How can we represent biological relativity in logical notation?

Organism *a* is adapting relative to organism *b*

Aab

Organism *b* is adapting relative to *a*

Aba

Organisms *a* and *b* are adapting relative to each other

Aab & Aba

This schema is unsatisfactory because it describes the situation from an indeterminate outside perspective: a and b are said to be adapting relative to each other without regard to the observer describing the situation. Relativity applies to all the perspectives in question (with special focus on any observer perspective) and hence we need a way to include the observer perspective. This means we need to take into account how the observer is adapted such that the observer(s) can be compared to the organisms in question.

To remedy this problem let quantifiers range over organisms and include witnesses to identify the specific organisms in question:

For any organism x, for any organism y, there exists an organism z and there exists an organism u such that x is adapted relative to y according to organism z, and y is adapted relative to x according to organism u.

(∀x)(∀y)(∃z)(∃u)A[xyzu]

Unfortunately this formulation is insufficient because witness z is logically dependent upon both x and y (as is u as well) and we want z to only witness x and u to only witness y: as both z and u are dependent upon both x and y, both x and y must be chosen before selecting z and u. This means that organisms x and y are selected (logically) independent of the witness organisms defeating the purpose of having those witnesses.

Getting around this difficulty is not trivial in first order logic. There is no way in first order logic to linearly order the four quantifiers such that z only depends on x and u only depends on y (Kolak & Symons p.249 [p.40 of the pdf]). Independence Friendly logic suffices though :

(∀x)(∀y)(∃z/∀y)(∃u/∀x)A[xyzu]

This statement says that for any organism x, for any organism y, there exists an organism z that does not depend on y and an organism u that does not depend on x, such that organism x as witnessed by z, and organism y as witnessed by u, are adapted relative to each other.

However, though this statement gets very close to describing biological relativity, if we consider how the witnesses *witness* the organisms, i.e. how z witnesses the organism x, there is a problem. By stating that z witnesses x and that z is independent of y, the statement āx is adapted relative to y as witnessed by zā is nonsense: since z is independent of y it could not be a witness to āx adapting relative to y.ā Likewise for u.

The solution is simple enough though:

(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & A[x,y])

By letting x=z, making z independent of x and dependent on y, z witnesses y from the perspective of x without requiring x to be chosen before z. Likewise for u: if y=u, u is logically independent of y and u is dependent on x, then u may be chosen before y, u is dependent as a witness to the choice of x and witnesses x from the perspective of y. Perhaps more prosaically: x and y are adapting relative to each other, as witnessed by organisms z and u (who have the equivalent adaptations respectively to x and y), and it is not necessary to predetermine what those adaptations are.