There are three basic operators on boolean expressions:

• The unary not-operator ~
• The binary and-operator &
• The binary or-operator |

In addition, round brackets ( and ) can be used to group expressions. In the absence of brackets, the precedence of the operators corrresponds to the order in which they are listed above. For instance, ~p() | q() & r() is identical to (~p()) | (q() & r()).

As the first of these two exercises shows, the "or"-operator is actually redundant, since we can express it using a combination of "and" and "not". (Alternatively, we can also express the "and" in terms of "or" and "not", so also "and" is redundant once we have the other two.) Of course, there are good reasons for nevertheless having all three operators. One reason is that formulas can be written in a more compact and legible way if we have "or" available as an abbreviation for the more complex formula that uses "and" and "not". Another reason is that "or" appears to a primitive of human thought: it is part of how knowledge is structured in the human brain and therefore its presence in the language is necessary to allow natural representations of this knowledge.

The second of the two exercises above illustrates an important caveat: in everyday speech, we often use the same word for different concepts, counting on our common-sense to figure out which meaning is intended. Compare, for instance, the following two uses of the word "or".

• One detective to another: "The suspect must have left his fingerprints on the gun or on the bullets."
• The robber to the victim: "Give me the money or I'll shoot!"

In the first example, the intended meaning of the word "or" is the "inclusive or" (which has three models), while in the second example it is the "exclusive or" (which has only two models). These different uses of the same word are often entirely unproblematic, since people are typically very good at figuring out which meaning is intended. When building formal representation of knowledge, however, it is important to be aware of the fact that such different meaning exist, and that each operator in the formal language of course only has a single meaning (that of "inclusive or", in this case).

This section introduces two new boolean operators. Similar to the "or", these operators are redundant, in the sense that they can be seen as abbrevations for expressions involving the basic boolean operator. However, they too seem to correspond to how knowledge is structured in our brains and are therefore important to have in the formal language.

The first is the implication operator, which is written as F => G. We read this as "F implies G" or more simply as "if F then G". Similar to "or", the "if … then …" construct is also used in different meanings in natural language. The implication operator captures the meaning that "if" has in the following example:

• One member of the bomb squad to another: "If the drawing we found in the suspect's apartment is correct, then the red wire is the one we should cut."

There are four situations, of which three are possible according to this statement:

• The drawing is correct and the red wire is the right one.
• The drawing is incorrect and the red wire is not the one to cut.
• The drawing is incorrect but the red wire is the one to cut anyway.

Therefore, this statement only rules out one situation, namely this one:

• The drawing is correct and the red wire is not the one to cut.

This immediately also shows us that F => G can indeed be seen as an abbreviation for an expression in the basic boolean operators, namely ~(F & ~G). As we learned from an earlier exercise, this expression is equivalent to ~F | G. In other words, as long as the "if"-part of the statement is not true or the "then" part is true (or both are true), the entire statement is true.

Again, it is important to know that this is the meaning of "if … then" that is captured by the => operator. The case that might create confusion is the "or both" case, i.e., the fact that it is possible that the "if"-part of the statement is false, but the "then"-part is true anyway. To illustrate, consider the following other use of "if":

• A natural number is called prime if it is strictly greater than one and does not have any divisors other than one and itself.

Here, the word "if" is being used with a different meaning, that does not correspond to the => in FO. The difference here is that the statement is intended to define what prime numbers are, so it should not be possible that the "if"-part is false for some number, but that this number is prime anyway. While it should be clear to a typical reader that this is the intended meaning of the statement, mathematicians may prefer to be more explicit about this, in order to make absolutely sure that the statement will be unambiguously understood. In this case, a mathematician may say:

• A natural number is called prime if and only if it is strictly greater than one and does not have any divisors other than one and itself.

The use of the "if and only if" removes any room for misunderstanding. Such a statement is called an equivalence and is denoted in FO by a double arrow <=>. An expression F <=> G is true if (and only if) F and G have the same truth value. In other words, the following two situations are possible:

• A number is prime and satisfies the condition of being >1 and having no non-trivial divisors.
• A number is not prime and does not satisfy this condition.

The following two situations are not possible:

• A number is not prime even though it satisfies the condition.
• A number is prime even though it does not satisfy the condition.

This last case is the one which distinguishes the equivalence prime <=> condition from an implication prime <= condition. The reason for the choice of a double-headed arrow as the symbol for equivalence is that F <=> G is identical to (F => G) & (F <= G). In other words, F and G are equivalent if and only if F implies G and also G implies F.

(As a side remark, the IDP-Z3 software also allows to write F <=> G as F = G. It is slightly unusual to write an equivalence in this way, but it nevertheless of course makes perfect sense.)

In our previous examples, it was always pretty clear whether the word "if" was intended as an implication or an equivalence. The following example is more debatable:

• Detective to witness: "If you don't testify in court, a dangerous criminal will go free."

Does the detective also intend to say that if the witness does testify, the criminal will not go free? This is a question for linguists or psychologists, but not for us. All that matters from a knowledge representation perspective is that once we know what the intended meaning of the statement actually is, we are able to translate it to an expression in the formal logic.