A minimal baseline of expressive capability that a logical fragment or logic must possess to capture a specified class of phenomena or properties; the least set of syntactic constructs and semantic guarantees required for adequacy with respect to a target specification.

Principle

Floors are determined by the minimal closure properties and operators necessary to express essential distinctions (for example, the ability to state atomic facts, use Boolean combinations, or quantify existentially), and are sensitive to the model class and the intended properties to capture.

Demonstration

If one wishes a logic to characterize reachability in arbitrary directed graphs, the logic must at least provide a mechanism for transitive closure or an equivalent fixed-point operator; without such an operator no fragment of first-order logic can serve as the required baseline—this required capability is the expressivity floor for reachability.

Misapplication

Treating the floor as a numerical threshold or as implying that any logic meeting the floor will be efficient; a floor only asserts minimal expressive adequacy, not succinctness, decidability, or low computational cost.

Consequence

Identifying a floor clarifies design choices: any candidate logic that fails the floor cannot represent the target phenomena and must be extended, while satisfying the floor enables further trade-off analysis between expressivity, succinctness, and complexity.

Reversal

Removing constructs below the floor (for example forbidding Boolean combinations or existential quantification) yields a logic incapable of expressing the target class; conversely, enriching a logic above the floor increases capability but may incur complexity costs.

Boundary

A floor is relative to a chosen target specification and model class; it does not imply universality across tasks and excludes considerations such as algorithmic efficiency or practicality of inference in the logic that meets the floor.

Semantic Tension

There is tension between setting a low floor (which preserves simplicity and tractability) and raising the floor (which guarantees richer expressibility but risks complexity blow-up); floors interact with ceilings, since a floor that approaches the ceiling leaves little room for intermediate logics.

Synthesis

An expressivity floor specifies the minimal syntactic and semantic features a logic must have to meaningfully capture a class of properties: it is a design criterion ensuring adequacy for a task, which then guides whether to keep the logic minimal or to strengthen it knowing the trade-offs involved.