Quasi-quotation - How IT Works

How It Works

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:

1. Any lowercase Roman letter (with or without subscripts) is a wff of L.
2. If φ is a wff of L, then '~φ' is a wff of L.
3. Nothing else is a wff of L.

Interpreted literally, rule 2 does not express what is intended. For '~φ' (that is, the result of concatenating '~' and 'φ', in that order, from left to right) is not a wff of L, because the Greek letter 'φ' is used as a metavariable and thus cannot occur in wffs. In other words, our second rule says "If the sequence of symbols φ is a wff of L, then '~the sequence of symbols φ' is a wff of L. Because φ stands for a sequence of symbols instead of the proposition that the sequence might denote in the object language, φ isn't the kind of thing that can be negated. Rule one tells us that lowercase letters of the object language (such as 'p' and 'q') denote well-formed formulas, and thus our rule 2 needs to be changed so that φ indicates such a letter or sequence of symbols in the first instance, but is replaced by that letter or sequence of symbols in the second instance.

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 wff of L, then ┌~φ┐ is a wff of L.

The quasi-quotation marks '┌' and '┐' are interpreted as follows. Where 'φ' denotes a wff of L, '┌~φ┐' denotes the result of concatenating '~' and the 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 wff of L, then '~p' is a wff of L.

Similarly, we could not define a language with disjunction by adding this rule:

2.5. If φ and ψ are wffs of L, then '(φ v ψ)' is a wff of L.

But instead:

2.5'. If φ and ψ are wffs of L, then ┌(φ v ψ)┐ is a wff of L.

The quasi-quotation marks here are interpreted just the same. Where 'φ' and 'ψ' denote wffs of L, '┌(φ v ψ)┐' denotes the result of concatenating left parenthesis, the wff denoted by 'φ', space, 'v', space, the 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 wffs of L, then '(p v q)' is a wff of L.