Overdetermination of the symbol =

The symbol = is seen in every corner of logics, mathematics, and programming languages. Let us take a tour on how this specifc symbol is used and referred to, and see how “overdetermined” it currently is. Starting from basic formal logic - specifically the Law of Identity, loaded with Mathematical categories like equivalence relations, isomorphisms, and so on, finally toward practices and theories in and about programming languages.

Determination of = in logics

In classical formal logic, the Law of Identity says:

$$A = A$$

Remarks

• There is no further determination of A except it’s an empty letter thats holds an intentionality toward some “logical proposition”, and has dual occurrences.
• There is a short-circuit between the arbitrariness of the letter and the universally-quantifiability of the proposition.
• Intentionality toward other letters (potentially B, C, D and so on) is not explicit.
• In addition to holding the identity of A, = reflects an internal tension of A.

There is already a minimum level of contradiction in the dual occurrences of A. Questions arise naturally: What’s the difference between the A in lhs and the A in rhs? If A is self-consistent (or self-persistent), then for what and by what it has to identify with itself?

Such identity is free (with no cost). There is no locativity and linearity, let alone temporality and further physicality. More about Identity is discussed in another post.

Determination of = in mathematics

The most familar and basic determination of = in mathematics is as “equivalence relation”, which satisfies:

• Reflexivity: $A = A$
• Symmetricity: $A = B$ iff $B = A$.
• Transitivity: if $A = B$ and $B = C$, then $A = C$.

For now let us suspend any references to foundations of mathematics like Set Theory.

Determination of = in PL

In modern programming languages, we can write programs like:

let f = \x -> (x, x) in
  let a = 42 in
    let r = f a in
      r

Where = intuitively means “binding” or “assignment”. Clearly, none of three (reflexivity, symmetricity, transitivity) are applicable to the = above. This is despite tons of “features” of PL that keep overdetermining the symbol =.

Intrusion of locativity

The triad (A before, A after, identification/alienation of A).

Minimum level of spatiality (or temporality, causality?).

See Jean-Yves Girard.

(under construction)

As postulate, or alsob

Ad hoc postulation. Reduction to a common third.

(under construction)

As exchange, or transaction

Exchange of commodity. Linear Logic.

(under construction)

The dialectic of identity

(under construction)

References

1. https://www.youtube.com/watch?v=0mrC0kIr_aQ