I’ve been doing a re-read of George Boole’s work “Laws of Thought”. It’s one of the foundational works that define Boolean Algebra. This, of course, leads directly to the modern field of Computer Science.
I’m going to post my notes for this work on my blog. I’m skipping Chapter I, since it’s quite lacking in any kind of philosophical content.
These are notes. I won’t try to explain all of Boole, it’s really rough stuff. These are really just my public notes to self ( in case anyone might need some notes ). That being said, corrections are more then welcome!
Remember, these are just reading notes / highlights. I’ll include as much as I can, but no more! :)
I’m going to refer to the text in the following format:
(Chapter.Point ID.How far into the block, percentage)
So, something from Chapter III, Proposition II, halfway down would be:
(III.PII.50)
If the last part ( percentage into the text ) is omitted, the whole block is significant.
Don’t freak out! It’s actually easy enough to read!
I have also read the mark &c as et cetera. Seems like it fits, and et c. or etc. in Latin could be done as & c ( or &c). This could be wrong, but it works syntactically.
Chapter II ( Signs and their Laws )
Of signs in general, and of the signs appropriate to the science of logic in particular; also of the laws to which that class of signs are subject
1.: Boole points out that language is fuzzy, and lacking. He makes the point that there are multiple meanings to things, and that we need to beware of that. I have interpreted (II.1) as Boole paying a bit of homage to Aristotle, being sure to avoid the critique that he’s making an equivocation of dissimilar terms.
2.: Boole points out that words are “signs”, creating a fixed interpretation for the rest of his work. This is basically the same as (II.PI.1st)
Preposition I.
1st: Literal Symbols as x, y (&c) represent things as subjects of conception. I have read this as allowing you to do assignment, such as “x = men”, as he does later on.
2ed: Signs of operation, such as +, -, and x stand for the operations we think of ( from Algebra ).
3ed: The sign of identity is denoted by =. I think that identity is a poor term to use, it should really be called the sign of assignment. You can think of some of this through the lens of group theory, so I’d take care to not mix up terms.
However, even though the signs are the same as algebra, do not mix it up! Boole is very clear that normal algebraic rules may not apply. (II.PI.3ed.90).
Class I
Appellative or descriptive signs, expressing either the name of a thing, or some quality or circumstance belonging to it.
The Class I symbols represent so-called spatial-temporal “stuff”. The usage may or may not imply it’s existence, the fact that it does or does not exist has no place in this system. That is to say it’s truth neutral. Take, as example, the following to both be true and equivalent.
Water is a fluid thing
and
Water is fluid
Boole also uses a second example ( as he often does ). This is about using phrases to represent a single class. He compares the Greek “βαθυδίνης [ὠκεανός]” to the phrase “Deep-eddying ocean”, both of which refer to the same “thing”.
He goes on to describe how “good” and “great” fit in. When combined with a “thing”, “good” and “great” become “virtual”, and the phrase ( such as “good men” ) could be represented by a single sign. (II.C1.5.75-100)
Since we can now define anything as a “sign”, we can just replace a single “sign” with letters. (II.C1.6.0).
Boole uses the example: Represent all men as the symbol x, such that every man is contained within x, or simply “The class of men”.
This hits home as a very Aristotelian or Platonist perspective, replace “class” with either Form or Universal. It would not be harmful to think of Boole’s “Class” as an Universal, or Form.
Next, assign y to the Class of “Good”, such that y is “All good things” ( ST:TNG, FTW ). We may now combine terms such that xy is the mark for “All good men”.
As his second example, Boole suggest x stands for “Sheep”, and y being all things “White”, such that xy contains all white sheep.
It should also be noted that xy and yx are functionally identical.
If we then add z ( to be horned things ), xyz is the collection of all horned, white, sheep.
Keep in mind that zxy, zyx and xyz (&c) are all identical ( multiplication is commutative! ). (II.C1.6.10 — II.C1.7.50).
Nextly, we may consider the following case. Let x be some Class of things which contains y as a subset. We can now see that xy = y.
If we consider the case where x and y both refer to the same Class, we can now see the situation before, again. xy = y. We could also re-write this as yy = y, or y^2 = y. (II.CI.9).
This is all to cover the following case:
x is defined by the Class of Men.
y is defined by the Class of the Good.
xy = xyy
That is to say that the class of “Good Men” is identical to the Class of “Good Good Men” ( All Good Men are in fact, Good )(II.CI.9.75). It should also be noted that this is a Law of Language ( the product of the Laws of Thought ).
Class II
Signs of those mental operations whereby we collect parts into a whole, or separate a whole into parts.
Class II symbols are those which join or divide things. The examples Boole uses are “and” and “or”. Take, as example, “Trees and minerals”. Given x as “Trees” and y as “minerals”, that phrase can be expressed as x+y.
As another example, consider the following.
x is defined as Men
y is defined as Woman
z is defined as European
Let’s consider “Men and Woman”. This, of course, translates directly to x+y.
Now, let’s add in another symbol. Let’s say “European Men and European Woman”. This can be represented as zx+zy.
These rules also follow some of the normal rules of algebra, so let’s consolidate the x and y, and rewrite as z(x+y) ( otherwise written as “Men and Woman who are European” ).
Another great Class II is the word “except”. Let’s consider the following.
x is defined as all Men
y is defined as all Americans
The concept of “All men except Americans” can be expressed as x-y.
Or, more accurately, “All men except American Men”, which can be expressed as x-yx.
Let’s lastly consider the following case.
x is defined as all Men
y is defined as all Americans
z is defined as all Programmers
We can express something such as “Male Programmers, except American Programmers” simply as zx-zy, or z(x-y) ( in this case, it also follows normal algebra ).
Class III
Signs by which relation is expressed and by which we form propositions.
All verbs fall under Class III symbols. Boole uses, as an example the two words “is” and “are”.
Let’s take as example the phrase “Cæser conquered the Gauls”. We can rewrite it without complaint as “Cæser is he who conquered the Gauls”. Boole makes the argument that you can not understand what is being talked about unless you understand the components ( separated by the Class III ). Consider the following:
fig 1.
(1) (2)
+-------+----+------------------------------+
| Cæser | is | he who | conquered the Gauls |
+-------+----+------------------------------+
^ ( or One who has )
You can not understand fig 1. without first understanding (2) ( That is to say: you can not understand “Cæser is he who conquered the Gauls” without understanding what it means to “conquer the Gauls” ). (II.CIII.12.90)
You can think of is as that of assignment, so that it directly translates to = in this system. (II.CIII.13)
Let’s consider Boole’s next example.
x is defined as the Stars
y is defined as the Suns
z is defined as the Planets
We can now represent “The stars are the suns and the planets” as x = y + z.
Given that the formula above is correct, we can now make the assumption that x - z = y is also true ( read: All Stars, except Planets are Suns ) (II.CIII.13.5). This is just showing that the algebraic rule of transposition is valid, in this case. Boole proves that at (II.CIII.13.40).
The catch ( as presented at (II.CIII.14.0) ) is shown. I’ll present that below.
Consider the case where members of x contain property z, and members of y also contain z. y and x are not pointing to the same thing, and do not share all members in common.
If we phrase the relationship as zx = zy. We can not use a z-inverse ( such as division, in this case ) to re-form the equation as x = y. That, of course, is false.
In the words of Boole himself:
In other words, the axiom of algebraists, that both sides of an equation my be divided by the same quantity, has no formal equivalent here. I say no formal equivalent, because, in accordance with the general spirit of these inquiries, it is not even sought to determine whether the mental operation which is represented by removing a logical symbol,
z, from a combinationzx, is in it’s self analogous with the operation of division in Arithmetic.
Last Notes
To qualify the bit just above, the deduction above is also only true when z is not 0, because if z = 0, then x = y is true.
Now, if we take a look at the law at ( xx = x or x^2 = x ) at (II.CI.9), we can deduce that we may only admit two numbers into this system of logic. 0 and 1 ( as 1 x 1 = 1, as well as 0 x 0 = 0 which holds in the case of xx = x )(II.CIII.15.50).