[[Semantics MOC]]
# Liar paradox
The **Liar paradox**[^ep], 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
>$(\lambda_{L})$ This sentence is a lie ^LL
but it is often more expedient[^con] to treat
> $(\lambda)$ This proposition is false ^L
[^ep]: Sometimes referred to as **Epimenides' paradox**, however the Cretan's statement “All Cretans are liars” is only paradoxical if one regards a liar as someone who never tells the truth.
[^con]: This Zettel follows the stipulations set forth in 1989, [[@barwiseLiarEssayTruth1989#Important terms and stipulative definitions|The Liar]]
## Variations and related sentences
See 1989, [[@barwiseLiarEssayTruth1989|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**
> $(\lambda_{S})$ This proposition is a lie or ‘gappy’ ^LS
### Truth-teller
The following sentence can be decided to be true or false without any apparent contradictions
> $(\tau)$ This proposition is true ^T
### Liar cycle
Taking bare self-reference to be the fundamental problem with [[#^L]] is shown to be erroneous by the liar cycle
> $(\alpha_{1})$ The proposition expressed by [[#^A2]] is true. ^A1
> $(\alpha_{2})$ The proposition expressed by [[#^An]] is true. ^A2
> $\vdots$
> $(\alpha_{n})$ The proposition expressed by [[#^B]] is true. ^An
> $(\beta)$ The proposition expressed by [[#^A1]] is false. ^B
### Contingent Liar
> $(\gamma)$ Max has the three of clubs and this proposition is false. ^C
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
> $(\alpha_{1}' )$ Max has the three of clubs. ^A1c
> $(\alpha_{2}')$ The proposition expressed by [[#^Bc]] is true. ^A2c
> $(\beta')$ At least one of the propositions expressed by [[#^A1c]] and [[#^A2c]] is false. ^Bc
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.
> $(\delta)$ If this proposition is true, then Max has the three of clubs. ^D
Using this proposition, one deduces via [[Modus ponens]] and [[Conditional proof]] that Max has the three of clubs.
> [!check]- Proof*
> Assume the antecedent of [[#^D]], i.e. that the proposition it expresses is true.
> Then we have [[#^D]] and its [[Hypothetical proposition|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.
> <span class="QED"/>
### Gupta's puzzle
[[Anil Gupta]] presented the following counterexample to a treatment of the Liar by [[Paul Kripke]]. Imagine two people $R$ and $P$ are making claims about a card game played between Max and Claire, where Claire has the ace of clubs.
Thus [[#^R1]] is false and [[#^P2]] is true.
$R$ claims
> $(\rho_{1})$ Max has the ace of clubs. ^R1
> $(\rho_{2})$ All the claims made by $P$ are true. ^R2
> $(\rho_{3})$ At least one of the claims made by $P$ is false. ^R3
and $P$ claims
> $(\pi_{1})$ Claire has the ace of clubs. ^P1
> $(\pi_{2})$ At most one of the claims made by $R$ is true. ^P2
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.
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#state/tidy | #lang/en | #SemBr