A Dual-Threshold Probabilistic Knowing Value Logic

We introduce a dual-threshold probabilistic knowing value logic for uncertain multi-agent settings. The framework captures within a single formalism both probabilistic-threshold attitudes toward propositions and high-confidence attitudes toward term values, thereby connecting probabilistic epistemic logic with classical knowing value logic. It is especially motivated by privacy-sensitive scenarios in which an attacker assigns high posterior probability to a candidate sensitive value without guaranteeing that it is the true one. The main idea is to separate the threshold domains of propositional and value-oriented operators. While $K_i^\theta$ ranges over the full rational threshold interval, the knowing-value operator $Kv_i^\eta(t)$ is restricted to $(\frac{1}{2},1]$. This high-threshold restriction has a structural effect: once $\eta>\frac{1}{2}$, two distinct values cannot both satisfy the threshold, so uniqueness becomes automatic. Over probabilistic models with countably additive measures, $Kv_i^\eta(t)$ is interpreted as non-factive high-confidence value locking. We establish sound axiomatic systems for the framework and develop a two-layer construction based on type-space distributions and assignment-configuration mappings. This resolves the joint realization problem arising from probabilistic mass allocation and value-sensitive constraints, and yields a structured weak-completeness theorem for the high-threshold fragment.