Quasi-quotation is particularly useful for stating formation rules for
formal languages. Suppose, for example, that one wants to define the
well-formed formulas (wffs) of a new formal language,
L, with only a single logical operation,
negation, via the following
recursive definition: • Any lowercase
Roman letter (with or without subscripts) is a well-formed formula (wff) of
L. • If φ is a well-formed formula (wff) of
L, then '~φ' is a well-formed formula (wff) of
L. • Nothing else is a well-formed formula (wff) of
L. Interpreted literally, rule 2 does not express what is apparently intended. For '~φ' (that is, the result of
concatenating '~' and 'φ', in that order, from left to right) is not a well-formed formula (wff) of
L, because no
Greek letter can occur in well-formed formulas (wffs), according to the apparently intended meaning of the rules. In other words, our second rule says "If some sequence of symbols φ (for example, the sequence of 3 symbols φ = '~~
p') is a well-formed formula (wff) of
L, then the sequence of 2 symbols '~φ' is a well-formed formula (wff) of
L". Rule 2 needs to be changed so that the second occurrence of 'φ' (in quotes) be not taken literally. Quasi-quotation is introduced as shorthand to capture the fact that what the formula expresses isn't precisely quotation, but instead something about the concatenation of symbols. Our replacement for rule 2 using quasi-quotation looks like this: :2'. If φ is a well-formed formula (wff) of
L, then ⌜~φ⌝ is a well-formed formula (wff) of
L. The quasi-quotation marks '⌜' and '⌝' are interpreted as follows. Where 'φ' denotes a well-formed formula (wff) of
L, '⌜~φ⌝' denotes the result of concatenating '~' and
the well-formed formula (wff) denoted by 'φ' (in that order, from left to right). Thus rule 2' (unlike rule 2)
entails, e.g., that if
p is a well-formed formula (wff) of
L, then '~
p' is a well-formed formula (wff) of
L. Similarly, we could not define a language with
disjunction by adding this rule: :2.5. If φ and ψ are well-formed formulas (wffs) of
L, then '(φ v ψ)' is a well-formed formula (wff) of
L. But instead: :2.5'. If φ and ψ are well-formed formulas (wffs) of
L, then ⌜(φ v ψ)⌝ is a well-formed formula (wff) of
L. The quasi-quotation marks here are interpreted just the same. Where 'φ' and 'ψ' denote well-formed formulas (wffs) of
L, '⌜(φ v ψ)⌝' denotes the result of concatenating left parenthesis, the well-formed formula (wff) denoted by 'φ', space, 'v', space, the well-formed formula (wff) denoted by 'ψ', and right parenthesis (in that order, from left to right). Just as before, rule 2.5' (unlike rule 2.5) entails, e.g., that if
p and
q are well-formed formulas (wffs) of
L, then '(
p v
q)' is a well-formed formula (wff) of
L. == Scope issues ==