Quasi-quotation - A Caution

A Caution

It does not make sense to quantify into quasi-quoted contexts using variables that range over things other than character strings (e.g. numbers, people, electrons). Suppose, for example, that one wants to express the idea that 's(0)' denotes the successor of 0, 's(1)' denotes the successor of 1, etc. One might be tempted to say:

• If φ is a natural number, then ┌s(φ)┐ denotes the successor of φ.

The expanded version of this statement reads as follows:

• If φ is a natural number, then the result of concatenating 's', left parenthesis, φ, and right parenthesis (in that order, from left to right) denotes the successor of φ.

This is a category mistake, because a number is not the sort of thing that can be concatenated (though a numeral is).

The proper way to state the principle is:

• If φ is an Arabic numeral that denotes a natural number, then ┌s(φ)┐ denotes the successor of the number denoted by φ.

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.