*D.M. Gabbay and L. Maksimova*

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0005
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter contains a proof of Lyndon's interpolation property (LIP) for quantified extensions of basic modal logics K, T, D, K4, and S4, and for some others, including the propositional S5 has ...
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This chapter contains a proof of Lyndon's interpolation property (LIP) for quantified extensions of basic modal logics K, T, D, K4, and S4, and for some others, including the propositional S5 has LIP. At the same time, the quantified extension of S5, as well as other systems satisfying the Barcan formula has neither Lyndon's nor Craig's interpolation, nor Beth's property. Some examples of propositional modal logics, which have CIP but do not possess LIP are found. Craig's interpolation property is proved for a number of propositional modal logics, including the Grzegorczyk logic Grz, its extension Grz.2, and the provability logic G. A class of so-called L-conservative formulas is defined, which can be added to a propositional logic L as new axiom schemes without loss of interpolation. It is proved that the interpolation properties are preserved by transfer from predicate logics without equality to their extensions with equality.Less

This chapter contains a proof of Lyndon's interpolation property (LIP) for quantified extensions of basic modal logics K, T, D, K4, and S4, and for some others, including the propositional S5 has LIP. At the same time, the quantified extension of S5, as well as other systems satisfying the Barcan formula has neither Lyndon's nor Craig's interpolation, nor Beth's property. Some examples of propositional modal logics, which have CIP but do not possess LIP are found. Craig's interpolation property is proved for a number of propositional modal logics, including the Grzegorczyk logic Grz, its extension Grz.2, and the provability logic G. A class of so-called L-conservative formulas is defined, which can be added to a propositional logic L as new axiom schemes without loss of interpolation. It is proved that the interpolation properties are preserved by transfer from predicate logics without equality to their extensions with equality.

*Joseph Mazur*

- Published in print:
- 2016
- Published Online:
- January 2018
- ISBN:
- 9780691173375
- eISBN:
- 9781400850112
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691173375.003.0014
- Subject:
- Mathematics, History of Mathematics

This chapter discusses what Rafael Bombelli called dignità, which translates to the English word “dignity” and is equivalent to what we refer to as “exponents.” It first considers Bombelli's ...
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This chapter discusses what Rafael Bombelli called dignità, which translates to the English word “dignity” and is equivalent to what we refer to as “exponents.” It first considers Bombelli's L'Algebra (1579), which introduces a new kind of notation for the unknown and its powers. Written in Italian, L'Algebra used equality in a different sense than we do. For Bombelli, the higher powers meant hierarchies of dignity. He was not only inventing genuine symbols when depicting dignità as little cups holding numbers, but also inventing words that were new to mathematics. The chapter also examines how the problem of finding the roots of polynomials became the problem of factoring polynomials. Finally, it looks at René Descartes's idea of using numerical superscripts to mark positive integral exponents of a polynomial in his La Géométrie.Less

This chapter discusses what Rafael Bombelli called *dignità*, which translates to the English word “dignity” and is equivalent to what we refer to as “exponents.” It first considers Bombelli's *L'Algebra* (1579), which introduces a new kind of notation for the unknown and its powers. Written in Italian, *L'Algebra* used equality in a different sense than we do. For Bombelli, the higher powers meant hierarchies of dignity. He was not only inventing genuine symbols when depicting *dignità* as little cups holding numbers, but also inventing words that were new to mathematics. The chapter also examines how the problem of finding the roots of polynomials became the problem of factoring polynomials. Finally, it looks at René Descartes's idea of using numerical superscripts to mark positive integral exponents of a polynomial in his *La Géométrie*.