Monday, July 5, 2021

Disjunctive predicates

I have found myself thinking these two thoughts, on different occasions, without ever noticing that they appear contradictory:

  1. Other things being equal, a disjunctive predicate is less natural than a conjunctive one.

  2. A predicate is natural to the extent that its expression in terms of perfectly natural predicates is shorter. (David Lewis)

For by (2), the predicates “has spin or mass” and “has spin and mass” are equally natural, but by (1) the disjunctive one is less natural.

There is a way out of this. In (2), we can specify that the expression is supposed to be done in terms of perfectly natural predicates and perfectly natural logical symbols. And then we can hypothesize that disjunction is defined in terms of conjunction (p ∨ q iff ∼(∼p ∧ ∼q)). Then “has spin or mass” will have the naturalness of “doesn’t have both non-spin and non-mass”, which will indeed be less natural than “has spin and mass” by (2) with the suggested modification.

Interestingly, this doesn’t quite solve the problem. For any two predicates whose expression in terms of perfectly natural predicates and perfectly natural logical symbols is countably infinite will be equally natural by the modified version of (2). And thus a countably infinite disjunction of perfectly natural predicates will be equally natural as a countably infinite conjunction of perfectly natural predicates, thereby contradicting (1) (the De Morgan expansion of the disjunctions will not change the kind of infinity we have).

Perhaps, though, we shouldn’t worry about infinite predicates too much. Maybe the real problem with the above is the question of how we are to figure out which logical symbols are perfectly natural. In truth-functional logic, is it conjunction and negation, is it negation and material conditional, is it nand, is it nor, or is it some weird 7-ary connective? My intuition goes with conjunction and negation, but I think my grounds for that are weak.

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