A blog about belief

Definition of universality

In Philosophy on September 4, 2011 at 7:33 pm

It is not only in a Kantian system that universal claims play a role in ethics.  An ethical system that finds its roots in human nature could also involve statements that are universal with respect to humanity, and if we’re thinking that way we would do well to clarify what our notion of universality entails.  It isn’t enough to call a statement “universal” iff it begins with a universal quantifier, because any statement can massaged into that form: Px becomes ∀y (y = x → Py).  A similar type of manipulation would be saying something like, it is true of everyone that they like Jerry Lewis when they are of French ancestry.  Such a statement does, at the linguistic surface, say the same thing about everyone, but it would be facetious to call it a universal.  Statements would seem to qualify for universality only when their implications affect each member of the universal set in an in-some-way equitable fashion.  One way of formalizing this fuzzy definition would be to make universality about properties, not predicates, but I’d prefer to avoid bringing in the ontological baggage of properties.  The definition I have in mind is epistemic.  To call a statement universal is to say that, essentially, it can be falsified by new information that’s discovered about any one member of the universal set.  Formally, a statement ∀x Px is universal iff

∀x ⋄(¬Px ∧ ∀y(y ≠ x → Py))

I mean ⋄ to be interpreted in terms of branching-time semantics, with ⋄p meaning it is possible that, at some time in the future, p.  The formula is epistemic in the sense that it gives us ways of falsifying our original statement – specifically, it asserts that there are as many different possible ways of the statement being falsified as there are elements in the universal set, with the effect of excluding statements that, like the facetious example given above, don’t really say something about every one of those elements*.  An obvious way of falsifying the statement that everyone of French ancestry likes Jerry Lewis would be to find a French person who doesn’t like Jerry Lewis, but ∀x Px with P defined such that Px is true iff x is not such a person would only meet the first conjunct of the formula if it is possible for any given person x to be the eventual falsifying case.  Although it is reasonable to claim that it is possible for anyone to become a detractor of Jerry Lewis in the future, it is certainly not possible for any given person to become French by ancestry.  The second conjunct of the formula, ∀y(y ≠ x → Py), is meant to exclude the trickery of defining Px such that its truth value has nothing to do with x – defining it, for instance, as true for any x iff all people of French ancestry presently like Jerry Lewis.  The negation of that Px for any given x could possibly become true in the future, and then would trivially suffice to falsify the claim that everyone of French ancestry likes Jerry Lewis.  But ∀x Px with P defined in that way would not satisfy our formula because this P necessarily has the same truth value for every x at any given time.  Note that the formula excludes universality in cases where coordination of the members of the universal set is necessarily true.  Thus, if at any given moment it is necessary that either all grues are green or all grues are red, then we cannot say that the statement of their common redness at this moment is a universal statement.  The same would hold if it were necessary that all grues are red at all times, period.  Universality, as I’ve defined it, only applies when it is possible that things be otherwise.  This is a definition for a world without a priori truths.

*Actually, for full generality, an exception would have to be made for objects in their last moment of existence.  However, bringing in the machinery needed to formalize that exception would needlessly complicate this post.  We would also run into trouble if the universal set is infinite, but assuming we are talking about extant people this is not a problem.

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