A relation comparing two formal languages or formalisms in which one is said to be at least as expressive as the other when every property, class of structures, or predicate definable in the second can be defined or simulated in the first.

Principle

Expressiveness is measured by definability or faithful simulation: L1 is at least as expressive as L2 if there exists a systematic encoding or interpretation that maps each L2-definable property to an L1-definable property, typically yielding a pre-order (reflexive, transitive) on languages.

Demonstration

Propositional logic is strictly less expressive than first-order logic because there are first-order properties (e.g., “there exist two distinct elements”) that cannot be expressed propositionally; conversely, many modal logics are less expressive than the fragment of first-order logic obtained by the standard translation of modal formulas into first-order formulas over frames.

Misapplication

Treating relative expressiveness as equivalent to decidability, complexity, or succinctness; for example, concluding that because a language is more expressive it must be undecidable, or confusing expressiveness with the brevity of formulas.

Consequence

When one language is at least as expressive as another, meta-theorems, definability results, and some semantic distinctions can be transferred via the encoding; it guides choice of formalism for specification and analysis.

Reversal

The inverse relation identifies strict inexpressiveness: L1 is less expressive than L2 when there exist properties definable in L2 that no encoding into L1 can capture, demonstrating genuine qualitative gaps between languages.

Boundary

Applies to formal languages and formalisms with a clear notion of definability or satisfaction; it does not directly compare implementation performance, proof search complexity, or pragmatic succinctness unless these are explicitly part of the definability criterion.

Semantic Tension

Competes with notions such as decidability and succinctness: a more expressive language may sacrifice decidability or require longer formulas, so expressiveness must be balanced against algorithmic and pragmatic concerns.

Synthesis

Relative expressiveness is the preorder on formalisms induced by the existence of systematic encodings or simulations that preserve definability; it identifies when one language can faithfully capture the descriptive resources of another while noting trade-offs with decidability and complexity.