Propositional calculus
Syntax

Given a set of atomic propositional variables p 1 {\displaystyle p_{1}}, p 2 {\displaystyle p_{2}}, p 3 {\displaystyle p_{3}},..., and a set of propositional connectives c 1 1 {\displaystyle c_{1}^{1}}, c 2 1 {\displaystyle c_{2}^{1}}, c 3 1 {\displaystyle c_{3}^{1}},..., c 1 2 {\displaystyle c_{1}^{2}}, c 2 2 {\displaystyle c_{2}^{2}}, c 3 2 {\displaystyle c_{3}^{2}},..., c 1 3 {\displaystyle c_{1}^{3}}, c 2 3 {\displaystyle c_{2}^{3}}, c 3 3 {\displaystyle c_{3}^{3}},..., a formula of propositional logic is defined recursively by these definitions:

Writing the result of applying c n m {\displaystyle c_{n}^{m}} to ⟨ {\displaystyle \langle } A, B, C, … ⟩ {\displaystyle \rangle } in functional notation, as c n m {\displaystyle c_{n}^{m}} (A, B, C, …), we have the following as examples of well-formed formulas:

What was given as Definition 2 above, which is responsible for the composition of formulas, is referred to by Colin Howson as the principle of composition. It is this recursion in the definition of a language's syntax which justifies the use of the word "atomic" to refer to propositional variables, since all formulas in the language L {\displaystyle {\mathcal {L}}} are built up from the atoms as ultimate building blocks. Composite formulas (all formulas besides atoms) are called molecules, or molecular sentences. (This is an imperfect analogy with chemistry, since a chemical molecule may sometimes have only one atom, as in monatomic gases.)

The definition that "nothing else is a formula", given above as Definition 3, excludes any formula from the language which is not specifically required by the other definitions in the syntax. In particular, it excludes infinitely long formulas from being well-formed.
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