negation, not, \+
The concept of logical negation in Prolog is problematical, in the sense that the only method that Prolog can use to tell if a proposition is false is to try to prove it (from the facts and rules that it has been told about), and then if this attempt fails, it concludes that the proposition is false. This is referred to as negation as failure. An obvious problem is that Prolog may not have been told some critical fact or rule, so that it will not be able to prove the proposition. In such a case, the falsity of the proposition is only relative to the "mini-world-model" defined by the facts and rules known to the Prolog interpreter. This is sometimes referred to as the closed-world assumption.

A less obvious problem is that, depending again on the rules and facts known to the Prolog interpreter, it may take a very long time to determine that the proposition cannot be proven. In certain cases, it might "take" infinite time.

Because of the problems of negation-as-failure, negation in Prolog is represented in modern Prolog interpreters using the symbol \+, which is supposed to be a mnemonic for not provable with the \ standing for not and the + for provable. In practice, current Prolog interpreters tend to support the older operator not as well, as it is present in lots of older Prolog code, which would break if not were not available.

Examples:

?- \+ (2 = 4).

true.

?- not(2 = 4).

true.

Arithmetic comparison operators in Prolog each come equipped with a negation which does not have a "negation as failure" problem, because it is always possible to determine, for example, if two numbers are equal, though there may be approximation issues if the comparison is between fractional (floating-point) numbers. So it is probably best to use the arithmetic comparison operators if numeric quantities are being compared. Thus, a better way to do the comparisons shown above would be:

?- 2 =\= 4.

true.