# 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](https://ampersandtarski.gitbook.io/documentation/the-language-ampersand/terms/semantics-visualized/semantics-visualized). The systematic translation is given in the following table:

| Formally     | Natural language template                                           |
| ------------ | ------------------------------------------------------------------- |
| `a (r\s) b`  | For every `x`: if `x r a`then `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](https://ampersandtarski.gitbook.io/documentation/the-language-ampersand/terms/semantics-in-logic/residual-operators) explains them in logic. [Click here](https://ampersandtarski.gitbook.io/documentation/the-language-ampersand/terms/semantics-visualized/semantics-visualized) for visualized examples about residual operators.