Squeezing argument and the plurality of informal notions

Kreisel's squeezing argument (1967) shows that there is an informal notion of validity which is irreducible with respect to both model-theoretic and proof-theoretic validity of First-Order Logic (FOL), but it is coextensive with both formal notions. Besides Kreisel's informal notion of validitu, it is possible to provide other informal notions which also are coextensive with those formal notions. We will argue that the existence of more than one informal notion squeezed by the formal definitions of validity of FOL shows that the definitions of first-order validity are underdeterminated with respect to informal notions. We will argue that these issues are not only found on FOL. As an example, we will the case of the intuitionist logic.