Call me a classicist if you wish, but I find that old stuff, like modus ponens and modus tolens helpful in navigating the macroscopic environment in which I dwell, and
http://www.youtube.com/watch?v=J-3VxOqHI-4
Oh, this takes me back. I have a book on Logic by Lewis Carroll, the guy that wrote "Alice's Adventures in Wonderland". He also happened to be a Mathematician, possible pederast of a girl named Alice, Anglican Deacon and Photographer (or so I am informed by the Great Wiki Wiki in the Sky).
Actually, Charles Lutwidge Dodgson was his real name, Lewis Carroll being his pseudonym. The book I have of his on logic is a very interesting read. It is almost plodding, methodically creating higher level logic concepts so that I had to take a break after about the 15th page when first reading it. Interesting reading though. I definitely endorse it, if I could remember it's name.
Oh well. Yes, always fun to bring up Latin Terms. I think in equations though, can't help it. Here is a synopsis for everyone else in case they are interested in what is involved in Modus Ponens and Modus Tollens.
Modus Ponens
(P -> Q, Q) /\ P
Modus Tollens
(P -> Q, !Q) /\ !P
Pretty simple really. I hope the symbols are self-evident enough.
/\ means 'replace'
-> means implies.
(P,Q) means P and Q.
[P,Q] means P or Q. // for completeness
!P means not P.
P and Q are two prepositions, statements that are considered either true or false.
If P and Q are considered as per Fuzzy Logic things can get more interesting perhaps. One of the ideas behind Fuzzy Logic is that base 2 arithmetic called in mathematical terms Z_2 is all you need for classical logic, but perhaps it would be interesting to extend this idea by generalizing the elements to be real and between the range of 0 and 1. That is what the inventors of Fuzzy Logic did. {Huh, just had a weird idea, can there be Fuzzy Quantum Logic? Odd idea I guess.}
We know that for Z_2 the AND logic function is a mapping of the following kind:
AND: Z_2 x Z_2 -> Z_2.
The function is very simply expressed as
AND(x, y) = xy.
If, for instance, you make a logic table of the above you get
x y AND(x,y)
0 0 0
0 1 0
1 0 0
1 1 1
which is correct. Doing the same for real numbers in the range of 0 to 1 inclusive makes sense
therefore. Let us define the set I as follows,
I = {i: 0 <= i <= 1}.
The Fuzzy AND, let's call it FAND, then has the mapping definition
FAND: I x I -> I.
(I am making up this FAND, FOR...etc. business as my own notation).
First off FNOT. This function is most easily defined in first trying to generalize ! as
FNOT: I -> I,
FNOT(x) = 1 - x.
The Fuzzy OR is FOR and all the rest follow suit. But now that FAND is defined how to find FOR? We know from First Order Predicate Logic (1OPL) that
!(P,Q) = [!P,!Q].
This would mean that
FNOT(FAND(x,y)) = 1 - xy = FOR(1-x, 1-y)
The solution for FOR is therefore
FOR(x,y) = 1 - (1-x)(1-y).
There appears to me to be some further generality in formulation allowed if one also considers that the functions FAND and FNOT probably only need to follow some rules so that consistency between 1OPL and Fuzzy Logic. This consistency requirment is that when a fuzzy predicate is considered that is only ever 0 or 1 then the two logics become equivalent. I will have to think about that to give any answer how that could possibly work.
Some time ago there was a nice Scientific American Journal on Fuzzy Logic. Included in it was how Fuzzy Logic could be made to be related to fractals through the use of feedback. Imagine you have a fuzzy predicate with value x belongs to I.
Classically we know that (P,!P) = 0. In terms of Fuzzy Logic however it is possible to have
FAND(x, FNOT(x)) != 0, for some x.
The reason why is simple.
FAND(x, FNOT(x)) = x(1-x).
Let 0 < x < 1, then 0 < 1 - x < 1. Finally, since both of the previous set of inequalities is true,
0 < x(1-x) < 1.
The function
f(x) = x(1-x)
can then be generalized so that the domain set is from the complex numbers. Define C to be
C = {z: |z| < 1 and z belongs to Z}, Z = "complex numbers".
We could then have f be redefined so that
f: Z -> Z,
f(z) = z(1-z).
You can start at a point in the Argand Plane and then iterate f over and over again until you see what the orbit thus created is attracted to. Perhaps just look at a series of values
|z|, |f(z)|, |f(f(z))|, etc.
where |z| = |a + ib| = a^2 + b^2, the modulas squared of the complex number z.
I am not sure if I remember correctly but I think the series above will converge on either 0 or 1. You can then use this fact to create a computer program that gives you a pretty picture of black and white shapes inside the unit circle of the Argand Plane. I do not recall from the article or from memory if there is much beyond it than that. You can change up the formula of the function f(z) based on other fuzzy functions and see how that turns out as well, if that is your sort of thing.
It is hard to explain Quantum Computing in a few paragraphs. If you know Linear Algebra well enough, then it should at least be possible to give a short overview of some of the highpoints. Consider a complex vector space Z x ... "Cartesian Product n-times" x Z = Z^n {n can be infinite}.
Let ? mean "belongs to".
Two vectors |U>, |V> ? Z^n will have an inner product defined as so
<U|V>
It is possible to think of the inner product in a completely abstract way (part of the point of using Dirac Bracket Notation), but fot those who want a concrete representation consider |V> as an n x 1 matrix (column vector with complex components), and <U| as the complex conjugate of the transpose, which is also known as the adjoint, of |U>, which is a 1 x n matrix (row vector with complex components).
There are two things you do in QM. You, so it is called, collapse the wave-function {I do not like the fuziness in thinking this inspires in people, there is no collapsing, there is just a series of identical tyoe measurements being done to give you statistics whith which you compare to experiment for confirmation of theory with experiment, that is what is really happaning}, or you let it evolve unitarily.
Unitary Operators are what you need to know to understand Quantum Computing (or, at least a fair portion of the literature on the subject). The point is that the vector space we are considering restricts the vectors to the set
<V|V> = 1.
Such vectors are called wave-vectors. A wave-function is just a wave-vector with infinitely dense components, one for each point in space in which the wave-function is defined. If we consider linear superposition of vectors |V_n> that form a basis for the vector space (Hilbert Space), then one can have
|Psi> = \sum_{n} Psi_n |V_n>
Each of the |V_n> is of unit length
<V_n|V_n> = 1.
Therefore since |Psi> must also be of unit length (the sum of all probabilities must sum to 1),
<Psi|Psi> = \sum_{n} |Psi_n|^2 = 1.
If we are in the Schroedinger Picture the wavefunctions evolve over time according to the equation
|Psi(t)> = e^(-iHt/hbar) |Psi(0)>.
H in the above is the Hamiltonian Operator. The point is that the wavefunction can be changed over time according to the equation above. Part of the definition of the Hamiltonian is the Potential Operator.
H = T + V
T being the kinetic energy operator and V the potential energy operator. We have control over V, so we can control what H is over time (do not worry, do it in steps and then you will not have to worry about seperation of variables to get to the Time Independent Schroedinger Wave Equation needed for formulating the Quantum Evolution Equation in the Schroedinger Picture given above for |Psi(t)>).
So, of course, e^(-iHt/hbar) is a Unitary Operator. Remember that a unitary operator U is one in which
In Quantum Computing one often considers qubits. The simplest example of a qubit is a spin up or spin down electron. Spin is a very important part of our universe it seems (you can not understand anything of the Standard Model for instance if you do not understand spin, let alone isospin and so on).
So the Hilbert Space of electron spin is a two dimensional complex vector space. When you make a measurement though you only ever have one spin or the other. The probability of having spin up |+> for instance if the wave-vector is |V> is
P_V(+) = |<+|V>|^2.
The same works for spin down,
P_V(-) = |<-|V>|^2.
I guess I could go into other topics related to Quantum Computing (some of the algorithms perhaps), but this post is already too long. I guess I should go sit down with a pad and paper and try and figure out if Fuzzy Quantum Logic could make any sense or not.
This has nothing to do with consciousness per se, but I felt like writing some about these topics on this forum since there was some peripheral mention of these subjects.
Drugs forum has a huge archive. And yeah, everyone has endocannabinoids to degree. Fascinating new area of research in how our bodies consciousness produces consciousness altering chemicals like this. There are many more that are not even cannabinoids that have been found now. DMT is also endogenous.
) when there is no reason to suppose it is necessary?