For example, one can use quasi-quotation to illustrate an instance of substitutional quantification, like the following:
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:
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:
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.
The expanded version of this statement reads as follows:
The proper way to state the principle is:
It is tempting to characterize quasi-quotation as a device that allows quantification into quoted contexts, but this is incorrect: quantifying into quoted contexts is always illegitimate. Rather, quasi-quotation is just a convenient shortcut for formulating ordinary quantified expressions—the kind that can be expressed in first-order logic.
As long as these considerations are taken into account, it is perfectly harmless to "abuse" the corner quote notation and simply use it whenever something like quotation is necessary but ordinary quotation is clearly not appropriate.