Quasi-quotation
Encyclopedia
Quasi-quotation is a linguistic device that facilitates rigorous and terse formulation of general rules about linguistic expressions while properly observing the use–mention distinction. It was introduced by the philosopher and logician Willard van Orman Quine
in his book Mathematical Logic, originally published in 1940. Put simply, quasi-quotation enables one to introduce variables that stand for a linguistic expression in a given instance and are used as that linguistic expression in a different instance.
For example, one can use quasi-quotation to illustrate an instance of substitutional quantification, like the following:
Quasi-quotation is used to indicate (usually in more complex formulas) that the φ and "φ" in this sentence are related things, that one is the iteration
of the other in a metalanguage
.
s (wffs) of a new formal language, L, with only a single logical operation, negation
, via the following recursive definition
:
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:
But instead:
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.
that range over things other than character strings
(e.g. number
s, 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:
The expanded version of this statement reads as follows:
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:
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.
Willard Van Orman Quine
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition...
in his book Mathematical Logic, originally published in 1940. Put simply, quasi-quotation enables one to introduce variables that stand for a linguistic expression in a given instance and are used as that linguistic expression in a different instance.
For example, one can use quasi-quotation to illustrate an instance of substitutional quantification, like the following:
-
- "Snow is white" is true if and only if snow is white.
- Therefore, there is some sequence of symbols that makes the following sentence true when every instance of φ is replaced by that sequence of symbols: "φ" is true if and only if φ.
Quasi-quotation is used to indicate (usually in more complex formulas) that the φ and "φ" in this sentence are related things, that one is the iteration
Iteration
Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.-Mathematics:Iteration in...
of the other in a metalanguage
Metalanguage
Broadly, any metalanguage is language or symbols used when language itself is being discussed or examined. In logic and linguistics, a metalanguage is a language used to make statements about statements in another language...
.
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 formulaWell-formed formula
In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
s (wffs) of a new formal language, L, with only a single logical operation, negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
, via the following recursive definition
Recursive definition
In mathematical logic and computer science, a recursive definition is used to define an object in terms of itself ....
:
- Any lowercase Roman letter (with or without subscripts) is a wff of L.
- If φ is a wff of L, then '~φ' is a wff of L.
- 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
Concatenation
In computer programming, string concatenation is the operation of joining two character strings end-to-end. For example, the strings "snow" and "ball" may be concatenated to give "snowball"...
'~' 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
Denotation
This word has distinct meanings in other fields: see denotation . For the opposite of Denotation see Connotation.*In logic, linguistics and semiotics, the denotation of a word or phrase is a part of its meaning; however, the part referred to varies by context:** In grammar and literary theory, the...
in the object language
Object language
An object language is a language which is the "object" of study in various fields including logic, linguistics, mathematics and theoretical computer science. The language being used to talk about an object language is called a metalanguage...
, φ isn't the kind of thing that can be negated. Rule one tells us that lowercase letters of the object language (such as
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
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
A caution
It does not make sense to quantify into quasi-quoted contexts using variablesVariable (programming)
In computer programming, a variable is a symbolic name given to some known or unknown quantity or information, for the purpose of allowing the name to be used independently of the information it represents...
that range over things other than character strings
String (computer science)
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....
(e.g. number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s, people
People
People is a plurality of human beings or other beings possessing enough qualities constituting personhood. It has two usages:* as the plural of person or a group of people People is a plurality of human beings or other beings possessing enough qualities constituting personhood. It has two usages:*...
, electrons). Suppose, for example, that one wants to express the idea that
- If φ is a natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
, 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
Category mistake
A category mistake, or category error, is a semantic or ontological error in which "things of one kind are presented as if they belonged to another", or, alternatively, a property is ascribed to a thing that could not possibly have that property...
, because a number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
is not the sort of thing that can be concatenated (though a numeral
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
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
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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.