Our topic is one that involves the interface between natural language and mathematical logic. First-order predicate language/logic does a good job approximating many parts of (English) speech, i.e., nouns, verbs and prepositions, but fails decidedly when it comes to, say, adjectives. In particular, it cannot account for the quite different ways in which the adjectives green and big modify a noun such as chair. In the former case, we can easily view a world in which the class of green chairs is the intersection of the class of green things with the class of chair-things. By contrast, the way big modifies a noun depends on the noun itself: a big chair is microscopic when compared to the smallest of galaxies. We investigate logical languages inspired by this phenomenon; particularly those with variables ranging over individuals and with variable-binding operators akin to generalized quantifiers

We discuss simple subtheories of Peano arithmetic over languages which include the monus function. The system $\mathrm{ZDL}$ corresponds with $\mathrm{PA}^-$. The choice of language permits our theories to have special universal axiomatizations; their classes of models have corresponding model theoretic properties.

Peirce's systems of existential graphs (Alpha: classical propositional calculus; Beta: first-order purely relational logic; Gamma: modal calculi and second-order logic) are presented both from an historical perspective and succinctly for the modern reader. Peirce's alternative continuum, with its main non-Cantorian properties (genericity, reflexivity, modality), is also presented both historically and synthetically. The blend of Peirce's existential graphs and his non-Cantorian continuum gives rise to a thoroughly original logical approach to the "labyrinth of the continuum". We explain why such an approach was set aside in the main developments of logic in the $\,\hbox{\tiny XX}^{\hbox{\tiny th}}\,$ century, and we hint to a possible renewal of interest for Peirce's continuity logic from the viewpoint of contemporary developments in category theory and geometric logic.