What Smith means by the syntax of propositional logic are
- the basic symbols of the language and
- how those symbols can be combined to make sentences of the language.
These sentences and only these sentences are considered well formed formulas (wff).
On the top half of page 41 he defines the syntax.
The basic symbols of propositional logic (PL) are
- basic propositions
- punctuation symbols (parentheses)
These basic symbols can be combined to make sentences (wff). He has three rules for how to make wff from the basic symbols:
- Basic propositions are by themselves wff. This is the "base clause" of this recursive definition.
- Propositions (basic or compound) with connectives used in the ways specified are wff. This is the "recursive clause" of the recursive definition.
- Nothing else is a wff.
The above definition is called "recursive" because if you are given a wff with connectives you can break it apart examining each of the wff connected by the connectives. Each of these wff may have connectives which can be broken down. This process continues until you reach the basic propositions without any connective (i.e. is recursive). At this point you can stop.
Smith, N. J. (2012). Logic: The laws of truth. Princeton University Press.