Wednesday, September 03, 2008

Logic Standing Firmly on its Own Two Shoulders

I have thought of an amusing and elegant variation of my grounded arithmetical logic (a variation on standard lambda logic, which I discussed here, here, and almost everywhere inbetween).

The logic supposedly gets its grounding from the possession of a black box that executes programs represented as lambda terms-- the idea being that they literally are programs, so cannot fail to be grounded  (despite the fact that no logical description of the concept of computation is complete, which makes it sound impossible to represent logically).

But, this idea is odd; it sounds as if the logic is getting grounding from an external black box that it is hooked to. What is needed is an embodiment of procedural-ness. But, there is a natural source of this from within the logic: deduction is a procedural task, which is itself Turing-complete; so, it can serve as a representation for arbitrary computations, just as well as lambda-reduction can.

So, the idea is to ground the logic in its own deduction system. This is not a radically new idea; provability logic is a well-established field of study. But, in any case, the idea is useful here if we don't want the logic's groundedness to rely on a lambda-evaluator that is outside of the logic itself.

This is satisfying for another reason, as well. I talked previously, somewhat, about the "reason" that my version of lambda logic is more semantically capable than plain first-order logic: it is able to represent concepts with a closed-world assumption in addition to the standard open-world assumption. What this means is, the logic can define a predicate by stating some rules that hold for the entity, and then adding "that's all". Statements that don't follow from the rules are automatically false. This concept offers an explanation for where unprovably true statements come from: the "that's all" operator has a meaning that cannot possibly be covered by an associated proof method. (Or, at least not one that terminates in finite time.) So it is the origin of much mathematical mystery.

Unfortunately, it cannot be the source of all mysteries; it allows us to talk about arithmetical truths, but not set-theoretical ones.

Anyway. The second nice feature of a logic grounded in its own deduction system is that it allows the "that's all" concept to be put directly to use. We can define a predicate by saying it is true if and only if it is provably true via a given set of statements.

This construction would have some interesting behavior. The first thing to realize is the all-encompassing reach of the closed-world assumption. The construction should not be used, for example, when defining some property of bicycles: if we discover a new bicycle, we will be unable to describe it with that property. It is better suited for defining properties of sets that can be completely pre-specified, like the natural numbers. We will not happen upon any new numbers, so we are safe. To apply predicates to bicycles while still using the closed-world construction, we would need to do something fancy such as define a class of all possible bicycle-types, define a closed-world predicate on that, and then apply the predicate indirectly to actual bicycles by classifying them as instances of the abstract bicycle-types.

The strength of the system could be increased even further by adding truth-predicates via one of the methods in this post, for example... but set theory would still not be grounded in the system.