1. The relationship between languages of formal logic and natural language is complex and delicate. Some of the formal languages such as the language of the classical predicate calculus or that of the Montagovian intensional logic are traditionally seen as useful tools for semantic analysis of natural language, while some others, like the languages of some many-valued or modal logics are quite detached from any natural means of communication. There is, however, a sense in which all logical languages worth the name must be anchored in the structures of natural language: formal systems whose languages lack this anchoring simply do not deserve, strictly speaking, the title logical. The point is that we can construct an abundance of formal systems that will employ "languages" furnished with something called "consequence" or "deduction system", or a kind of "semantics", but it is only to the extent that these artifacts reflect what we (in natural language) call consequence, inference or semantics that calling them thus becomes more than a deliberate fiat Logic has been conceived to help us recognize and sustain the proper modes of argumentation, reasoning, justifying or proving, which inevitably take place in natural language (where natural language is not meant to be opposed to such devices as the language of ordinary mathematics, which is also to a large extent natural, but to artificial languages, constituted entirely by means of definitions). Natural language is, we claim, the principal area where meaning(s) as such get constituted‘ and therefore it must be the relationship to natural language that is the measure of all logical things (of things which are logical, that they are, and of things which are not logical, that they are not). From this viewpoint, the assessment of the relationship between a given calculus and natural language is a crucial issue. We should have criteria for evaluating how fruitfully (if at all) we can use the calculus as a grid to be superimposed over natural language to explicate its inferential structure or its semantics...