Syntax

We have already visited, informally, the syntax of propositional languages: we did so in Chapter 1: Preliminaries. Here we need to be a bit more precise; we need to specify the alphabets of such languages, as well as the grammar that is used to build formulas from these alphabets. The alphabet of a propositional language is composed of a part that is allowed to vary between langauges, and a part that is invariant. The part that varies is a set of propositional variables; this set is a subset of

$$\{P_1, P_2, P_3, P_4, \ldots\}$$

By convention, we sometimes refer to the first three symbols in this set as P, Q, and R. The alphabet also contains the five truth-functional connectives we briefly discussed in Chapter 1, namely, $\neg, \vee, \wedge, \rightarrow, \leftrightarrow$.

Also, to ease the symbolization of English sentences and phrases in the propositional calculus, we allow propositional variables to be written in helpful ways. For example, in order to represent the sentence

$$\mbox{William is a crook}$$

we might represent this sentence with the propositional variable W_c. W_c would then be equivalent to some P_i, but we might not bother to inquire as to what number the subscript takes on.

The grammar for propositional languages is composed of the following rules:

• All propositional variables P_i are wffs (well-formed formulas).
• If $\phi$ is a wff, then $\neg\phi$ is a wff.
• If $\phi$ and $\psi$ are wffs, then $\phi \star \psi$ is a wff as well, where $\star = \vee, \wedge, \rightarrow, \leftrightarrow$.

We shall refer to $\cal L$$_{PC}^\infty$ as the propositional language composed of all those wffs that can be obtained by this grammar from an alphabet that includes all the propositional variables P_i.