1. Two fundamental theorems
2. The world's funniest
Well.. I am still paying back my walk of over two weeks ago, and so I
am still not
feeling very well, and also I did not sleep enough last night, so I am
going to use a logical note.
And in order not to loose most of my readers, I throw in the world's
Two fundamental theorems
The following I never saw in print, though it is - I
think - quite neat.
The following uses HOL - Higher
Order Logic - but that is quite natural. In fact I assume very
1. The distinction between predicates and subjects:
Only subjects are identified as names, and predicates then defined as
anything that when added to zero or more names makes a statement. I
will distinguish them as lower and upper case, respectively.
2. Ordinary bi-valent logic: This has rules for &,
V, ~, and = which are precise forms of
and, or, not and equals. Such
rules may be as follows, with x
and y and z as variables, as are p and q,
and a is a constant. The
appended ' means that the predicate it is appended to is a
p p |-
~~p |- p x=y, F'x |- F'y
q q |- pVq
p&q pVq, ~pVq |- q
This suffices for
propositional and predicate logic.
3. Quantifiers for
predicates and for subjects:
(Ex)F'x -||- F'a with a new
(EF)Fa -||- G'a with G'a
(F)Fa =def ~(EF)~Fa
This suffices for higher order and first order logic. And
"a new" says that a must be new to the proof, while
"(Ex)" is "for some x" and "(x)" is "for all x". And "=def" is
"is by definition the same as".
4. The principle of abstraction: |x:F'x = the
x such that x is F' and
a e |x:F'x -||- F'a
This also introduces "e" for "is a": a is a x
such that x is F' if and only if a is F'.
One may use much more time to state the above, but the above is
adequate, and it permits the following two fundamental theorems:
Assumption of indirect proof
Fx IFF ~(xex)
4. aea IFF ~(aea)
a for x
5. Contradiction and T1
If one objects against introducing a logical particle, one can start
Otherwise, the proof is the
Assumption of indirect
(~(a=|x:xea)) Fx IFF xea
(~(aea IFF aea))
a for x
This is quite neat. The first in effect says there is no Russell-set
(defined as a set z such that ( z = |x: ~(xex) ), and the second in
effect says that every subject does have some defining property.
Maybe I should write these out, using "the Russell-set" for "the set of
things that are not elements of themselves", and use "set" for
T1. Some properties do not define a (consistent) set.
For suppose all do. Then the one used for the Russell-set must define a
set. But then it is an element of itself if and only if it is not an
element of itself, which is a contradiction.
T2. Every (consistent) set has a defining property.
For suppose not. Then there is a (consistent) set that is not defined
by any property, and so the property of not being an element of that
set cannot define it either. But then the set is an element of itself
iff it is not an element of itself, which is a contradiction.
Note that the two theorems are just the same, except for having
subjects and predicates interchanged.
There are people who do not like logic. Actually, there
are far more people who do not like logic than who do like it. (See: Stupidity.)
In any case... for those who do not like logic, and also for those who
do, here is the
world's funniest joke:
Two hunters are
out in the woods when one of them collapses. He doesn't seem to be
breathing and his eyes are glazed. The other guy whips out his phone
and calls the emergency services. He gasps, "My friend is dead! What
can I do?" The operator says "Calm down. I can help. First, let's make
sure he's dead." There is a silence, then a gun shot is heard. Back on
the phone, the guy says "OK, now what?"
If you don't like it that
much, you might consider that it is originally by Spike MIlligan,
who was voted to be "the funniest persion the last 1000 years", though
he also was a - real - manic depressive.
No, none of the previous paragraph was a joke. Here is a runner up in
the same contest that landed the above one Number One:
A woman gets on a
bus with her baby. The bus driver says: "That's the ugliest baby that
I've ever seen. Ugh!" The woman goes to the rear of the bus and sits
down, fuming. She says to a man next to her: "The driver just insulted
me!" The man says: "You go right up there and tell him off – go ahead,
I'll hold your monkey for you."
Still not a laugh on your
face? One more time:
Australian beer like making love in a canoe?"
"Because it is
fucking close to water, mate!"
I know that one from Monty Python, and dealt with it in my discussion of
humour (in Dutch). I think it is the best on this page.
If you still haven't smiled, I give up. And the same if I did.
ME/CFS (that I prefer
to call M.E.: The "/CFS" is added to facilitate search
is a disease I have since 1.1.1979: