A crocodile takes a child but promise the parent he will return the child if the parent guesses what the crocodile will do. The parent responds that the crocodile will not give the child back.

What does the crocodile do in response? How would the crocodile get around this using an alternate form of logic? — Marchesk

The answer to this is the same answer I gave to The Pinocchio Paradox. The stated rule ("I will return the child if and only if you correctly guess what I will do") is one that cannot be followed without exception.

So the crocodile's claim ("I will return the child if and only if you correctly guess what I will do") is false.

That's the standard presentation of the Principle of Explosion, which is that any well-formed sentence (Q in this case) can be proved from a contradiction. The contradiction is Line 1, and that's a proper, formal contradiction, which is the sentence 'L and not-L' for any well-formed sentence L. The derivation of Q is valid.

I agree with everything in that post. What I am saying is that the Liar sentence in natural language does not give us line 1 in that proof. The Liar sentence is not of the form 'L and not-L', and attempts to derive a sentence of that form from the Liar sentence make untenable assumptions.

With math and logic it's a matter of using the axioms and the rules of inference to determine what follows from what.

There's nothing like either of that for liar-like statements. — Michael

I don't know how you can say that. The contradiction in the liar statement stems from following the rules of logic. If not, then why is it considered an issue?

I don't know how you can say that. The contradiction in the liar statement stems from following the rules of logic. If not, then why is it considered an issue? — Marchesk

What I'm saying is that the truth (or falsity) of that statement isn't derived from some set of axioms. You don't say "the liar sentence is true because it follows from these true sentences" because there are no true sentences from which that sentence can be derived. Liar-like statements have no truth-maker (even in principle).

The Liar sentence is not of the form 'L and not-L', and attempts to derive a sentence of that form from the Liar sentence make untenable assumptions. — andrewk

Which form of the liar statement?

I am lying.

This sentence is false.

This sentence is not true.

The next sentence is true.
The previous sentence is false.

Any of them.

Edit: sorry, thought that was directed at me.

Any of them. — Michael

The third one is the "strengthened" liar paradox. Consider:

"This sentence does not express a true proposition."

If it's true, then you're back at the same contradiction.

The third one is the "strengthened" liar paradox. Consider:

"This sentence does not express a true proposition."

If it's true, then you're back at the same contradiction. — Marchesk

It's not true. The sentence isn't truth-apt.

It really is a straightforward proof by contradiction. If it being either true or false leads to a contradiction then it must be neither true nor false.

It's not true. The sentence isn't truth-apt.

It really is a straightforward proof by contradiction. If it being either true or false leads to a contradiction then it must be neither true nor false. — Michael

That seems like cheating. Now you can just remove any paradoxical statement by saying it's neither true or false. Let's try this as a result:

"This statement is neither true or false."

Does that lack a truth value? I say no, it has a truth value if what you stated is true. And then you're right back at the liar paradox.

Surely, if it were that easy, the liar paradox wouldn't have remained a puzzle to philosophers for 2300 years!

But you can't get out of it by stating that if it leads to another statement resulting in contradiction. — Marchesk

Are these contradictions?

"this statement doesn't correspond to some obtaining state-of-affairs" doesn't correspond to some obtaining state-of-affairs.

"this statement doesn't follow from some specified set of sentences" doesn't follow from some specified set of sentences.

I don't think they are.

So why would this be a contradiction?

"this sentence isn't true" isn't true.

Of course, this could all be made clearer if we could explain what it even means for a sentence to be true. I've provided two possible answers. Are there any others you could suggest?

But we do say things like that on occasion. For example, "This party is not a party", meaning it's a party in name only. I'm pretty sure I have said something akin to "this chair is not a chair" when being forced to sit on something uncomfortable that served as a chair. I've also said, "I'm not myself today", which would seem to be a violation of the law of identity, but clearly it's not meant to be taken in literal terms. — Marchesk

That's very true, but what is at issue here is the semantics, what is meant by the statement. In your examples, what is expressed, is that the thing is misnamed. The get together of people should not be called a party, the thing you are sitting on should not be called a chair. That is the meaning expressed.

Also, there is this very big counter to the claim that the liar sentence is without meaning:

"This sentence is meaningless."

Which would be true if the liar sentence is meaningless, but then we get ourselves into another regress. — Marchesk

The liar sentence is not without meaning, it is used as an example, this use indicates meaning. So people like philosophers bring it up to discuss, and this is its context, which gives it its meaning. But there's very little difference between it and other self-contradicting, self negating examples, like "the square circle". It just demonstrates that we can say things, which according to the placement of words, appear to be meaningless due to self-contradiction, and ask others what is meant by this.

That is the meaning of the liar sentence, it is used by philosophers to demonstrate that we have the capacity to use words in this way. It should be of no surprise to anyone, because we have the capacity to deceive. And this means that we can say things which are completely different from what our intentions are. So when I say "this circle is square", or "this sentence is false", what I mean by this (what my intentions are, or what I am doing with those words), is something different from what the words appear to mean, just like common forms of deception. Again, we have exposed that separation between what is meant by the author (the author's intentions, what the author is doing), and what the words, on their own, appear to say. And this simply demonstrates our ability to deceive.

"This sentence is not in Italian", which is not meaningless, but is in the same form as the liar sentence. — Marchesk

One problem is thinking that whether a sentence is what I called "substantive" is simply a matter of its form. I wouldn't say that it is.

But arguably, "This sentence is not in Italian" isn't the same form as "This sentence is false." As I mentioned, the "is false" part of "This sentence is false" is just taking the truth value assignment and making it explicit/making it part of the sentence--or would-be sentence in that case, since "This sentence" isn't a sentence. So that wouldn't be the same form in that case. The same form would be "This sentence is not in Italian is true." (Or more conventionally, "It is true that this sentence is not in Italian.") Of course, this brings up possible ambiguities, at least contra conventions, over just what a sentence's form is, too.

"This sentence is meaningless."

Which would be true if the liar sentence is meaningless, — Marchesk

Why would it hinge on the liar sentence? At any rate, if someone assigns no meaning to that sentence, then "This sentence is meaningless" is simply true. Otherwise it's false. That's no paradox.

At any rate, if someone assigns no meaning to that sentence, then "This sentence is meaningless" is simply true. — Terrapin Station

If the sentence is meaningless, then it can't be true.

If the sentence is meaningless, then it can't be true. — Marchesk

I understand why you're saying that, but it seems wrong. However, I have to think about it further to explore how I'd analyze the problem there.

[Later;] Ah--okay, so I'd say that it simply amounts to reading the sentence as saying, "This sentence isn't making any claim other than that it's making no claim." The sentence not being meaningless in that case would be it making some claim other than that.

If the sentence is meaningless, then it can't be true. — Marchesk

The sentence is not meaningless, because Jaydison posted it in the op with intent, purpose, to discuss it's meaning. Therefore it must have meaning, Jaydison meant something with it. But, as I explained in my last post, what was meant by it, what Jaydison was doing with it, is something completely different from the meaning which appears from a reading of the words. What appears is some form of meaninglessness. Therefore the sentence is actually posted as a form of deception. And this can be seen as the essence of all such sentences which appear to be paradoxical, they are simply posted as a form of deception.

As Michael said, none of them are in the required form to be a contradiction. The required form is 'L and not-L'. None of those sentences contain an 'and'.

I don't understand your position.
For example if there is not a good formal definition for contradictions does this mean proofs by contradiction fail?

Who is saying there is no formal definition of a contradiction? Not me. I gave a definition in both of my last two posts.

You will have to forgive me, as I said I was having trouble understanding.

I was talking about this post in particular.

I agree, we do use it, and I should have included the word 'necessarily' in that sentence you quoted ('does not necessarily a contradiction make ....'). But my understanding of what it means for a natural language sentence to contain a contradiction is that it is equivalent* to a formal sentence that contains one.

I have never seen any other definition of contradiction that is sufficiently objective to enable one to determine in all cases whether the definition is satisfied.

* 'equivalent' in the sense that the user of the sentence would not object to the translation as inaccurate, if the meaning of the translated sentence were explained to them. — andrewk

I mistook it to mean there was no formal definition of a contradiction.

You also posted this and I was hoping you would get into the weeds a bit and delve into nuances you mention.

It's not a contradiction unless one can set the contradiction out as a deduction in a formal logical language. And my point is that, in the process of trying to do that, one encounters obstacles that lead one to realise that there is no contradiction there. There's a bunch of nuances to this that I've left out for the sake of brevity. But we cannot even begin to consider them until somebody makes the attempt to formalise the sentence.

A bunch of natural language words does not a contradiction make, no matter how much it may feel as though they do. — andrewk

The main nuance I had in mind is that one can, by using certain higher-order logical languages, formally express a version of the Liar Sentence. However the languages in which one can do that are known to be inconsistent, meaning that one can derive contradictions in them without even introducing any non-logical axioms.

Those contradictions are closely related to the set in Russell's Paradox of set theory, as is the Liar Sentence. Because one cannot trust a language in which contradictions can be proved (as demonstrated in Marchesky's post above), such languages are excluded from the serious consideration of higher-order logics. That is done by putting constraints on what constitutes a well-formed sentence in the language. Those constraints make it impossible to formalise the Liar Sentence in the language.

There is a pretty good introductory explanation of this in this note from Washington Uni.

Thanks

Saul Kripke (1975) suggested a solution to the problem by introducing a third-value "undefined", still, it is still problematic (see "strengthened liar" or "revenge of the liar")

Alfred Tarski, on the other hand, worked on the assumption that self-reference (proved as the diagonal lemma) is inevitable and that truth is binary. He introduced the hierarchy of languages that is in order for us to talk about a language we should another language (a meta-language). The problem is Tarski aimed at introducing truth to formalized language, and claimed that we cannot introduce truth to ordinary language without resulting into a contradiction. (see 1944 Tarski)

'This sentence' is a phrase (not a proposition) that refers to the sentence 'This sentence is false'. Thus, we would get a sentences like ''This sentence is false' is false' and so on. Let us take not that we are not assigning truth to phrase 'This sentence' but to the sentence it refers to: the sentence 'this sentence is not true'.

the problem here is that it is agreeable the every instance of usage of truth takes the form 'x is true', where 'x' is any truth-bearer such as sentence, proposition etc. Thus, 'x is true' somehow serves as a basic definition of truth. However, as we can see above, there is an instance which leads us to a contradiction.

The idea here is simple. If giving the LP either the value of "true" or the value of "false" results in an inescapable contradiction, we can avoid the Paradox by saying that the LP has neither value, thus preventing the contradiction. There are many arguable problems here. Firstly, this would seem to require abandoning the Law of the Excluded Middle or else the Principle of Bivalence. Now, I've no qualms with dropping Classical Logic in favor of a Non-Classical Logic, but I get the feeling many people would not like that. — MindForged


Do we have to abandon classical logic when we claim that the sentence "go away" is neither true nor false? — Michael

Hello,

I do not think that we have to abandon classical logic because of the "liar",
but I like playing around with non-classical logics.
One exotic logic is my favorite:
The "logic of reflection" by Ulrich Blau
which i developed and extended to an alternative logic, the "layer logic".

Link to the logic of Ulrich Blau:
https://ivv5hpp.uni-muenster.de/u/rds/blau_review.pdf

Link to "layer logic" "Trestone":
https://www.researchgate.net/post/What_do_you_think_of_layer_logic-and_the_use_of_a_new_dimension_to_come_around_contradictions

In layer logic a hierarchy of truth layers is used (similar to the hierarchy of types of Bertrand Russell).
Proposals to not have truth values any more,
but only proposals in connection with a layer have a truth value.

The liar statement here has the following form:
L:= For all n= 0,1,2,3,... "This statement is true in layer n+1 if it is not true in layer n and else it is false"

As all statements are u=undefined in layer 0,
we get for layer n=0:
"This statement is true in layer 0+1 if it is not true in layer 0 and false else"
Therfore: "This statement is true in layer 1"
For n=1: "This statement is true in layer 1+1 if it is not true in layer 1 and false else"
Therfore: "This statement is false in layer 2"
We see: The liar statement L is undefined in layer 0, true in layer 1,3,5,7,...
and false in layer 2,4,6,8,...

So the truth-value is alternating with the layers.
A classical statement would have only one truth-value in all layers 1,2,3,4,5,...,
so we can see, that L is a non-classical statement.
In layer logic it is not paradox, but an ordinary statement.

Layer logic is a little bit cumbersome,
but has amazing advantages:
Nearly all paradoxes can be solved similiar to the liar.
With layer logic a "layer set theory" can be defined,
where the diagonalization proof of Cantor is no longer valid
and where the Russell set and the set of all sets are ordinary sets.
Even natural numbers and an arithmetic can be defined.
A small set back: The prime factorisation of natural numbers could differ in layers.
But on the other hand it is probable, that the proofs of Gödel`s incompleteness theorems are valid no more.

Layer logic is a kind of a third way between classic logic and constructivistic/intuitive logic.
It is a logic with three truth-values, but the layers are the most important part.
Indirect proofs are allowed in layer logic, but only within a layer.
As in most classic proofs there are different layers involved if transformed to layer logic,
those indirect proofs are mostly not valid any more.

Unfortunately nobody up to now has seen a layer in reality,
but as long as the idea does not lead to contradictions it remains an interesting idea.

Yours
Trestone