Monday, May 21, 2012

Quantification without Quantifiers (or Epsilons)

I believe I've  found a way to say everything that one would normally say in predicate logic using quantifiers, but without actually using any, and also without using Hilbert’s epsilon operator. (See "Hilbert's Epsilon Calculus", here.) To begin with, instead of saying that something holds for all x by binding x with a universal quantifier, one can say it by capitalizing all of x’s occurrences in a sentence that would, in “standard notation”, be in the scope of the quantifier. And instead of saying that something holds for some y by binding y with an existential quantifier, one can say it by leaving all of y’s occurrences un-capitalized. Thus, in “variable notation”—so called because it expresses quantification by modifying variables—instead of ∀x (Fx) one would write FX, and instead of ∃y (Gy) one would write Gy.

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.