Formalization of natural language

From statement to formula

The lesson on the language of propositional logic built a language without meaning: strings of circles and disks, a formation rule and some abbreviations. The lesson on induction on formulas proved that each formula is read in only one way. Now, that language was built for a purpose: to represent the form of the statements with which we reason in everyday life, in the sciences, in law and in mathematics, so that the correctness of a piece of reasoning can be studied in the form itself, independently of its subject matter. This lesson builds the bridge between the two worlds. It is not a mechanical bridge: natural language is rich, ambiguous and dependent on context, and the formal language is poor, precise and rigid. It follows that to formalize is to interpret, and that every formalization must be justified.

A statement, in the sense of this lesson, is a declarative sentence that, in its context, is true or false: “Water boils at a hundred degrees at sea level”, “Ann arrived late”, “$7$ is prime”. Questions, commands and wishes (“Did Ann arrive?”, “Close the door”, “If only it would rain”) are not statements. A statement is compound when it contains other statements joined by particles such as “not”, “and”, “or”, “if… then”; it is atomic when it cannot be decomposed in that way. Propositional logic does not analyse the inside of atomic statements: for it, “Ann arrived late” is an indivisible piece, which can only be true or false.

The letter convention

The formulas of the base language are strings of circles and disks; the $n$th variable is $v_n = \sigO^{n}\sigI$, that is, $n$ circles followed by a disk. Writing such strings is as safe as it is tiring, and for formalizing it is convenient to have short names. We adopt the following convention, which holds in this lesson and in the rest of the course unless expressly stated otherwise:

If so, the Latin letters $p$, $q$, $r$, $s$ and $t$ are names of five particular variables, in the same way that $7$ is the name of a particular number: $p$ does not vary, but always designates the string $\sigO\sigI$. The Greek letters $\varphi$, $\psi$, $\chi$ and $\theta$, on the other hand, remain metavariables, as in the lesson on the language: they designate arbitrary formulas, and a statement written with them, such as $(\varphi \downarrow \psi)$, is a schema that covers infinitely many formulas. A difference that matters follows from this: $(p \downarrow q)$ is a single formula, the string $\sigI\sigO\sigI\sigO\sigO\sigI$; $(\varphi \downarrow \psi)$ is a form of formulas, of which $(p \downarrow q)$, $(p \downarrow p)$ and $(q \downarrow (p \downarrow r))$ are instances. Some texts use $p$ and $q$ as metavariables; not here, because in formalizing we need two different letters to designate, with certainty, different variables, and a metavariable does not offer that guarantee: nothing prevents $\varphi$ and $\psi$ from designating the same formula.

The dictionary of a formalization

Formalizing a statement consists of three operations: identifying its atomic statements, assigning a variable to each one and rebuilding the compound statement with the connectives. The assignment made in the second operation is called the dictionary of the formalization, and it must always be declared, because without it the formula says nothing. The dictionary obeys two rules: the same atomic statement always receives the same variable, and different atomic statements receive different variables. The first guarantees that the formula records the repetitions in the text; the second, that it does not invent identities that the text does not assert.