A Pure Definition of Time and Its Three Logical Arrows

This paper proposes a purely logical and reductive definition of time grounded in the asymmetric nature of necessary conditions. Starting from minimal primitives—events (as changes of state), negation, and conjunction—we define an asymmetric necessary-condition relation (termed ``cause'' with a precise stipulative meaning). Time is identified with the partial order induced by this relation. Three Logical Arrows of Time are derived: 1. First Logical Arrow: Mutual necessity between distinct events is logically impossible, forcing strict asymmetry in the necessary-condition relation and providing an intrinsic direction for time. 2. Second Logical Arrow: The conjunction of all causes of an event B constitutes its minimal sufficient condition—any other sufficient set must contain (or entail) at least this set—and since no subsequent event shares an identical minimal sufficient past, the causal past grows strictly with time. 3. Third Logical Arrow: In a dynamic world W = O + E (O a finite-capacity embedded observer, E the environment), new events in E force O to incorporate novel details. This unbounded accumulation exceeds O's capacity (a consequence of O being a proper subsystem), compelling irreversible loss of earlier microstate information and yielding increasing observer-relative effective entropy. These arrows unify time's structure, direction, causal completeness, and thermodynamic asymmetry in a nearly analytic framework.