3  wffs in SL

The rules for wffs in SL, or for the construction of a wff\(_{\text{SL}}\), have three components:

• a rule defining an atomic formula in SL as a wff\(_{\text{SL}}\)
• a set of rules that recursively define compound well-formedness in relation to well-formed parts, i.e., in relation to one or more wffs\(_{\text{SL}}\)
• a rule stipulating no other ‘formulas’ are wffs\(_{\text{SL}}\)

Collectively, these rules will tell us how to use each of our three symbol types from the SL alphabet:

• upper-case letters,
• logical connectives, and
• parentheses and brackets.

3.1 Atomic formulas in SL

We begin by defining atomic formulas in SL:

An upper-case letter is an atomic formula in SL

And, as our first rule, we allow that such formulas are well-formed:

1. An atomic formula in SL is a wff\(_{\text{SL}}\)

3.2 Compound formulas in SL

We now recursively define well-formedness for logical connectives in SL, and, by implication, for parentheses.

2. If \(\phi\) is a wff\(_{\text{SL}}\), \(\neg \phi\) is a wff\(_{\text{SL}}\)
3. If \(\phi\) is a wff\(_{\text{SL}}\) and \(\psi\) is a wff\(_{\text{SL}}\), \((\phi \vee \psi)\) is a wff\(_{\text{SL}}\)
4. If \(\phi\) is a wff\(_{\text{SL}}\) and \(\psi\) is a wff\(_{\text{SL}}\), \((\phi \wedge \psi)\) is a wff\(_{\text{SL}}\)
5. If \(\phi\) is a wff\(_{\text{SL}}\) and \(\psi\) is a wff\(_{\text{SL}}\), \((\phi \to \psi)\) is a wff\(_{\text{SL}}\)
6. If \(\phi\) is a wff\(_{\text{SL}}\) and \(\psi\) is a wff\(_{\text{SL}}\), \((\phi \leftrightarrow \psi)\) is a wff\(_{\text{SL}}\)

3.3 Final rule

And, finally, we stipulate that no other ‘formulas’ (that is, strings of symbols in the alphabet of SL) are wffs\({_\text{SL}}\).

7. No other formula is a wffs\({_\text{SL}}\)

3.4 Complete set

1. An atomic formula in SL is a wff\(_{\text{SL}}\)
2. If \(\phi\) is a wff\(_{\text{SL}}\), \(\neg \phi\) is a wff\(_{\text{SL}}\)
3. If \(\phi\) is a wff\(_{\text{SL}}\) and \(\psi\) is a wff\(_{\text{SL}}\), \((\phi \vee \psi)\) is a wff\(_{\text{SL}}\)
4. If \(\phi\) is a wff\(_{\text{SL}}\) and \(\psi\) is a wff\(_{\text{SL}}\), \((\phi \wedge \psi)\) is a wff\(_{\text{SL}}\)
5. If \(\phi\) is a wff\(_{\text{SL}}\) and \(\psi\) is a wff\(_{\text{SL}}\), \((\phi \to \psi)\) is a wff\(_{\text{SL}}\)
6. If \(\phi\) is a wff\(_{\text{SL}}\) and \(\psi\) is a wff\(_{\text{SL}}\), \((\phi \leftrightarrow \psi)\) is a wff\(_{\text{SL}}\)
7. No other formula is a wff\({_\text{SL}}\)