A blog about belief

Definition of universality, pt. 2

In Philosophy on September 14, 2011 at 1:52 pm

Recently, I started looking for a way to formalize what seems to me essential to the definition of “universality,” that a universal claim must say something meaningful (meaning falsifiable) about every element of the set across which it quantifies – that, to put it another way, if what it says about the elements is contingent, it cannot be contingent in such a way that certain elements will be excluded in all possible circumstances.  We could do this by requiring that if the universal claim has the form ∀x (Qx → Px), it must be possible that Q be true of any x; but I also want to cover cases in which the contingency is “hidden” in the definition of a predicate.  What I came up with this this: a statement ∀x Px is universal iff

∀x ⋄(¬Px ∧ ∀y(y ≠ x → Py))

What this states is that any element x of the universal set can in some possible future be a lone counterexample to the claim that ∀x Px.  But we still do not specifically require that Px says something “about” x; all we are really guaranteeing is that there are, numerically, as many ways of falsifying the universal as there are elements x.  I noted in the last post that this method breaks down when the universal set is infinite (because, of course, one can exclude some elements of an infinite set without changing its cardinality), but there is a similar problem that can arise in finite cases.

Suppose that there are 100 balls, 50 of them red and 50 green.  Let us assume that the color of a given ball is fixed, and that no ball can be both red and green.  Suppose also that each of the balls can be striped, and that each can be spotted, and that a ball can be both spotted and striped at once or be neither.  Now consider the claim that all of the green balls are both spotted and striped.  This claim does not really “say” something about all 100 of the balls, but we can finagle it into the form ∀x Px such that P meets the condition given above, like so.  Let f be a bijection from the set of all red balls onto the set of all green balls.  Such a function must exist because the number of red balls is equal to the number of green balls.  If x is a green ball, let Px be true iff x is spotted; if x is a red ball, let Px be true iff f(x) is striped.  ∀x Px is true, then, iff all of the green balls satisfy P (meaning that all the green balls are spotted) and all of the red balls satisfy P (meaning that all of the counterparts of the red balls, i.e. of all the green balls, are striped).  This is equivalent to our original statement of the claim.  Under this definition, P can be false of any x while being true of everything else.  If a red ball x fails to be spotted, Px will be false, and if a red ball x fails to be striped, Py will be false of its green counterpart y.  The claim only “says” something about 50 of the balls, but because there are two different ways in which a red ball can counter our claim (either by failing in spottedness or by failing in stripedness), we can find 100 ways in which the claim could be disproven.

Instead of requiring that any presently-existing x could in the future serve as a counterexample while the rule continues to apply universally elsewhere, let us now require that any presently-existing x could in the future serve as a counterexample in the absence of any other objects.  Formally,

∀x ⋄(¬Px ∧ ∀y(y = x))

It is easy to see that the example claim about the red and green balls does not meet this modified standard.  If it is possible that all balls be extinguished except for one, our definition of P is incoherent inside the modal qualifier because the bijection f does exist in all possible futures.  We cannot define an arbitrary counterpart relation that can be applied across time if things we want to make counterparts are existentially independent.

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