Claims, Mirages and Horizons: One Criterion for the Liar and Continuum Hypothesis

I propose one criterion for which sentences make claims: a sentence makes a claim only if it bottoms out in things and relations. A sentence can fail in two ways. It can loop and never arrive, as the liar and Russell's "R contains R" do; these are mirages. Or it can wait on something that nothing supplies, as the continuum hypothesis does inside ZFC; when every consistent settling of the open facts would make it a claim, it is a horizon. Claims are true or false, with no third value. The base construction is Kripke's least fixed point under the strict (weak Kleene) scheme, with generalizations over what someone said ranging only over the sayings that are claims. Verdicts about non-claims are made one level up. The framework keeps the strict rule, yet gives everyday values to sentences such as "everything Jones says is true" when Jones also uttered a liar, where Kripke's and Feferman's systems leave gaps. The formal results cover the paradoxes built with truth and saying; Russell's paradox gets the same verdict by the same criterion, stated informally. I defend the framework as better in practice, not as the only possible one.