## Sunday, January 29, 2012

### Grounding

It is normal to think that a disjunctive proposition that is true is grounded in each of its true disjuncts.

This may be false. Let p be the proposition that 2+2=4. Let q be the infinite disjunction p or (p or (p or ...)). Then q is its own second disjunct. Moreover, q is true. But surely what q is grounded in is not q itself but p.

For the same reason, it does not appear correct to say that a conjunction is always partly grounded in at least one of its conjuncts. For instance, take the infinite conjunction: p and (p and (p and ...)). The second conjunct is the conjunction itself, but it does not seem that this conjunction is even partly grounded in itself.

When I came up with the disjunction example today, I thought that I could weaken the disjunctive grounding principle to say that a disjunction is grounded in at least one of its disjuncts. But even that is not clear to me right now. For perhaps we could construct a complex infinite disjunction such that each disjunct is the disjunction itself. But I am less sure that such a disjunction really does exist.

On reflection, I wonder if my examples work. Maybe there is no disjunctive proposition p or (p or (p or ...)), but only the disjunctive proposition ...(((p or p) or p) or p).... In other words, the direction of the infinite nesting may matter. The difference is that the latter disjunctive proposition has a starting point: we take p and disjoin p to it infinitely often. The former one does not.

Another interesting case is: "2+2=4 or the proposition expressed by this sentence is true."