Abstract
This paper studies the meanings and uses of two universal quantifiers in French, tout and chaque. It proposes a semantics based on proofs, accounting for the correlation between the use of tout and prescriptivity and the use of chaque and descriptivity. From the perspective of proof theory, prescriptive statements correspond to universal quantification rules of deductive statements already present in the work of Aristotle: if a property is settled for a variable of which nothing special holds, then it holds for any individual. The assertability of a descriptive statement, instead, relies on a precise knowledge of the domain of quantification and the verification for each of the individuals in the domain of the truth of the statements. The universal quantification thus relies on conjunction. The quantification via tout expresses a rule whose validity implies the validity of the same rule expressed by a chaque-statement, as the validity of the descriptive statements is a consequence of the validity of the prescriptive one. The contrary does not hold. This explains that tout can be replaced by chaque, but chaque cannot be replaced by tout.