Circular Logic

Count the number of cards in your graveyard as Circular Logic resolves to determine how much the controller of the target spell has to pay. Circular Logic is still on the stack at that time. It doesn't count itself.

A spell cast for its madness cost is put onto the stack like any other spell. It can be countered, copied, and so on. As it resolves, it's put onto the battlefield if it's a permanent card or into its owner's graveyard if it's an instant or sorcery card.

If you choose not to cast a card with madness when the madness triggered ability resolves, it's put into your graveyard. Madness doesn't give you another chance to cast it later.