In C, or any programming language, what does it mean for a condition to be true? What does it mean for a condition to be false?

What does it mean to say that something is "true"?

For the present purposes, the following are considered categories of truth.

• Logical truth (integer numbers)
• Reality truth (statistical reasoning, floating point approximations of real numbers)
• Human truth (everything else)

These areas can overlap but the distinction is, nonetheless, useful.

Logical truth: symbol manipulation involving integer values, which is exact but may involve paradoxes. This includes coded information, programming languages, digital computers, etc.

Mathematics is concerned with logical without any connection to reality. (Mathematicians such as Hilbert made this decision in the early 1900's).

In the 1920's, Hilbert (1928) proposed finding a consistent mathematical system that will allow all possible truths
to be decided. This would allow the automatic, or mechanical, proving of all possible truths.

Hilbert invented/discovered the Hilbert curve, a monster curve that was not fully understood until the introduction of fractals a half-century later. Logical truth involves mathematical and computer/information logic using two discrete logic values (e.g., true and false, 1 and 0) and symbol manipulation using rules to determine which claims are "true" or "false".

Computer logic and programming involves logical truth.

Hilbert and his problems and the separation of math with reality and philosophy.

Note that generalizations of logical truth can involve paradoxes, incompleteness, undecidability, etc.

Reality truth: approximations using all math values, probability and statistics, etc., which may use logical truth. Traditional science involves reality truth.

Reality truth involves what exactly is real, or reality. Reality truth involves probability, statistics, approximations, unknowns, etc. The traditional scientific method involves reality truth. So does, to some extent, the traditional legal process.

Opinion truth: all other types of truth, which may or may not use logical and reality truth, but need not do so.

Human truth can be whatever one or more humans decide it is. Thus, there are no precise rules for defining human truth other than it is not necessarily logical truth or statistical truth.

Advice/adage:

• Logic: If the law is on your side, pound the law.
• Reality: If the facts are on your side, pound the facts.
• Opinion: If neither is on your side, pound the table.

In a programming languages course, one is primarily concerned with logical truth. There may or may not be any connection of logical truth with reality truth and/or human truth.

Thus, a Boolean condition that is either true or false may or may have no connection to "reality".

Arithmetic operations (addition, subtraction, etc.) involve a potentially infinite number of integer values.

• -∞, ... , -2, -1, 0, 1, 2, ... , ∞

Note: In math, integers are infinite. In computers, integers are of finite size.

Only two values are needed for two-value logic, so it must be easier than integer arithmetic.

Often, 1 is used for true and 0 is used for false.

Thus, truth table logic using two values must be simpler than arithmetic operations and logic.

Logical operations, sometimes called Boolean operations after George Boole (Founder of Boolean logic) (1820-1898), are operations that involve two values, true and false.

Two values are needed for two value logic. Here are some examples.

• True and False, t and f
• 1 and 0
• yes and no, y and n, yea and nay
• on and off
• high and low, h and l

Double-edged swords have a true edge (very sharp) and a false edge (not so sharp).

• If you are not sad/mad, you are happy.

• If you are not happy, you are sad/mad.

Magnets have opposite polarities, called poles.

Magnetic media, such as fixed/floppy disks, use one polarity for 1 and the other for 0.

Optical media, such as CD-ROM's, use high or low reflectivity as two values.

The high or low current or voltage can be used as 0 and 1.

The literal names true and false are the simplest form of what are called logical expressions.

A literal literally expresses a value and cannot be simplified. In this case, the expression true expresses the value true and the expression false expresses the value false.

Be aware that mathematicians often call an expression a formula while computer scientists often call a formula an expression.

Since we have two distinct values,

• true is always true,
• true can never be false,
• false is always false, and
• false can never be true.

The C programming language does not directly support Boolean values except as conditions (for if, while, etc.).

Even in other languages, it is often best to use 0 for false, 1 for true, and allow for additional integer values for additional states.

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