Dual combinators emerge from the aim of assigning
formulas containing $\leftarrow$ as types to combinators. This
paper investigates formally some of the properties of combinatory
systems that include both combinators and dual combinators.
Although the addition of dual combinators to a combinatory system
does not affect the unique decomposition of terms, it turns out
that some terms might be redexes in two ways (with a combinator as
its head, and with a dual combinator as its head). We prove a
general theorem stating that no dual combinatory system
possesses the Church-Rosser property. Although the lack of
confluence might be problematic in some cases, it is not a problem
per se. In particular, we show that no damage is inflicted
upon the structurally free logics, the system in which dual
combinators first appeared.
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