Liar paradox
The Liar paradox1, in its various formulations, gives an example of a sentence which fails to be either true or false. In it’s most basic form it reads
This sentence is a lie
but it is often more expedient2 to treat
This proposition is false
Variations and related sentences
See 1989, The Liar, pp. 20ff. for a “budget of Liar-like paradoxes”.
Strengthened liar
If one attempts to resolve the Liar by rejecting the Law of excluded middle and claiming ^L is neither true nor false, but rather ‘gappy’, then a similar paradox presents itself in the strengthened Liar
This proposition is a lie or ‘gappy’
Truth-teller
The following sentence can be decided to be true or false without any apparent contradictions
This proposition is true
Liar cycle
Taking bare self-reference to be the fundamental problem with ^L is shown to be erroneous by the liar cycle
The proposition expressed by ^A2 is true.
The proposition expressed by ^An is true.
The proposition expressed by ^B is true.
The proposition expressed by ^A1 is false.
Contingent Liar
Max has the three of clubs and this proposition is false.
Intuitively, if Max does not have the three of clubs then ^C is simply false, otherwise we get the same situation as ^L.
Contingent Liar cycle
Combining the Contingent Liar and Liar cycle we get
Max has the three of clubs.
The proposition expressed by ^Bc is true.
At least one of the propositions expressed by ^A1c and ^A2c is false.
If Max does not have the three of clubs, then ^A1c is false and ^A2c and ^Bc are true, but if Max does have three three of clubs there is no unproblematic true/false assigment.
Löb’s paradox
The following, sometimes called Curry’s paradox, is closely related to Löb’s theorem in proof theory.
If this proposition is true, then Max has the three of clubs.
Using this proposition, one deduces via Modus ponens and Conditional proof that Max has the three of clubs.
Proof*
Assume the antecedent of ^D, i.e. that the proposition it expresses is true. Then we have ^D and its antecedent, whence Max has the three of clubs by Modus ponens. Therefore by Conditional proof, if ^D is true, then Max has the three of clubs, which is precisely what ^D claims. Thus ^D is true, and once again applying Modus ponens Max has the three of clubs.
Gupta’s puzzle
Anil Gupta presented the following counterexample to a treatment of the Liar by Paul Kripke. Imagine two people
Max has the ace of clubs.
All the claims made by are true.
At least one of the claims made by is false.
and
Claire has the ace of clubs.
At most one of the claims made by is true.
The natural way to reason here is as follows. Since ^R2 and ^R3 contradict each other, at most one of these can be true. Since the claim made by ^R1 is false, ^P2 is true. Therefore ^P2 expresses a truth while ^R3 does not.