How can we represent relativity in logical notation?

a is moving relative to b

Mab

b is moving relative to a

Mba

a and b are moving relative to each other

Mab & Mba

This schema is unsatisfactory because it describes the situation from an indeterminate outside perspective: a and b are moving 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.

To remedy this problem let quantifiers range over perspectives and include witness individuals to identify the specific perspectives in question:

For any perspective x, for any perspective y, there exists a perspective z and there exists a perspective u such that x is moving relative to y according to witness z, and y is moving relative to x according to witness u.

(∀x)(∀y)(∃z)(∃u)M[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 perspectives x and y are selected independent of the witness perspectives 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)M[xyzu]

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

However, though this statement gets very close to describing relativity, if we consider how the witnesses *witness* the perspectives, how z witnesses the perspective x, there is a problem. By stating that z witnesses x and that z is independent of y, the statement ‘x is moving relative to y as witnessed by z’ is nonsense: since z is independent of y it could not be a witness to ‘x moving relative to y.’ Likewise for u.

The solution is simple enough though:

(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & M[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 move relatively to each other, as witnessed by z and u (who have the equivalent perspectives, respectively to x and y), and it is not necessary to predetermine what those perspectives were.

A time variable rounds everything out nicely:

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

So, at time t (say now) let’s let u be your (the reader’s) perspective and z be my (the author’s) perspective. Then this statement describes our current motions as relative to each other because my perspective depends upon y, which is your perspective and your perspective depends on x, which is my perspective. Success!

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- Daniel Kolak and John Symons, “The Results are In: The Scope and Import of Hintikkaâ€™s Philosophy[PDF],” Quantifiers, Questions and Quantum Physics, in Daniel Kolak and John Symons, eds., Quantifiers, Questions, and Quantum Physics, Dordrecht, The Netherlands: Springer 2004 pp. 205-268 ISBN 1-4020-3210-2
- To see a (distant) predecessor to this paper, see my “Discourse, Perspectival and Public Identifications, and Tautology“

Ack! How did I not realize that that time variable needs to be relativized?