Expressions in formal and natural languages have a lot in common. Both kinds of expression are finite combinations of symbols that are well-formed according to particular rules of syntax. Symbols and expressions in formal languages denote formal objects (numbers, sets, functions). Symbols and expressions in natural language denote mental objects (individuals, groups, properties). Sentences in a natural language can be likened to formulae in a formal one: both are expressions that make some assertion. When we assert something in natural language, it is usually about either an individual or a situation. We might be saying that one or more individuals have a particular property. Or we might be saying that a particular proposition holds in one or more situations.
The semantics both of formal and of natural languages is compositional. The meaning of a complex expression is systematically derived from the meanings of its constituents. Of particular interest is the interaction of functions and their arguments. This interaction is exemplified in mathematical expressions like sin(x) and linguistic expressions like “all dragons”. Functions (which we will call operators) map between things. x has some value; sin(x) may have a different value. “dragons” means one thing; “all dragons” means something else.
Possibly the simplest operator is the negative operator, realised in English as “not”. It maps from the meaning of one expression to its logical complement. Quantifiers are another kind of operator, and are of great use. They allow us to speak generally. We may quantify over individuals using quantificational determiners (some, all). Or, we may quantify over situations (actual or non-actual) using modal verbs (may, must), intensional verbs (think, want), and quantificational adverbs (sometimes, always).
Operators can only operate on material in their scope, which we can clarify with brackets. For instance:
All dragons (breathe fire)
The meaning of “all dragons breathe fire” depends only on the meanings of “all dragons” and of “breathe fire”.
A mathematical example may help to clarify. Below, the variable x is in the scope of the sin operator and the variable y is in the scope of the cos operator:
sin(x) + cos(y)
The value of sin(x) depends only on the sine function and the value of x. Likewise, the value of cos(y) depends only on the cosine function and the value of y.