I saw someone react to the Liar Paradox on YouTube so I'd thought I'd share some old thoughts of my own that form something of a first impression or first pass at the problem. I haven't read much on this topic, so I still have much to learn.
Part 1: Start with the Barber Paradox
A barber shaves all and only those people in the town that don't shave themselves. Question: Does the barber shave himself? If the barber shaves himself, then the 'only' part of the rule is violated, because the barber only shaves those people who don't shave themselves. If the barber does not shave himself, then the 'all' part of the rule is violated, because the barber shaves all people who don't shave themselves. So either way, the rule will be violated. The rule is impossible.
If we force the rule to go through, then the barber does and does not shave himself. Contradiction. Though there is another option: Keep the rule, but it applies to everyone except the barber. So we end up with the following inconsistent set:
1) Self-reference;
2) The Rule;
3) The Law of Non-Contradiction.
You can only keep two out of these three. One of them has to go.
Option A: If you keep 2 and 3, then self-reference is impossible.
Option B: If you keep 1 and 3, then the rule is impossible.
Option C: If you keep 1 and 2, then there are true contradictions.
Part 2: Option C doesn't work
I have the following issue with Option C: You can verify that it doesn't work! At the end of the day, the barber will have either shaved himself or not. That's undeniable. So no matter what, Option C doesn't work, at least not in this case. But whatever necessary truths there are that would ground (or constitute) the necessary truth that Option C doesn't work, I don't see why those same necessary truths wouldn't apply in parallel cases.
So let's consider two parallel cases: Set Theory and Propositions. These are known as Russell Paradoxes after Bertrand Russell.
Part 3: Russell Paradoxes
Set Paradox: There is a set that contains all and only those sets that don't contain themselves.
This is why it's nice to start with a concrete example like the barber paradox. With the set paradox, it's harder to see that Option C must be rejected. But if we can and must reject Option C in the barber case, I don't see why we shouldn't do the same in this case. So again, either the set does not self-refer OR the set's rule is not true because it's impossible. Just as there can be no self-referenced barber that satisfies the barber rule, for the exact same reason there can be no self-referenced set that satisfies the set rule. The barber is impossible, and so is the set. What's good for the goose is good for the gander.
Proposition Paradox: There is a proposition about all and only those propositions that aren't about themselves.
Same deal. Thanks to the barber paradox we know that Option C doesn't work. And there are no relevant differences between the barber paradox and the set or proposition paradoxes. At least, this is true if we
discover the principle behind the impossibility of Option C for the
barber case and see that it applies just as well to these parallel
cases.
Part 4: The principle behind the impossibility of Option C
The question now is, why is it the case that Option C is impossible? I can see that it is the case that Option C is impossible, because I can see that at the end of the day the barber will have either shaved himself or not, which means that either the rule has been violated or that the rule doesn't apply to the barber. But why is it that I can see such a thing? What makes Option C impossible?
Here's one explanation: Option C is impossible because true contradictions are impossible; the Law of Non-Contradiction is necessarily true.
Part 5: Option A (eliminate self-reference) or B (eliminate the rule), or leave it open?
Now how do we decide whether to go for Option A and say that self-reference is impossible, or B and say that the rule is impossible? One option would be to leave it open and say you can pick either one.
But we can force the issue by forcing self-reference:
Barber*: There is a barber that, including himself, shaves all and only those people in the town that don't shave themselves.
Set*: There is a set that, including itself, contains all and only those sets that don't contain themselves.
Proposition*: There is a proposition that, including itself, is about all and only those propositions that aren't about themselves.
Now my hand is forced and there is only one option: Option B. The rule itself is impossible.
It doesn't seem like there is a similar way to force the rule. Consider:
Barber**: There is a barber that, including himself, shaves all and only those people in the town that don't shave themselves, and this rule is necessarily true.
This doesn't force my hand in a similar way, because the rule is still just false. By tacking on the "necessarily true" part at the end, you're just adding another false bit to the rule.
But if self-reference is impossible, then the "including himself" bit is impossible. If that bit is included in the original rule, then you'd have a case where both self-reference and the rule are impossible, which amounts to choosing both Option A and Option B, which is not possible.
You could say that the "including himself" bit adds a second rule, and so Option A involves denying the self-reference rule while Option B involves denying the original rule.
So as far as I can tell, Option A and Option B are both available. Insofar as self-reference is necessary, then Option B is selected, and insofar as the original rule is necessary, then Option A is selected. But I don't see a way to force the truth of either the self-reference rule or the original rule. A couple thoughts:
Thought a) Intuitively, it feels like self-reference is easier to accept over the rule, because the rule itself is ad hoc while self-reference is a more stable or familiar concept. But I don't have a strong enough grasp of this to make an argument for treating self-reference as necessary.
Thought b) You could combine the self-reference rule with the original rule to get one rule with three components: the self-reference component, the 'all' component, and the 'only' component. This changes the Options from above to what's below:
Option A*: Deny the rule by eliminating the self-reference component.
Option B*: Deny the rule by eliminating the 'all' component.
Option C*: Deny the rule by eliminating the 'only' component.
Option D: Accept all components of the rule and reject the Law of Non-Contradiction.
Just as with Option C, Option D is necessarily rejected in the barber case. The only explanation I can think of that explains why Option D is impossible is because contradictions are impossible. But if the impossibility of contradictions is what explains the necessary rejection of Option D, then that same explanation carries over to parallel cases.
There is a theoretical Option E, which is to accept A*, B*, and C* and deny all components of the rule. But that would fail to accurately describe what happened to the barber. The options can be framed as descriptions:
Description A: The barber successfully shaves all and only those people who don't shave themselves other than himself. The barber is an exception to the rule because self-reference is impossible in this case, because self-reference in this case entails a contradiction and contradictions are impossible, and anything that entails an impossibility is itself impossible.
Description B: The barber shaves all and only those who don't shave themselves, including himself. The barber does not shave himself. So really, he shaves almost all and only those people who don't shave themselves.
Description C: The barber shaves all and only those who don't shave themselves, including himself. The barber shaves himself. So really, he shaves all but not only those who do not shave themselves.
Description D: The barber shaves all and only those who don't shave themselves, including himself. The barber is both shaved and unshaved in the same sense at the same time.
(I suppose there is a bonus option: The barber is excluded from the category 'people' so that he does indeed shave all and only those people who do not shave themselves. But I take that to be clearly false: the barber is necessarily a person.)
Description D necessarily gives a false description. But do any of the other descriptions necessarily give a false description?
Following 'Thought a)', you could argue that as a matter of description, self-reference is taking place. You could imagine a kind of phenomenal self-reference in the form of an intentional self-reference. The barber says to himself, "I am going to shave all and only those people who don't shave themselves, including myself." There is now a mental, intentional moment of self-reference. Because of that, if you describe the situation as lacking self-reference, your description is incorrect. But does an intention of self-reference necessitate self-reference? You could describe the situation in terms of the barber intending and failing to self-reference.
Purely intuitively, I suspect there's a way to get necessary self-reference out of the description, which would mean that Option A gives a necessarily false description. For example, if intentional self-reference entails necessary self-reference, then you can have your necessary self-reference at least that way. But I don't have a way to show this at the moment. So as far as I can tell, Description A, B, or C can apply to the barber paradox, the set paradox, and the proposition paradox.
Part 6: The Liar Paradox
Liar: This sentence is false.
The Law of Non-Contradiction explains why Description D is necessarily false. So if we take the idea seriously that there are no true contradictions, what becomes of the liar paradox? I see a number of options:
L1: Failed Reference Problem: 'This sentence' fails to refer to anything, because, like with Option A above, self-reference is impossible here because it entails a contradiction. So the liar is claiming that some indeterminate thing is false. But indeterminate things cannot be false. So the Liar is false.
Or, because 'This sentence' fails to refer to anything, the Liar sentence fails to be about anything, and thus fails to express a proposition, because propositions are necessarily about something.
L2: Fragment Problem: 'This sentence' refers to the literal sentence fragment 'this sentence', in which case the liar is saying that the sentence fragment 'this sentence' is false. But sentence fragments do not express propositions, and only propositions and their expressions can take on a truth value. So the Liar is false.
L3: Nesting Problem: 'This sentence' refers to the sentence 'This sentence is false.' But that means we can substitute any instance of 'this sentence' with 'This sentence is false.' But that means we have an infinite regress. When we unpack it once, we get: This sentence is false is false. When we unpack it again, we get: This sentence is false is false is false. When we fully unpack it, we get an infinite number of "is false" phrases at the end, and so the sentence never completes. Assuming propositions must complete, the Liar fails to express a proposition and so is neither true nor false.
L4: Aboutness Problem: This sentence is false. False about what? False about its being false. False about its being false about what? False about its being false about its being false. False about its being false about its being false about what? Again, this goes on forever, and the sentence has this infinite regress of intentionality. But propositions don't have regressing intentionality. So the Liar fails to express a proposition.
This applies to the 'Truth Teller': This sentence is true. True about what?
L5: The Meta Problem: For every proposition there is a meta proposition about that proposition. For example:
Proposition: The earth revolves around the sun.
Meta proposition: <The earth revolves around the sun> is true.
Meta meta proposition: [<The earth revolves around the sun> is true] is false.
Meta meta meta proposition: ([<The earth revolves around the sun> is true] is false) is true. Or: It's true that it's false that it's true that the earth revolves around the sun.
We can call these meta(1), meta(2), and meta(3). The truth value of a meta(1) proposition depends on the truth value of the corresponding proposition. That dependence relationship only goes one way; propositions can never have a truth value that depends on the truth of the meta proposition.
If 'This sentence is false' is translated as 'The meta proposition of this sentence is false', then the Liar is either false because it claims there is a false meta proposition when there isn't one, or fails to express a proposition because propositions cannot have their truth value depend on their meta proposition.
So it seems to me there is at least one way, if not several ways, that the Liar fails to express a proposition, or expresses a false proposition. I don't see any need to give up the Law of Non-Contradiction.
Part 7: Paradoxes of Law
While I'm at it, why not address paradoxes of law.
The following is my paraphrase of Kane B's formulation, wherever he got it from, from one of Kane B's videos:
Sidney is an Australian aboriginal. Like many aboriginals, she is a housekeeper. The owner of the house becomes fond of her and writes her into his will. After he dies, Sidney becomes the owner of his land.
There are two Australian laws:
Law(1): If you own land, you have a right to vote.
Law(2): If you are an aboriginal, you do not have a right to vote.
Normally, aboriginals cannot own land, as they are not allowed to purchase it. And normally, no one gifts land to aboriginals. But in this unique case, Sidney is both a landowner and aboriginal, and so Sidney both has and does not have a right to vote.
Because a right is a legal status granted by law, it's undeniable that Sidney has the legal status 'Can Vote' and it's undeniable that Sidney does not have the legal status 'Can Vote'. So we have a true contradiction: It's both true and false that Sidney has a right to vote.
But we see a similar problem as with the barber paradox: At the end of the day, it's undeniable that Sidney will either be allowed to cast a ballot or not. If Sidney is allowed to cast a ballot that counts, then she has a right to vote. If Sidney is forbidden from voting, or if her vote is tossed on account of her being aboriginal, then she does not have a right to vote.
What would be impressive would be showing Sidney casting a vote and not casting a vote in the same sense at the same time. But that can't be shown, because it's impossible, because contradictions are impossible. Instead, all we have is this suspicious notion of "legal status".
I don't know how jurisprudence systems interpret the notion of 'legal status' or how they interpret what laws are really saying or doing. At the very least, we can say that if a law makes a false claim or implies acceptance of a false claim, then that doesn't make that claim true.
If we interpret the laws as saying:
Law(1*): If you own land, you have a right to vote. That is, you can cast a vote without being physically stopped or you can vote without having your vote tossed.
Law(2*): If you are an aboriginal, you do not have a right to vote. That is, you will be physically stopped from casting a vote, or if you do vote your vote will be tossed.
Now it's undeniable that one of the laws will be proven false when Sidney goes to vote.
Instead of interpreting rights as these magical legal
statuses that people have, if rights are interpreted as descriptions of
what will happen given varying circumstances or how the state will treat
a citizen, then the paradox resolves.
Another interpretation:
Law(1**): If you own land, you have a right to vote. That is, you can vote without violating any laws.
Law(2**): If you are an aboriginal, you do not have a right to vote. That is, there are no laws granting you the right to vote.
In this case, both laws are false. Sidney cannot vote without violating any laws and there is a law granting Sidney the right to vote. So both laws contain falsehoods with Sidney acting as the exception to both laws.
Another interpretation of law is that laws are something like performative commands under fictional canons. So if I say "Harry Potter defeated Voldemort", I say something true relative to a fictional canon, but not true in any absolute sense. If I write my own fanfic where "Voldemort defeated Harry Potter" is true, then that claim is true relative to a fictional canon as well. But that doesn't mean there's a true contradiction that Harry Potter defeated and failed to defeat Voldemort.
So Sidney could have a right to vote according to a canon established by one law, and not have a right to vote according to a canon established by another law. But just like fictions, there is no absolute truth to either canon. I'm aware of Kathleen Stock's idea of collective fiction. So law might work in that way, as a collective fiction. State and country borders, and legal entities being treated as persons, are examples of legal fictions. Legal fictions correspond to social facts, facts about how people behave as if something is true. So while there is no objective border that delineates one state from another, we will still behave as if such a border exists. Likewise, while there is no objective fact that someone has or does not have a right, we will behave as if that person has or does not have that right.
Another way to impress me would be this: When lawmakers discover that Sidney's situation creates a contradiction within the law, they leave the contradiction. After all, if contradictions are not fundamentally problematic, lawmakers can just leave the contradiction there. Sure, it's true that Sidney has a right to vote. And sure, it's true that she doesn't. What's the problem? Haven't you heard, there are true contradictions, and this is one of them. Of course, that's not what happens. Lawmakers or a judge would issue a ruling and resolve the contradiction, either by having one law trump the other or by changing the laws. If laws are meant to dictate behavior, then it is exactly because we cannot behave as if a contradiction is true that is why the contradiction cannot be left in the law. The idea that contradictions are fundamentally problematic makes sense of all of this.
(Imagine what that behavior would look like and how absurd it would be. Sidney casts her ballot. When folks are asked whether Sidney voted, they will say "Yes she did, but also no she didn't." And imagine that her vote happens to be the tie-breaking vote. "Yes, the election was a tie. It's also the case that the election was not a tie." The elected official would be both elected and unelected, and they would have to both implement and not implement a tie-breaking procedure because there both was and was not a tie. Imagine if through that procedure the rival candidate wins the tie-breaker. Then you'd have two officials, both elected and non-elected, both serving and not serving the same governing role. Any laws they veto would be both vetoed and not-vetoed. And so on.)
The point is that what laws are really doing metaphysically needs to be explained. Given the surface-level explanations here, there's no true contradiction in the Sidney case.
(As an aside, I see how a representational theory of truth can work neatly with fictions: By treating a fiction as a backdrop for external properties, then you can measure internal properties against that backdrop. So the internal properties within the sentence "Harry Potter marries Hermione" fails to match with the "external" properties of the canon established by the Harry Potter books. But the properties of the canon itself fail to match the external properties of the real world. So that's how fictional claims can be true and false in different ways at the same time.)
I don't have a citation for this but I think it's a fun story. There's an interview on YouTube featuring JC Beall where he mentions that Graham Priest doesn't take Liar paradoxes to be the best candidate for true contradictions, but takes paradoxes of law to be better candidates. Beall implies that he disagrees with this and thinks paradoxes of law are no good, or at least not as good as Liar paradoxes. So while it's tempting to think that all logicians who deny classical logic in favor of some kind of subclassical or paraconsistent logic are on the same team, in typical philosophy fashion there can still be a great deal of disagreement within that space. So even philosophers who defend subclassical logics can agree with my conclusions about one category of paradoxes while disagreeing with my conclusions about the other category.