Double bondings and feedback loops

Consider the double bonding A ⇒ B and B ⇒ A, or A ⇔ B. Double bonding is also known as the biconditional or XNOR connective in formal logic.

In a double bonding, the two fields A and B are co-extensive. If these 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 possibility 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.

-oOo-

References

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).

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