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CRAM Math Logical Thinking

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After knowing that algebra is a formal language, the next thing to know is that algebra is logical.

The same advice on going slowly and reading this out loud still applies, as it really helps in making sense out of it all when you hear it.

Algebra as deductive logic

To say that algebra is deductive logic is the same as saying that everything in algebra can and should be known, and anything in algebra is related to everything else you know about algebra.

A deduction is a conclusion that must come from the premises or facts that are known at the time you begin to work the problem; these premises lead to your conclusion when certain rules of deduction are used.
An example is the conclusion that follows from the following premises:
  • If it is raining, then Andrea is carrying her unbrella.
  • It is raining.
The inescapable conclusion is:
  • Andrea is carrying her umbrella.
When a deductive argument is in proper form, and all of the premises are true, then it is inescapable that the conclusion must be true. The argument is then said to be sound.
People who disagree with sound deductive arguments are inescapably unsound.

The beauty of algebra is that being deductive logic, it means that everything in algebra is either a definition that is accepted as true because it is the definition of something algebraic (such as the properties of algebraic objects or the axioms of the deductive logic you are using), or it is a deduction that is directly or indirectly derived from those definitions.

Corollary: There is no magic in algebra.
Nothing just appears magically in algebra. If you run into something new, then it is either a new definition of some algebraic object that you haven't seen before, or you must have skipped something earlier.
If it is something you skipped earlier, GO BACK or BE LOST on going forward. (A MAJOR CAUSE OF THRASHING)

A simplified deductive logic

It is possible to spend an entire semester on a college-level symbolic logic course, but we can strip that down to a minimum to get going with CRAM Math.

We can begin by looking more closely at the previous example about Andrea's umbrella in the rain.
Premises (often cited as "Given"):
  • If it is raining, then Andrea is carrying her unbrella.
  • It is raining.
Conclusion (often signaled with the word "Therefore"):
  • Andrea is carrying her umbrella.

There is a pattern here. We can get symbolic by using single capital letters stand for entire sentences, so that it is easier to concentrate on the logical pattern of the argument.

Let A stand for "It is raining."
Let B stand for "Andrea is carrying her umbrella."

The "If... then..." statement is a hypothetical statement, and it has a special form in deductive logic.

Let A \to B stand for "If A then B."
We can also read A \to B as "A implies B."

Now we can summarize the argument using only symbols, so long as we remember what A and B stand for.

A \to B (given)
A (given)
Therefore (often shown by stacked cannon balls "\therefore"):
B (conclusion)

One more time, using only the symbols we have just learned:

A \to B (given)
A (given)
\therefore B

This is pretty much what a deductive argument looks like.

Notice that the conclusion is also contained somehow in the premises. This is part of the "no magic" property of deductive logic. The conclusion just did not appear out of thin air but is actually contained in the premises.

The deductive rule of inference: Modus Ponens

Any deductive argument that is the above pattern is known as Modus Ponens.

For the last time, listing the meanings of our capital letters and using "Modus Ponens" or "MP" as the reason for making our conclusion:
Let A stand for "It is raining."
Let B stand for "Andrea is carrying her umbrella."
A \to B (given)
A (given)
\therefore B (MP)
Because A is true in this problem, we know that Andrea is carrying her umbrella in this problem.

So what?

Solving algebra problems assigned as homework or dropped in our laps during quizzes and exams is a logical step-wise process of re-writing the facts we are given into a simple statement of our solution. The deductive nature of this process guarantees us that as long as we state the givens and use the deductive process to reach a conclusion, our answers must be right!

Formally-trained mathematicians and teachers of algebra identify and recognize these re-writing processes as proofs and know they cannot be argued with unless one is trying to be unsound.
Proofs are sound only when the proof is in valid form and the premises are true; these are the two necessary conditions for having a sound argument. If any of the premises are shown to be not true, or if the argument's deductive form is invalid in any way, you don't really have a proof.

THIS MEANS THAT IF YOU USE THIS PROCESS CORRECTLY, YOU CANNOT BE WRONG!!!

If you learn and use CRAM Math, it is completely possible for you to become an "A" student, even if you have been flunking out up until now. I have seen it happen, where students who were flunking out actually passed Algebra I with "B" and "A" semester grades after getting an "F" in the previous semester.
This cannot happen unless you learn and practice what you learn. There are no miracles, only hard work, but it does pay off in better grades when you use CRAM Math as a guide to avoid thrashing in frustration.

Observation about logic leading to programming

Much of what goes on in a computer program is actually decision-making. A series of implications can form the basic logic of a program. As the program runs over time, various conditions become true, and the program reacts to things that the user provides as input, whether it is a keystroke on the keyboard, the movement of a mouse or joystick, or the press and release of the joystick button.

All of the "reactions" of animated characters in computer games are actually the evaluations of algebraic expressions in a computer program that are mapped to your computer's input devices.

Reference

CRAM Math Fundamentals

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