# Harrop formula

In intuitionistic logic, the **Harrop formulae**, named after Ronald Harrop, are the class of formulae inductively defined as follows:[1][2][3]

- Atomic formulae are Harrop, including falsity (⊥);
- is Harrop provided and are;
- is Harrop for any well-formed formula ;
- is Harrop provided is, and is any well-formed formula;
- is Harrop provided is.

By excluding disjunction and existential quantification (except in the antecedent of implication), non-constructive predicates are avoided, which has benefits for computer implementation. From a constructivist point of view, Harrop formulae are "well-behaved." For example, in Heyting arithmetic, Harrop formulae satisfy a classical equivalence not usually satisfied in constructive logic:[1]

Harrop formulae were introduced around 1956 by Ronald Harrop and independently by Helena Rasiowa.[2] Variations of the fundamental concept are used in different branches of constructive mathematics and logic programming.

## Hereditary Harrop formulae and logic programming

A more complex definition of hereditary Harrop formulae is used in logic programming as a generalisation of Horn clauses, and forms the basis for the language λProlog. Hereditary Harrop formulae are defined in terms of two (sometimes three) recursive sets of formulae. In one formulation:[4]

- Rigid atomic formulae, i.e. constants or formulae , are hereditary Harrop;
- is hereditary Harrop provided and are;
- is hereditary Harrop provided is;
- is hereditary Harrop provided is rigidly atomic, and is a
*G*-formula.

*G*-formulae are defined as follows:[4]

- Atomic formulae are
*G*-formulae, including truth(⊤); - is a
*G*-formula provided and are; - is a
*G*-formula provided and are; - is a
*G*-formula provided is; - is a
*G*-formula provided is; - is a
*G*-formula provided is, and is hereditary Harrop.

## See also

## References

- Dummett, Michael (2000).
*Elements of Intuitionism*(2nd ed.). Oxford University Press. p. 227. ISBN 0-19-850524-8. - A. S. Troelstra, H. Schwichtenberg.
*Basic proof theory*. Cambridge University Press. ISBN 0-521-77911-1.CS1 maint: uses authors parameter (link) - Ronald Harrop (1956). "On disjunctions and existential statements in intuitionistic systems of logic".
*Mathematische Annalen*.**132**(4): 347. doi:10.1007/BF01360048. - Dov M. Gabbay, Christopher John Hogger, John Alan Robinson,
*Handbook of Logic in Artificial Intelligence and Logic Programming: Logic programming*, Oxford University Press, 1998, p 575, ISBN 0-19-853792-1