Formal models for concurrent and distributed systems describe machines; the people who operate them are either ignored or treated as external environment. Yet, key distributed systems -- notably grassroots platforms -- include people operating their personal machines (smartphones), and their faithful description must include the states of both people and machines and how they jointly effect system behaviour. Here, we propose volition-guarded multiagent atomic transactions -- executed atomically by machines and guarded by their people's volitions -- as a novel mathematical foundation for specifying systems consisting of people operating machines. Each agent's state consists of a volitional state and machine state; a transaction is enabled when the machine precondition holds and the guarding persons are willing. For example, befriending two people is guarded by both; unfriending, by either; voluntary swap of coins and bonds is guarded by both parties, while a payment is guarded by the payer. We develop the mathematical machinery to express safety and liveness of platforms specified in this framework, to implement one platform by another, and for an implementation to be resilient to faults; and provide example specifications of two grassroots platforms: social networks, and coins and bonds. These specifications are then used by AI to derive working implementations. We employ here a novel and simpler definition of `grassroots' that better captures the informal notion -- multiple instances can form and operate independently, yet may coalesce -- and show that the platforms specified here are grassroots under the new definition. We further introduce \emph{volitionally grassroots} protocols, in which two groups can become connected only by mutual consent -- the first transaction coupling them must be willed by a member of each -- and show that both platforms are volitionally grassroots.

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