The Liar Paradox
The Liar paradox is a statement such as:
This sentence is false.
The reasoning that makes this paradoxical is this:
If ‘This sentence is false.’ is true, then since it is stating that the sentence is false, if it is actually true that would mean that it is false, and so on.
Or, if ‘This sentence is false.’ is false, then since the sentence is stating that the sentence is false, if it is actually false that would mean that it is true, and so on.
If we assume that the ‘this’ does actually refer to the sentence it occurs in, if we try to substitute ‘this sentence’ by ‘This sentence is false’, we simply get:
‘This sentence is false’ is false.
and we can repeat the process, giving:
‘ ‘This sentence is false’ is false’ is false.
and so on, and there is no end to this process.
Because of this, the Liar paradox, as well as a host of similar paradoxes are commonly called ‘self-referencing’ or ‘circular’ paradoxes.
Analysis of the Liar Paradox
To see what is happening in the Liar paradox, it may be easier to first consider the following sentence:
This sentence has twelve letters.
Is the above sentence correct or incorrect? Because of the ambiguity of English, there is no definitive answer. On the one hand you could consider that the sentence is stating that “This sentence” is an object that contains twelve letters - in which case you would say that the sentence is correct. On the other hand, you could consider that “This sentence” is a name that refers to the expression “This sentence has twelve letters.” - and conclude that the sentence is incorrect.
Note that in either case, the reference is to an object, and the expression makes a statement, ‘has twelve letters’, regarding that object. The first case, where the object is “This sentence”, is straightforward. In the second case, however, “This sentence” is a name - but the thing that it refers to is also an object. The result is that the overall expression, “This sentence has twelve letters.” is at the same time an object in English, and also valid syntax of English.
Natural languages such as English have this characteristic that allows such self-referential sentences. In terms of a meta-language and a sub-language, English sentences such as the one above can be considered to be a mixture of two levels of language which cannot be separated into meta-language and sub-language, and because of that, they do not form logically valid propositions.
Bearing the above in mind, then for the case of the Liar paradox,
This sentence is false.
if we reject the possibility that the sentence is stating that the actual combination of symbols that is “This sentence” is false, that the only other possibility is that “This sentence” is a name, and the object that it refers to is “This sentence is false.”, so that “This sentence is false.” is at the same time an object in English, and also valid syntax of English - which means that the expression is not a logically valid proposition.
Various authors have suggested ‘solutions’ to such paradoxes, but the simple answer is that there is no solution to such paradoxes as long as a natural language such as English is being used - since the paradox arises from the ambiguity of natural language that allows symbol strings to be at the same time, objects of the language, and valid syntax of the language. Note that quotation marks (as used above) are often used as delimiters in an attempt to clarify when we are referring to a symbol string as an object, but they do not prevent the ambiguity that results from the mixture of meta-language and sub-language that occur in natural languages such as English. That ambiguity is the price we pay for the flexibility of natural language. An example of a failed attempt to construct a logically valid version of the lair paradox by using quotation marks can be seen on the web-page Quine’s Paradox.
Self-Reference and Incompleteness Proofs
In numerous incompleteness proofs, you will find the notion that a language A whose only objects are numbers is allegedly referring to its own formulas when it refers to numbers - because a meta-language has assigned numbers as names for the formulas of that language A. Of course, this notion is nonsensical, since the language A has no information regarding these names, since that information is not within that language A, but within a meta-language. Despite this obvious obstacle, such incompleteness proofs are accepted as valid proofs, although logical analysis always reveals some flaw in the proof. See Errors in incompleteness proofs and Analysis of incompleteness proofs.