Immediately a problem presents itself. In standard
notation the order of the quantifiers in a sentence can make a big
difference to its meaning. How can one make up for this loss if we
discard quantifiers? My solution is to use

*scope brackets*, ‘[**’**and ‘]**’**. For every distinct variable in a sentence there is a pair of scope brackets that determine the scope of the claim involving that variable. Where, in standard notation, a quantifier would be within the scope of another quantifier, in variable notation the scope brackets of the former quantified claim are inside the scope brackets of the latter quantified claim. The brackets should enclose no more than they need to.
Some examples will make this clear. Consider (1) and (2), and their counterparts (1’) and (2’):

1. ∀x ∃y

*(Rxy) 1’. [RX[y]]*
2. ∃y

*∀x (Rxy) 2’. [R[X]y]*
In (1’) the scope brackets for y
enclose only y, and the scope brackets for X enclose the entire
sentence, indicating that the existential claim is governed by the
universal claim. In (2’) the reverse is true, indicating that the
universal claim is governed by the existential claim.

If you want to see this idea worked out in more detail, please see the full article at Scholardarity, Quantification without Quantifiers (or Epsilons): A New Notation for Quantification in Predicate Logic.