Mark

Mark

Viña

Frame

Cuadra
Angiosperm
Angelus
Communications

Prov. 16:1 

"The preparations of the heart in man, and the answer of the tongue, is from the LORD."

Trotting Horse at Sunset, Dyersville, Iowa

 Chess: "Mark" "Viña" "Uva" "Cuadra" "Escuadra" "Frame" "Dorian Gray" "A Raisin in the Sun" 

Laws of Form 

 The position is simply this. In ordinary algebra, complex values are accepted as a matter of course, and the more advanced techniques would be impossible without them. In Boolean algebra (and thus, for example, in all our reasoning processes) we disallow them. Whitehead and Russell introduced a special rule, which they called the Theory of Types, expressly to do so. Mistakenly, as it now turns out. So, in this field, the more advanced techniques, although not impossible, simply don't yet exist. At the present moment we are constrained, in our reasoning processes, to do it the way it was done in Aristotle's day. The poet Blake might have had some insight into this, for in 1788 he wrote that 'reason, or the ratio of all we have already known, is not the same that it shall be when we know more.'
Recalling Russell's connexion with the Theory of Types, it  was with some trepidation that I approached him in 1967 with the proof that it was unnecessary. To my relief he was delighted. The Theory was, he said, the most arbitrary thing he and Whitehead had ever had to do, not really a theory but a stopgap, and he was glad to have lived long enough to see the matter resolved.
Put as simply as I can make it, the resolution is as follows. All we have to show is that the self-referential paradoxes, discarded with the Theory of Types, are no worse than similar
self-referential paradoxes, which are considered quite acceptable,in the ordinary theory of equations.
The most famous such paradox in logic is in the statement, 'This statement is false.'
Suppose we assume that a statement falls into one of three categories, true, false, or meaningless, and that a meaningful statement that is not true must be false, and one that is not false must be true. The statement under consideration does not appear to be meaningless (some philosophers have claimed that it is, but it is easy to refute this), so it must be true or false. If it is true, it must be, as it says, false. But if it is false, since this is what it says, it must be true. It has not hitherto been noticed that we have an equally vicious paradox in ordinary equation theory, because we have carefully guarded ourselves against expressing it this way. Let us now do so. We will make assumptions analogous to those above. We assume that a number can be either positive, negative, or zero.
We assume further that a nonzero number that is not positive must be negative, and one that is not negative must be positive. We now consider the equation Of course, as everybody knows, the paradox in this case is resolved by introducing a fourth class of number, called imaginary,
so that we can say the roots of the equation above are ± /, where / is a new kind of unity that consists of a square root of minus one.
What we do in Chapter 11 is extend the concept to Boolean algebras, which means that a valid argument may contain not just three classes of statement, but four: true, false, meaningless, and imaginary. The implications of this, in the fields of logic, philosophy, mathematics, and even physics, are profound.
What is fascinating about the imaginary Boolean values, once we admit them, is the light they apparently shed on our concepts of matter and time. It is, I guess, in the nature of us all to wonder why the universe appears just the way it does.
Why, for example, does it not appear more symmetrical? Well if you will be kind enough, and patient enough, to bear with me through the argument as it develops itself in this text, you will I think see, even though we begin it as symmetrically as we know how, that it becomes, of its own accord, less and lessso as we proceed.      

                                              G SPENCER-BROWN 
                                              Cambridge, England 
                                               Maundy Thursday 1972