Residual operators in natural language

The meaning of residual operators $/$, $\backslash$, and $\diamond$ 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.

Natural language templates

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 click here. The systematic translation is given in the following table:

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.

Other explanation

Would you like a different explanation of the residual operators? This page explains them in logic. Click here for visualized examples about residual operators.