The meaning of residual operators /, \, and â‹„ is best explained by means of examples.
Assume we have a relation, label[Contract*Colour], which contains the colour of labels on contracts. A fact "1834" label "blue" means that contract 1834 has a blue label.
Also assume another relation stored[Contract*Location], which gives the location where a contract is stored. Fact "1834" store "cabinet 42" means that contract 1834 is stored in cabinet 42.
The sentence: "All contracts with a blue label are stored in cabinet 42." is represented as "blue" (label\stored) "cabinet 42". Literally it says: For every contract, if it has a blue label, then it is stored in cabinet 42.
The sentence: "All contracts that are stored in cabinet 42 have a blue label." is represented as "blue" (label~/stored~) "cabinet 42". Literally it says: For every contract, if it is stored in cabinet 42, then it has a blue label.
The sentence: "All blue labeled contracts and no others are stored in cabinet 42." is represented as "blue" (label~<>stored) "cabinet 42". Literally it says: For every contract, if it has a blue label, then it is stored in cabinet 42 and if it is stored in cabinet 42, then it has a blue label.
There is a pattern to this. A computer can generate a literal translation from the formula to natural language. However, that translation looks clumsy, verbose and elaborate. It is up to you to turn that in normal language. For examples . The systematic translation is given in the following table:
Would you like a different explanation of the residual operators? explains them in logic. for visualized examples about residual operators.
Formally
Natural language template
a (r\s) b
For every x: if x r athen x s b.
a (r/s) b
For every x: if b s x then a r x.
a (r<>s) b
For every x: if a r x then x s b and if x s b then a r x.