Many-valued logic facts for kids

Kids Encyclopedia Facts

Many-valued logic (also called multi-valued logic) is a special kind of logic where statements can have more than just two possible answers, like "true" or "false."

Normally, in everyday logic, a statement is either completely true or completely false. For example, "The sky is blue" is true, and "The sky is green" is false. This is called two-valued logic.

But what if a statement isn't clearly true or false? What if it's "maybe," "unknown," or somewhere in between? Many-valued logic explores these possibilities. It allows for more than two "truth values." For instance, some systems use three values: "true," "false," and "unknown" or "possible." Others use even more values, or even an infinite number of values, like in fuzzy logic, which deals with things that are partly true, like "the water is warm."

History of Many-Valued Logic
Examples of Many-Valued Logics
Many-Valued Logic and Classical Logic
Functional Completeness
Applications of Many-Valued Logic
Research and Study
See also

History of Many-Valued Logic

For a very long time, people mostly followed the ideas of Aristotle, an ancient Greek thinker. He is often called the "father of logic." Aristotle believed that every statement was either true or false. He did think about future events, like "There will be a sea battle tomorrow." He wondered if such a statement was already true or false today, or if it was still undecided. However, he didn't create a new system of logic to explain this.

The idea of logic with more than two values really came back in the 20th century.

In 1920, a Polish logician named Jan Łukasiewicz started to develop systems with a third value, which he called "possible." He used this to think about Aristotle's puzzle of future events.
Around the same time, in 1921, an American mathematician named Emil Post also came up with ways to use more than two truth values.
Later, Łukasiewicz worked with Alfred Tarski to create logics with n truth values, where n could be any number greater than or equal to 2.
In 1932, Hans Reichenbach even created a logic with an infinite number of truth values!
Also in 1932, Kurt Gödel showed that some types of logic, like intuitionistic logic, couldn't be described with just a few truth values. He also created his own family of logics.

Examples of Many-Valued Logics

Many different systems of many-valued logic have been created. Here are a few examples:

Kleene's and Priest's Three-Valued Logics

Stephen Cole Kleene and Graham Priest both developed logics that add a third truth value, often called "indeterminate" or "unknown" (represented as $I$ ).

In these logics, you can still have "True" ( $T$ ) and "False" ( $F$ ). The tables below show how basic logical operations (like "not," "and," "or," "if...then," and "if and only if") work with this third value.

¬
$T$	$F$
$I$	$I$
$F$	$T$

∧	$T$	$I$	$F$
$T$	$T$	$I$	$F$
$I$	$I$	$I$	$F$
$F$	$F$	$F$	$F$

∨	$T$	$I$	$F$
$T$	$T$	$T$	$T$
$I$	$T$	$I$	$I$
$F$	$T$	$I$	$F$

→K	$T$	$I$	$F$
$T$	$T$	$I$	$F$
$I$	$T$	$I$	$I$
$F$	$T$	$T$	$T$

↔K	$T$	$I$	$F$
$T$	$T$	$I$	$F$
$I$	$I$	$I$	$I$
$F$	$F$	$I$	$T$

The main difference between Kleene's and Priest's logics is how they define what makes a statement a "tautology" (a statement that is always true). In Kleene's logic, only "True" is considered a winning value. In Priest's logic, both "True" and "Indeterminate" can be winning values.

Bochvar's Three-Valued Logic

Dmitry Bochvar also created a three-valued logic. In his system, the "indeterminate" value is "contagious." This means if any part of a complex statement is "indeterminate," the whole statement becomes "indeterminate."

∧+	$T$	$I$	$F$
$T$	$T$	$I$	$F$
$I$	$I$	$I$	$I$
$F$	$F$	$I$	$F$

∨+	$T$	$I$	$F$
$T$	$T$	$I$	$T$
$I$	$I$	$I$	$I$
$F$	$T$	$I$	$F$

→+	$T$	$I$	$F$
$T$	$T$	$I$	$F$
$I$	$I$	$I$	$I$
$F$	$T$	$I$	$T$

Belnap's Four-Valued Logic

Nuel Belnap combined ideas from Kleene's and Priest's logics to create a four-valued logic. Besides "True" ( $T$ ) and "False" ( $F$ ), he added:

B for "both true and false" (overdetermined)
N for "neither true nor false" (underdetermined)

$f \neg$
$T$	$F$
$B$	$B$
$N$	$N$
$F$	$T$

$f \land$	$T$	$B$	$N$	$F$
$T$	$T$	$B$	$N$	$F$
$B$	$B$	$B$	$F$	$F$
$N$	$N$	$F$	$N$	$F$
$F$	$F$	$F$	$F$	$F$

$f \lor$	$T$	$B$	$N$	$F$
$T$	$T$	$T$	$T$	$T$
$B$	$T$	$B$	$T$	$B$
$N$	$T$	$T$	$N$	$N$
$F$	$T$	$B$	$N$	$F$

Gödel Logics

Kurt Gödel created a family of logics where truth values are numbers between 0 and 1. For example, in a 3-valued Gödel logic, the values could be 0, 0.5, and 1. In an infinite-valued Gödel logic, any real number between 0 and 1 could be a truth value. Here, 1 means "true."

"And" (conjunction) is the smallest of the values.
"Or" (disjunction) is the largest of the values.
"Not" (negation) is 1 if the value is 0, and 0 if the value is greater than 0.
"If...then" (implication) is 1 if the first value is less than or equal to the second. Otherwise, it's the second value.

Łukasiewicz Logics

Jan Łukasiewicz also created logics with truth values between 0 and 1. For example, his three-valued logic used 0, 0.5, and 1. He also developed an infinite-valued logic where any real number between 0 and 1 could be a truth value. In these logics, 1 means "true."

"Not" (negation) is calculated as 1 minus the value.
"If...then" (implication) is the smaller of 1 or (1 minus the first value plus the second value).

Product Logic

Product logic also uses truth values between 0 and 1.

One type of "and" (conjunction) is found by multiplying the values.
"If...then" (implication) is 1 if the first value is less than or equal to the second. Otherwise, it's the second value divided by the first.
It also has a special "false" value, often called $\overline{0$ }.

Post Logics

Emil Post defined a family of logics where truth values are also numbers between 0 and 1, similar to Gödel and Łukasiewicz logics.

"Not" (negation) is 1 if the value is 0. Otherwise, it's the value minus a small fraction.
"And" (conjunction) is the smallest of the values.
"Or" (disjunction) is the largest of the values.

Many-Valued Logic and Classical Logic

In classical two-valued logic, the main goal is to keep "truth" safe. If you start with true statements and follow valid rules, your conclusion will also be true.

Many-valued logics aim to preserve a similar idea, but it's called "designationhood." This means that certain truth values are "designated" or "accepted" as good outcomes. For example, in a three-valued logic, the two highest values might be designated. The rules of inference then make sure that if your starting statements have designated values, your conclusion will also have a designated value.

For instance, in intuitionistic logic, the property preserved is "justification" instead of "truth." A statement isn't just true or false; it's either "justified," "flawed," or "we can't prove either." This means that the idea that a statement is either true or false (the law of excluded middle) doesn't always apply. If a statement isn't flawed, it doesn't automatically mean it's justified. You might just not have enough information to prove it either way.

Functional Completeness

"Functional completeness" is a cool property for logics. It means that if a logic has a certain set of basic operations (like "not," "and," "or"), you can use those operations to create *any* possible truth function.

Classical two-valued logic is functionally complete. This means you can build any logical statement using just "not" and "and," or "not" and "or." However, many-valued logics, especially those with infinite values, often don't have this property.

Applications of Many-Valued Logic

Many-valued logic isn't just for philosophers! It has practical uses, especially in computer science and electronics:

Making Computers Smarter: Sometimes, many-valued logic can help computers solve problems more efficiently. For example, when a computer program has multiple outputs, you can treat them as one many-valued signal, which can simplify the design.
Designing Better Chips: Many-valued logic is used to design electronic circuits that use more than just two signal levels (on/off).
- Less Wiring: If a signal can have four or more levels instead of just two, you need fewer wires to send the same amount of information. This can make computer chips smaller and more powerful.
- More Memory: Imagine a memory cell that can store two bits of information instead of one. This doubles the memory capacity in the same space!
- Faster Calculations: Some number systems, like "residue" or "redundant" systems, can speed up addition and subtraction in computers by reducing how much information needs to "ripple through" the circuit. These systems work well with many-valued circuits.
Testing Circuits: Many-valued logic is very important for testing computer chips for errors. Special values like "unknown," "0 instead of 1," or "1 instead of 0" are used to find problems in the circuit design.

Research and Study

There's an annual meeting called the IEEE International Symposium on Multiple-Valued Logic (ISMVL) where experts discuss new ideas and applications, especially in digital design. There's also a scientific journal dedicated to this topic called the Journal of Multiple-Valued Logic and Soft Computing.