construct {V} which
violates Regularity.

Without pairing, Regularity canNOT do the job, since V would satisfy
regularity even if it is in itself.

Since V is prevented from existence by Regularity and Pairing, and
since regularity imply that Ax( ~yey <-> y=y ) , then the set of all
sets that are not in themselfs is prevented from existence, therefore
Russell’s paradox doesn’t raise.

ZF(C) wihtout regularity doesn’t really avoid Russell’s paradox due to
its axiomatization per se, it rather uses the negation of Russell’s
paradox with separation to avoid the existence of the set of all sets
that are not in themselfs. So separation is not really a solution to
Russell’s paradox. Separation is a schema that confirm with the
negation of Russell’s paradox! But since it doesn’t prevent the
paradox by itself and using theorem in FOL other than the negation of
Russell’s paradox , then it remain questionable weather Separation
provides us with the limitation on comprehension sufficient to avoid
the paradox.

To clarify matters just suppose that the negation of Russell’s paradox
is not a theorem of FOL, would separation then prevent the existence
of the set of all sets that are not in themselfs? definitely NOT!

while Regularity and Pairing would prevent the existence of such set
even if the negation of Russell’s paradox is not a theorem of FOL with
identity

In Quine’s set theory, matters are different, the paradox is prevented
due to the axiomatization itself and the multisorted FOL that it uses,
so it is a more solid way of preventing Russell’s paradox.

Also my theory: DST (dual set theory) is presumed (by myself) to
prevent Russell’s paradox due to its axiomatization and its language
without refering to the negation of the paradox itself, so if DST is
consistent, then it affords a solution to Russell’s paradox
(in addition to Burali-Forti and Cantor’s paradox) that is simpler
than Quine’s.

Zuhair

The logical magician (logician, for short) is fond of surprising
unsuspecting friends with sentences such as, "I am lying". A friend is
likely to reply, "And what are you lying about this time?"

The logician will persist with something like, "No, I am lying right
now!" The perplexed friend will likely reply, "All right then, but
just what is it that you’re lying about right now?"

"Don’t you get it?" the logician will continue, "I thought you were
supposed to be intelligent. If I’m telling the truth then I’m lying,
and if I’m lying then I’m telling the truth. You do see that, don’t
you?"

If you don’t see, your friend is likely to show you the old logician’s
card trick. (All magicians have at least one card trick.) On one side
of a card there will be the sentence, ‘The sentence on the other side
of this card is false’, and on the other side there will be the
sentence, ‘The sentence on the other side of this card is true’. "You
do see, don’t you, that if the first sentence is true, then, according
to the false second sentence, which says the first sentence is true,
the first sentence is false? Is the paradox clear to you now?"

Although logicians rarely become violent, it is best to humor them
until you can find some excuse to get away from them. Head for a
sensible coffee shop, have a nice cup of coffee, and reflect on what
you have been told.

The truth is that logicians want you to believe that sentences can
refer to themselves, or to one another back and forth. But sentences
do not say or write anything. They come from the minds of the people
who, for one reason or another, speak and write them, and they express
the thoughts that such people have just been thinking.

For example, a helpful man who has just been asked if he has a
dictionary handy, might picture his own red dictionary on a cluttered
bookshelf and then reply, "My dictionary is on the shelf in my study".
If he had forgotten that he left the dictionary on his desk, then his
sentence would be false, but he would have made an honest mistake. On
the other hand, if someone tried to convince us that "Circles have
square corners", we should have serious doubts about either his
character or his sanity.

But what of the logician who shows you his paradoxical card? He wants
you to accept it at face value, as if the two sentences were referring
to one another right now. But we know that the card was once blank and
that the logician had been thinking about what he would write on the
card before he began writing. Then he wrote one of the two sentences,
e.g., ‘The sentence on the other side of this card is false’. Is this
really what the logician was thinking, when he knew that the other
side of the card was blank? No, the logician was engaging in an act of
deception. He had already planned to write the two sentences so they
would have the appearance of referring to each other.

Gene Ledbetter

You can ban people from using google forums. Pjmutn… the religious
spammer has banned "John Jones" who was attempting to stop his
spamming.

HERE’S WHAT TO DO:
Google can quickly ban people. Let’s get pjmutn banned from his
spamming. Simply report his post by clicking on his post, select ‘more
options’, then select ‘Report this message’, then say what he is
doing. (repeat posts, irrelevant religious material, SHOUTING).

     Joseph is said to have interpreted the dreams of the baker and the
cupbearer to refer to their immediate futures. He correcty predicted that
the royal baker would be iimpaled, and the royal cupbearer would be restored
to his position.

     But maybe he just guess, and the odds were fifty-fifty that one man
would live, and one man would die. On the other hand, why couldn’t both men
remain imprisoned (TT), or both men be freed (HH), or one man stay in
prison, and the other be released (H v T). As it happened, both were freed
from prison (HH), but one was executed, and one was restored (H v T).  So
the odds of Joseph guessing everything correctly were, what?  l/4 x l/2 x
1/2 =  l/16?  Difficult, but not impossible.(Not that I coulddo it, mind
you.)

   But if he was just guessing, why did he pick the cupbearer to give his
petition for clemency to?  Maybe it was a cadgy guess. The cupbearer would
be closer to Pharaoh, and have his ear, whereas the baker most likely
wouldn’t. So why not predict the cupbearer would live?

   But, wait. How could Joseph predict the release of the prisoners, and the
execution of the baker, and therestoration, rather than the dismissal, of
the cupbearer?

   Possibly they were thrown in the royal prison until Pharaoh could arrive
at a correct judgment of guilt; perhaps one was innocent,and the other was
guilty,and Pharaoh was not sure which to blame. Meanwhile, Pharaoh kept them
in prison for a year just to make sure that they both got some punishment.
The fact that Pharaoh got angry at both the cupbearer and the baker, and
threw them into prison, suggests either that he thought one was culpable,
but wasn’t sure which, or he thought they were both responsible together for
some sort of offense. The fact that Pharaoh resdtored the baker shows it was
not a case of conspiracy against Pharaoh.

   The usual punishment  for anyone convicted of an offense against Pharaoh
might have been excution by impaling; after all, one must set a public
example when it comes to poor etiquette in others. So Joseph would have
known that. He could have also known from the chief jailer that they would
be released in 3 days.

    Now, Joseph may have misinterpreted both dreams. They weren’t really
prophetic at all, but reflected the individuals dwelling on their crime. The
crime of the baker had been, as we can infer, to let birds poop on Pharaoh’s
special birthday cake, covered with white frosting and bits of chocolate. In
other words, he left the kitchen window open.
The crime of the cupbearer, as may be inferred from his dream, is that he
did not personally prepare the grape juice for the Pharaoh (who was god/man,
and therefore a teetotle, or maybe an adolescent, and his mother did not
allow him to drink wine.), or else he did not personally hand the cup to
Pharaoh. I suspect that the cupbearer completed the first half of the task,
but then he took the cup of grapejuice into the kitchen to personally
inspect the cake, but he had to go to the bathroom, and while he was gone
the birds pooped in Phraoh’s cup.
     After careful and prolonged thought, Pharaoh finally decided that it
was the baker’s fault for letting the birds into the kitchen, and that the
cupbearer could not be blamed for going to the bathroom.

To save you looking it up, Mendelson has the following system of prop.
calculus. Axioms all instances of

1. (A –> (B –> A))
2. ((A –> (B –> C )) –> ((A –> B) –> (A –> C )))
3. (( ¬B –> ¬A) –> ((¬B –> A) –> B))

with modus ponens as the only rule.

What are the shortest proofs — or indeed, what are any reasonable
proofs —  from these axioms (***without appeal to the deduction
theorem, or any other derived rule***) of

a)  ¬(A –> ¬B)  |-  A
b)  (A –> B)  |-  (¬B –> ¬A)

If someone with a taste for logic puzzles, or access to logic proving
software, or who remembers seeing worked examples somewhere can offer
answers, I’d be grateful!!!