Double bondings and feedback loops

Consider the double bonding A ⇒ B and B ⇒ A, or A ⇔ B.

In a double bonding in logic, the two fields A and B are co-extensive. If they are just two different names for the same thing, this is an innocent synonymy, as in the instances of nomenclatural synonymy in plant names. But if we consider them to be different (and by using the two names A and B we seem to be making that consideration), then it’s not at all innocent.

Then the statements A ⇒ B and B ⇒A together create a paradox where A and B are both identical and different; this can only be represented by an imaginary Boolean value as defined by Spencer-Brown (1969). The double bonding contains the seed of a feedback loop to an imaginary value.

This imaginary value can be approached more stealthily by making a series of bondings such as A ⇒ B, B ⇒ C, C ⇒ D and then adding D ⇒ A to create what Hofstadter (1979) called a strange loop. In other words, a function that re-enters itself, in this case at the fourth level.

The posssibility of double bondings as paradoxes or fallacies was noted by Lewis Carroll at the Mad Hatter’s tea party in Alice in Wonderland. Grammatically, “I see what I eat” could be equivalent to “I eat what I see.” But in English language syntax the order of antecedent and consequent expresses a convention that the first sentence means that Eat ⇒ See, but not that See ⇒ Eat.



Hofstadter, D. (1979) Gödel, Escher, Bach: An Eternal Golden Braid. (Basic Books: New York ).

Spencer-Brown, G. (1969) The Laws of Form. (Allen & Unwin: London).

Watsonia pillansii

Watsonia pillansii L.Bolus is widespread in the eastern (i.e. summer rainfall) part of South Africa at low and medium elevations. This wide geographic range is associated with variation in ecological requirements and plant size, but the flower colour is generally bright orange to orange-red.

Plants grown from seed recently imported from South Africa have unbranched stems to 1.2 m high bearing up to 22 flowers. They are evergreen, with new shoots appearing in late summer immediately after flowering and before the previous season’s leaves have died. Each flower has a cylindric tube 3.5 to 5 cm long and acute perianth lobes to 24 mm long that flare widely when fully open; the colour in this strain whose exact provenance is unknown is a rather weak orange-juice orange on the lobes and deeper on the outside of the tube. The anthers and pollen are cream.


Watsonia pillansii is related to W. schlechteri in the section Watsonia, subsection Grandibractea.

The species has been in cultivation in Australia since the 19th century. Cultivars that may be selections of W. pillansii include ‘Flame’ (marketed by Lawrence Ball in the 1940s) and ‘Watermelon Shades’ (Cheers, 1997). Watsonia ‘Beatrice’ or the Beatrice Hybrids is a group name for various natural hybrids of W. pillansii (Eliovson, 1968) that were exported to Britain, America and Australia in the early 20th century. The name comes from Watsonia beatricis J.Mathews & L.Bolus, which was a taxonomic synonym of W. pillansii.


Cheers, G. ed. (1997) Botanica. (Random House Australia).

Eliovson, S. (1968) Bulbs for the Gardener in the Southern Hemisphere. (Reed: Wellington).

Goldblatt, P. (1989) The Genus Watsonia. (National Botanic Gardens: Kirstenbosch)