#372 AP book of science--- High School or 1st Year College Logic // Teaching True Logic textbook series by Archimedes Plutonium This is AP's 372nd published book of science published on Internet, Plutonium-Atom-Universe,PAU newsgroup is this.

[Archimedes Plutonium]

High School or 1st Year College Logic // Teaching True Logic textbook series

by Archimedes Plutonium


This is AP's 372nd published book of science published on Internet, Plutonium-Atom-Universe,PAU newsgroup is this. Please read this textbook for free on Internet.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe

Preface: The largest single gap in modern day education is the teaching of Logic. It is the science of thinking straight, clear, and correctly. Math is the science of correct numbers, quantity, measure, and geometry figures. While Logic is the science of correct ideas and applying those ideas. The only place I have seen Logic as mandatory was at UC, University of Cincinnati, 1968-1972 where lawyers to be had to take Logic. I guess they realized that you need some logic training to be a good lawyer. All people who graduate from college or university should have had Logical training, especially scientists. And what I recommend is a required 2 years of Logic in college and university if you want a B.S. degree in a science.

Cover picture:

Table of Contents
-----------------------

Text
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plutonium dot archimedes at gmail dot com. Looking for a College or University press to hardcover publish all 372+ AP books of science, likely to become 500-600 maybe even 700 books by the time I die. E-books are too prone to unbalanced-unhinged censor-editors, who can easily make your books vanish by pulling a switch. Science should never have gatekeepers, who thwart access to true science.

 
|  /
| /
|/______ hardcover or paperback

PAU newsgroup is this.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe        
Archimedes Plutonium

[Archimedes Plutonium]

High School or 1st Year College Logic // Teaching True Logic textbook series

by Archimedes Plutonium


This is AP's 372nd published book of science published on Internet, Plutonium-Atom-Universe,PAU newsgroup is this. Please read this textbook for free on Internet.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe

Preface: The largest single gap in modern day education is the teaching of Logic. It is the science of thinking straight, clear, and correctly. Math is the science of correct numbers, quantity, measurement, and geometry figures. While Logic is the science of correct ideas and applying those ideas. The only place I have seen Logic as mandatory in school was at UC, University of Cincinnati, 1968-1972 where lawyers to be had to take Logic. I guess they realized that you need some logic training to be a good lawyer in arguing a case. All people who graduate from college or university should have had Logical training, especially scientists. And what I recommend is a required 2 years of Logic in college and university if you want a B.S. degree in a science. This is the largest gap in modern education from colleges and universities-- no training in how to think straight, clear, and correctly.

About this Logic textbook series: I should remark on this series of textbooks on logic I undertook starting 2025 and now taken almost 2 years to write. Normally for me I can write a textbook of science in a matter of months, however these logic textbooks have taken years. And the reason for that is because; writing logic textbooks has to be "logically written". The book itself requires logic to write correctly. I mean the order has to be correct. The ideas cannot conflict. Whereas writing say biology or even physics, I can jump in and out and here and there and be fine in doing such. Writing logic that will not pass. And I can safely say that my 5 textbooks on Logic are the first textbooks that are "written logically". But not only written logically, in that my 5 textbooks are the first logic textbooks that have all 4 of the most simple connectors of Logic correct. These 4 are the AND, OR, Equal-Not, If-->Then. All previous textbooks of logic had all 4 of those simple connectors in grave error. What is often called Boole and Jevons logic had all 4 of those connectors in serious error. So awful in error that Old Logic mixed up AND with OR, which is like saying 3 + 2 = 1 with 3 - 2 = 5. Old Logic of Boole and Jevons had OR be add while making AND be subtract. The errors of Old Logic are so terrible, I had no choice but writing the textbooks of Logic.

A final note on teaching, especially Logic, but in other courses like math or calculus or physics. Is that the education of students is often neglected in that the textbook or teacher or both are "over the heads of their students". I myself was a High School teacher of mathematics for awhile. And the one item that concerned me the very most in my teaching classrooms, was to never be over the heads of my students. To teach them an idea that I know they can grasp and learn from. Now this problem is an immense problem especially in colleges and universities because professors are not required to take study of "how to teach" and they end up in a classroom where all one sees is students taking notes, but not really learning anything, for the professor is way over their heads. In my opinion, a classroom is where the students are absorbing what the teacher is saying and doing, not the pencil and paper notebook. In this effort, I have written these 5 textbooks on Logic for the utmost concern that students can absorb everything I wrote. So much concern have I had of "not teaching over students head" that I wrote the High School logic textbook last, to see if the college and university textbooks were ---- simple enough for students----.

I can list many textbooks I encountered in college and High School that were over the heads of students in classrooms. My chemistry textbook in High School, along with PSSC physics textbook in High School were over our heads. In University, the Feynman Lectures on Physics were over the heads of undergraduates. Most math textbooks were written not to ease the students in learning, no, they were written that mostly obfuscates what is being talked about.

#366 Elementary Logic
#369 History of Logic
#370 Intermediate Logic
#371 Advanced Logic
#372 High School and 1st year College Logic textbook

Cover picture: A picture of Well-Defined "Charge" in Physics, telling us that a main duty of Logic is to have "well defined" ideas in mind.

Table of Contents
-----------------------

1) A definition of Logic.

2) The Structure of Logic compared to other sciences.

[Archimedes Plutonium]

Text
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A definition of Logic, and the easiest definition I can think of possibly, so that High School students can learn.

Definition of Logic
-----------------------

In mathematics we have numbers and geometry figures and we play around with these numbers such as add, subtract, multiply, divide and play around with geometry figures learning things about their sides and measurements.

In Logic we have ideas, and these ideas are similar to numbers of math, or geometry figures of math. And we can play around with these ideas by connectors of AND, OR, Equal-Not, If-->then, and learning new ideas.

Math is numbers and geometry figures, while Logic is ideas.

Math is the precision of numbers and geometry, while Logic is the precision of ideas.

That is the best definition of Logic for beginners in Logic. Later on, I elaborate and detail more what Logic is, but for the time being that is as easy and simple as can be expected.

But notice, that Logic is all about ideas, and of course, well math and numbers and geometry figures are ideas in and of themselves. This logically means, it means that Logic is a bigger subject than all of mathematics.

We then say that Logic is a larger set than is mathematics. Or we can say math is a subset of Logic.

However, Logic is not the largest set of knowledge, for Logic is just one science among many Sciences, and physics is the largest set in all of science.

So Logic is a subset of physics.

Math being a subset of Logic, means that everything found in math, must be found in Logic. And that Logic being a set that has all of math inside logic, means that logic must have something more that math, somethings that math does not have. For logic encompasses all language, and not just numbers and geometry figures, but the language itself that talks about math.

In the next chapter we discuss the Structure-of-Logic.

2) The Structure of Logic compared to other sciences.
   

We defined Logic as the world of ideas and how to make ideas clear, straight and correct. Now let us talk about the structure and the best way to do that is to talk about the structure of Math.

So Math has 6 operators of add, subtract, multiply, divide, calculus derivative, and calculus integral. Since math has those 6, Logic must have 6 at least that match math's 6. They are in order AND, OR, Equal-Not, If-->then, existential quantifier, universal quantifier. And in logic we call them connectors, while math calls them operators.

Since math is the precision use of its numbers and geometry figures and 6 operators. Then logic is the precision use of Ideas with its 6 connectors.

What this implies, is the best way to teach Logic, is to keep modeling logic up against math. Whatever is true in math is the best way to describe logic and how logic works.

So that AND is add, and OR is subtract, and Equal-Not is multiply, and If-->then is divide, and calculus derivative is the existential quantifier, and the calculus integral is the universal quantifier.

The structure of logic closely follows the structure of mathematics and so we model one with the other to find truth and conclusions.

In the history of logic, the logic was often compared to a "algebra", a algebra of logic.

[Archimedes Plutonium]

3) Teaching the rudimentary calculus derivative and integral for Logic.
--------------------------------------------------------------------------------------------


A proper teaching of Logic requires a small amount of understanding of calculus of mathematics in order to comprehend the Existential and Universal quantifiers. I have made this explanation as easy as possible.


The first connector in Logic to teach is the Existential quantifier which is the derivative in calculus. So here, in order to teach Logic properly, I have to stop and teach the derivative and integral of calculus. Do not worry, this is the most simple explanation of calculus you probably will ever see. And once you read and studied this, then, later, take calculus in college, you will breeze through it.

Logic, properly, can not be taught without knowing this rudimentary calculus.

Archimedes Plutonium Jun 8, 2026, 2:22:31 AM to Plutonium Atom Universe.

I start this textbook by referring to Mathematics as a model. For it is common-sense that Mathematics is the playing around with numbers. If we replace numbers with "ideas" and play around with ideas, is a short definition of Logic.

Math plays around with numbers. While Logic replaces numbers and plays around with "ideas".

And since math is a subset of Logic, for ideas encompass numbers of math and encompass geometry of math, we can expect that whatever exists in mathematics, must exist in Logic. Sort of like the idea that Chemistry is a subset of Physics. So whatever exists in chemistry, also exists in physics.

Now, since mathematics has 6 basic operators, means, Logic must have 6 basic operators, only we call them "connectors".

For Math we have add, subtract, multiply, divide, derivative, integral. For Logic we have AND, OR, Equal-Not, If-->then, Existential quantifier and Universal quantifier.

So to teach High School students beginners logic, I constantly refer to the mathematics.

Elementary Calculus
---------------------------

So at this moment in time of teaching Logic, we must start to learn what Calculus is. And I will teach you the easiest way to learn calculus.

For the integral is Fast Add which is multiply; while the derivative (differentiation) is fast subtract which is divide.

So here, I stop and teach function, integral calculus for multiply and derivative calculus for division.

It is impossible to properly teach Logic without teaching rudimentary calculus.

Existential quantifier which in math is seen as the calculus derivative (do not be scared, I will teach you this calculus supereasy)

Universal quantifier which is the math calculus called the integral (do not be scared-- it is easy)

Before I explain function, derivative, integral, let me show you pictures which sums it all up. These pictures are actually a proof of the biggest theorem-proof in calculus, called the Fundamental Theorem of Calculus.


From this:
        B
        /|
      /  |
 m /----|
  /      |
|A      |
|____|
a      b


The trapezoid roof has to be a straight-line segment (the derivative)
so that it can be hinged at a midpoint m, and swiveled down to form rectangle for
integral. The area of the rectangle is the integral.

To this:

__m__
|         |
|         |
|         |
---------
a        b


So, we have two items in Calculus for this theorem, we have a derivative, the straight line segment A to B with m in the middle. And we have the rectangle area which we call the integral.

We draw in m, the midpoint because that is where we have a hinge, we imagine a hinge there. In fact, some teachers will build this model in wood working class just to use in math class.

So, Calculus has two items-- the derivative which is the rooftop, the straightline. And the other item, the integral which is the rectangle area.

So, what is this theorem all about?

Well, it says that-- if you have a rectangle with a midpoint on its top side.

__m__
|         |
|         |
|         |
---------
a       b

That you can cut a right triangle from the midpoint

__m__
|  /      |
|/        |
|         |
---------
a       b

Cut that right triangle and swivel it up to make the trapezoid

        B
        /|
      /  |
 m /----|
  /      |
|A      |
|____|
a      b
Or, you can start with that trapezoid and swivel the right triangle downwards to make the rectangle


__m__
|  /      |
|/        |
|         |
---------
a       b

And, basically that is the Calculus at its most simple form. Where the slanted line (some call it a slope) is the derivative and the rectangle area is the integral. So, there, 15 year olds, you have just learned the fundamental basics of Calculus. Take a rectangle, swivel the right triangle and you have a derivative. Take the trapezoid, swivel the right triangle to form a rectangle area and you have the integral.

Basically, that is all that Calculus is.

Homework:: Take a sheet of paper, a used sheet, for no need to ruin a fresh sheet of paper. And find the midpoint and form the right-triangle and swivel up the right triangle forming the trapezoid.

Archimedes Plutonium Jun 12, 2026, 5:23:56 PM to Plutonium Atom Universe newsgroup.

I am trying to make the world's most simple explanation of Calculus, for High School students.

I am sure it must be a Geometry picture.

An explanation that even those who hate math, can understand.

Calculus is the science of motion, of change.

Calculus involves division in the derivative-- the slope, as we carve out the right triangle inside the rectangle and lift it up at the midpoint.

Calculus involves multiplication as the integral --- the rectangle (sometimes a square). For integral is area and area of rectangle is length times width.

Archimedes Plutonium Jun 13, 2026, 3:48:01 AM to Plutonium Atom Universe newsgroup.

So, I am striving for the most simple of all explanations of Calculus. The Finest- Simple Teaching of Calculus.

And what I need to accomplish that are 3 lines of thought. The first line of thought is what the true numbers of mathematics are. They are not the Reals with their continuum. You cannot have a calculus when the numbers form a continuum. You need empty space from one number to the next number.

The second line of thought to teach the most simple and easy Calculus is to well define what a Function is. Old Math did a good enough job in defining a function, but they should have carried it further to the idea that the Polynomial Function is the only valid function in all of mathematics, and that any of the other so called functions are just horrible silly and stupid aberrations. I say this because anyone that has completed first year college calculus, can testify along with me, that the Polynomial because it obeys the Power rule for derivative and integral--- just add or subtract 1 from exponent, yet every other so called (idiot function) has no easy rule--- especially the awful ugly trigonometry functions. So, well, Stewart's beloved Old Math Calculus book is approaching 1500 pages, and where some girls would have a hard time of even picking up the tome. When if you made POLYNOMIALS the only valid function in all of math, Stewart, I am guessing could have written a Calculus textbook covering all of what he covered in just 200-250 pages.

It is amazed me in Freshman Calculus class at University of Cincinnati, 1968-1972, how thoroughly easy Calculus is, if the only function was the Polynomial. Because of its charming easy Power Rules-- add or subtract 1 from exponent. And I am sure, although I as a teenager would not have known this in 1968, I am sure that the professors of math in most every college and university across the world would have encountered and known of the Lagrange Interpolation. This is a fact that any function that is not Polynomial, is easily transformed into being  a polynomial. Can we say the reverse of that is true??????
What I mean is can we say that any function can be transformed into a trigonometry function???? Or a logarithmic or exponential function????????

Is it true only for the Polynomial Function--- that hand me any obnoxious, obscene, dirty, defiling, pornographic function such as trigonometry, exponential, logarithmic, hyperbolic, and turn all other functions into that particular type of function, or is it the case that only POLYNOMIAL functions have that supreme feature and characteristic of turning stupid idiotic other functions into being another polynomial.

Third, is the picture diagram given above of the rectangle and a midpoint on top and then carve out a right triangle for which when hinged (up or down) is the derivative while the rectangle is the integral.

It makes sense, that true calculus is taught not in math classes in Colleges and Universities from math professors who know little to no logic, but that the true calculus is taught by a Logician in his 5 book series of Logic textbooks.

So, I need to cover 3 concepts (1) true numbers of math (2) what are functions and what is the Polynomial function (3) the geometry picture diagram of the proof of the Fundamental Theorem of Calculus.

I already have the picture diagram, and now talk about the true numbers and the function concept.

Archimedes Plutonium Jun 13, 2026, 4:43:12 AM to Plutonium Atom Universe newsgroup.

I am mindful that this is High School.

1) True numbers of mathematics. In the year 1900 a famous physicist announced to the world what would later be known as the birth of Quantum Mechanics. This physicist was Max Planck and what he discovered is that physics comes only in discrete quantities. The word "quantum" means discrete. We could today call it Discrete Mechanics.

Now if physics finds that the world of physics is all quantized, then what should have happened after the year 1900, is that all math professors and the college and university math departments should have payed attention to physics and asked the question--- are the numbers of mathematics quantized also???? But no, that never happened because mathematicians from year 1900 onwards were mostly dull and stupid as they further dived deeper into continuums of their Real numbers with rationals and irrationals all mixed together, cobbled together along with negative numbers, and where Paul Cohen dives deeper into quagmire of a "continuum hypothesis".

Not until 2013 does a mathematician, AP, while writing his book "True Calculus" does it become apparent that no Calculus can exist when the numbers of math are a continuum, for the simple reason, the derivative when hinged up at midpoint, ______must fall on the very next coordinate point that the function graph pinpoints_____. The derivative intercepts the next coordinate point of the function graph itself. If the numbers of math form a continuum, the derivative cannot fall on a ---- next coordinate point---.

So the true numbers of math have to have gaps and holes in between one number and the next number. The true numbers of mathematics are the Decimal Grid Numbers and the smallest of these is the 10 Grid, next comes the 100 Grid, next the 1000 Grid.

Here is a picture of the Decimal 10 Grid.

9.0, 9.1, 9.2, 9.3, 9.4, 9.5 9.6, 9.7, 9.8, 9.9, 10.0
8.0, 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9,
7.0, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, 7.9,
6.0, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9,
5.0, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9,
4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9,
3.0, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9,
2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9,
1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9,
0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9,

There are exactly 100 numbers not counting 0 in the Decimal 10 Grid.

If I wrote out the 100 Grid it would start with 0. 0.01, 0.02 and end in 99.98, 99.99, 100.
So, between 0 and 0.1 in 10 Grid, exists no other number but is empty space. Same for 0.1 to 0.2.

In order for Calculus to exist at all, you need empty space from one number to the next number. For the derivative in the picture diagram spans with a straight line the next point on the function graph. This is why physics needs calculus derivative for it predicts the next point of the function and is part of the function graph itself. Old Math was too stupid and they believed the derivative is a "tangent line to a point on the function graph". No, the derivative connects the previous point to the next point of the function graph and is part and parcel of the function itself.

Now we talk about the concept of Function.

Old Math got this correct in the idea that the Function is a correspondence of a given x-value to a unique one of a kind y-value.

I like to view a Function as a Motion, and calculus derivative as "in motion".

So in true math, there is first quadrant only for there are only positive numbers, no negative numbers. In Old Math, math professors had a difficult time of gaining fame and fortune. So, whenever a professor wants some fame and fortune, if they dream up some silly outlandish idea--- "hey, negative numbers exist". They get publicity and fame and fortune follows. Not content and happy to make a career in teaching just True Math, no, they want fame and fortune which then pollutes math.

In True Math we have only positive numbers and need for a graph to use only 1 quadrant.

Here is the start of 1st Quadrant where x and y axes are Decimal 10 Grid.

 ^
.4|
.3|
.2|
.1|
0|___________________>
       .1   .2   .3    .4

Now for teaching purposes, I will do the Integers only in 10 Grid and graph the function f(x) = x^2. We write a function as f(x), but I prefer to write a function as x^2 --> Y, or Y--> x^2.

The function x^2--> Y is a motion across the x-axis, starting at 0 and taking in every x-value number. Remember that the definition of function is there is a unique y-value given a x-value. A circle cannot be a function because many x values have 2 y-values. A half circle avoids this problem.

So looking at the function x^2 -> Y and we make a table in 10 Grid integers only.

x^2 -> Y
x       y
0      0
1      1
2      4
3      9
4      16
5      25
6      36
7      49
8      64
9      81
10    100

So we plug into the x^2 all the 10 Grid integer values on the x-axis and start making a table.

Then, we graph our table.

y-axis
^
|
|
|
                             
9                         /| 9
                            |
                           |
                           |
                         / |
                           |
                           |
                       /   |
                           |
                    /     |
                          |          
4            4/ |        |
                 |        |
           /     |        |
                 |        |
1    /  |1      |        |
   /     |       |        |
------------------------------------------------> x-axis
0      1       2       3

So let us summarize what a function is.

A function is a assigning a number on x axis with a unique number on y-axis. We denote a function with the symbol of an arrow ->. This symbol comes from the if-->then logic connector. The equality symbol = comes from equal-not in the logic connectors. We generally write a function with the arrow from the x axis to the y axis value, such as x -> Y, or x^2 +1 -> Y.
A function has a Y value side and the other side is the x value.
We use up every number in the x-axis of a Grid System to make our table. That is important, we use up every number on the x-axis to make our table. For this gives the function that of --- Motion---. Motion as it moves from 0 to every x value point.
Once we made our table we plot the function as a graph.
A function has a restriction, though. A function must have one y value for any given x-value. So for example a half a circle can be a function but not a full circle because most x values have two y values.