Trying to Understand George Boole's "Laws of Thought"

George Boole's Laws of Thought​

Hello. I was inspired to try and read George Boole’s Laws of Thought. I'm at a point where I'm stuck and near ready to quit. No you tube videos help. The link to the pdf of the book is: 15114-pdf.pdf (gutenberg.org). I'm so stuck that it's hard for me to even articulate what I'm stuck on. I got up to chapter 5, page 52 which states the following:

f(x) = ax + b(1-x) It's assumed in this expression that x can only have values of 0 and 1. f(1) = a, f(0) = b. Substituting back in we get f(x) = f(1)x +f(0)(1-x). I sort of follow this, but then he gets into very long expressions with f(x,y) which I can't follow at all, and haven't found anything explaining what he means. Also, I'm not sure what is significant about the form f(x) = ax + b(1-x). If you take away the f(x) part, I think the right side, ax + b(1-x) means all x's that have the property a plus all b's which are not x's. What's the reason for making it an equation with f(x)? What does it even mean for an expression like this to be a function of x? When I think of functions, I think of numerical functions like ## f(x) = x^2 ##.
Boole goes haywire then writing expressions involving f(x,y) which go on for three lines or more, then adding zeroes to them, etc. Can someone please help me understand what's on Boole's mind here and what is the general idea of what he is talking about?

Also, what was he thinking when he created a system of math with only 0's and 1's 80 years before computers were invented? What was the possible application of such a system back then? It's supposed to be a book that changed the world, but I can't find any source of information explanation what the light is behind all of it and what Boole's big insight was. At first I just had a feeling that this book was something great, and felt like I had some intuitive understanding of what he might have been getting at. Now I'm just totally lost.

Thanks

The sad truth is that if you don't understand a function like$$f(x) = ax + b(1-x)$$ then you're not ready to study a work such as this. The development of Proposition II (page 53 and 54) is fairly elementary mathematics and the general equation for ##f(x, y, z)## is self-explanatory. Just plug in all the possible values for ##x, y,z## to see the equality.

It looks like it does get harder after that.

If you are interested in Boolean ideas you could look for a text that presents the ideas in a more accessible format. In general, going back to original texts is not the best idea.

Boole developed his binary math for logic problems. It was common before his time to study philosophical logic in the context of syllogisms and the direct meaning of words like "and" and "or" and "not". Logic in that sense was used in proofs but not in solving logic problems.

Boole saw a connection between formal logic and algebra and thus following algebra laws logical systems could be simplified.

https://en.wikipedia.org/wiki/George_Boole

It's a mistake to interpret + and * as arithmetic operations in boolean algebra. As an example, the expression B=A+A is not 2A but B=A. As another example B = A + not A is equal to 1.

NoahsArk said:

Can someone please help me understand what's on Boole's mind here and what is the general idea of what he is talking about?

Perhaps this will help: http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PREPRINTS/aboole.pdf

PeroK said:

The sad truth is that if you don't understand a function likef(x)=ax+b(1−x) then you're not ready to study a work such as this. The development of Proposition II (page 53 and 54) is fairly elementary mathematics and the general equation for f(x,y,z) is self-explanatory. Just plug in all the possible values for x,y,z to see the equality.

It looks like it does get harder after that.

jedishrfu said:

It's a mistake to interpret + and * as arithmetic operations in boolean algebra. As an example, the expression B=A+A is not 2A but B=A. As another example B = A + not A is equal to 1.

I would understand the above equation if it were a regular equation, but not when it's a Boolean equation. As a regular equation it looks similar to the equation for the slope of a line except there is a -bx at the end. It almost doesn't make sense to have function of x when x can only either be zeroes and ones. If x is zero, the expression doesn't make sense. How can the expression "all x's which are also a's" have any meaning when there are no x's? "All horses that are white", where x= horses and white = a, makes no sense where x has a zero value since that would mean you have the color white left to describe nothing. What does all white horses even mean with a zero value for horses?

NoahsArk said:

I would understand the above equation if it were a regular equation, but not when it's a Boolean equation. As a regular equation it looks similar to the equation for the slope of a line except there is a -bx at the end. It almost doesn't make sense to have function of x when x can only either be zeroes and ones. If x is zero, the expression doesn't make sense. How can the expression "all x's which are also a's" have any meaning when there are no x's? "All horses that are white", where x= horses and white = a, makes no sense where x has a zero value since that would mean you have the color white left to describe nothing. What does all white horses even mean with a zero value for horses?

It was your choice to study that work! The impression I get from what @Stephen Tashi posted is that it's not an easy read.