Quasi-quotation or Quine quotation is a linguistic device in

formal languages In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sy ...

that facilitates rigorous and terse formulation of general rules about linguistic expressions while properly observing the

use–mention distinction The use–mention distinction is a foundational concept of analytic philosophy, according to which it is necessary to make a distinction between a word (or phrase) and it.Devitt and Sterelny (1999) pp. 40–1W.V. Quine (1940) p. 24 Many philos ...

. It was introduced by the philosopher and

logician Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

Willard Van Orman Quine Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century" ...

in his book ''Mathematical Logic'', originally published in 1940. Put simply, quasi-quotation enables one to introduce symbols 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 is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

of the other in a metalanguage. Quine introduced quasiquotes because he wished to avoid the use of variables, and work only with closed sentences (expressions not containing any free variables). However, he still needed to be able to talk about sentences with arbitrary

predicates Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, ...

in them, and thus, the quasiquotes provided the mechanism to make such statements. Quine had hoped that, by avoiding variables and schemata, he would minimize confusion for the readers, as well as staying closer to the language that mathematicians actually use. Quasi-quotation is sometimes denoted using the symbols ⌜ and ⌝ (unicode U+231C, U+231D), or double square brackets, ⟦ ⟧ ("Oxford brackets"), instead of ordinary quotation marks. Scott, D. and Strachey, C.: 1971, Toward a mathematical semantics for computer languages, Oxford University Computing Laboratory, Programming Research Group.

How it works

Quasi-quotation is particularly useful for stating formation rules for

. Suppose, for example, that one wants to define the

well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can ...

s (wffs) of a new formal language, ''L'', with only a single logical operation, negation, via the following

recursive definition In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set ( Aczel 1977:740ff). Some examples of recursively-definable objects include facto ...

: # Any lowercase

Roman letter The Latin script, also known as Roman script, is an alphabetic writing system based on the letters of the classical Latin alphabet, derived from a form of the Greek alphabet which was in use in the ancient Greek city of Cumae, in southern Italy ...

(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 In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenati ...

'~' and 'φ', in that order, from left to right) is not a well-formed formula (wff) of ''L'', because no

Greek letter The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...

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 In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...

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

It does not make sense to quantify into quasi-quoted contexts using variables that range over things other than character strings (e.g.

number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...

people A person ( : people) is a being that has certain capacities or attributes such as reason, morality, consciousness or self-consciousness, and being a part of a culturally established form of social relations such as kinship, ownership of prope ...

electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...

). 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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...

, then ⌜''s''(''φ'')⌝ denotes the successor of ''φ''. Suppose, for example, ''φ'' = 7. What is ⌜''s''(''φ'')⌝ in this case? The following tentative interpretations would all be equally absurd: # ⌜''s''(''φ'')⌝ = 's(7)', # ⌜''s''(''φ'')⌝ = 's(111)' (in the binary system, '111' denotes the integer 7), # ⌜''s''(''φ'')⌝ = 's(VII)', # ⌜''s''(''φ'')⌝ = 's(seven)', # ⌜''s''(''φ'')⌝ = 's(семь)' ('семь' means 'seven' in Russian), # ⌜''s''(''φ'')⌝ = 's(the number of days in one week)'. On the other hand, if ''φ'' = '7', then ⌜''s''(''φ'')⌝ = 's(7)', and if ''φ'' = 'seven', then ⌜''s''(''φ'')⌝ = 's(seven)'. 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 A category mistake, or category error, or categorical mistake, or mistake of category, is a semantic or ontological error in which things belonging to a particular category are presented as if they belong to a different category, or, alternativ ...

, because a

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 Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...

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—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

. 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.

References

Notes

Bibliography

*{{cite book , last=Quine , first=W. V. , title=Mathematical Logic , orig-year=1940 , edition=Revised , year=2003 , publisher=Harvard University Press , location=Cambridge, MA , isbn=0-674-55451-5

External links