The Liar's Paradox
"This statement is false"
If the above is true then it implies that it is false.
If the above is false then it implies that it is true.
A paradox? No.
The liar's paradox is a self-referencing statement that can be better written as:
If this statement is true then it is false.
OR
This statement implies that it is false.
OR
If A then not-A ("A" is any proposition).
OR
A implies not-A.
So
Consider A to be true, then not-A is false.
But "A implies not-A" will be false.
Consider A to be false, then not-A is true.
But "A implies not-A" will still be false.
Because the implication is only true when A implies A.
Hence it is a contradiction. The contradiction says that the implication is false and the proposition A in that implication is hence meaningless.
Consider "this statement is true"
OR
If this statement is true then it is true.
OR
This statement implies that it is true.
OR
A implies A.
OR
If A then A.
If A is true, then "A implies A" is true.
If A is false, then "A implies A" is still true.
So A implies A is true, it is always true, a tautology.
In an intuitive sense also, we may say that every statement implies itself. No statement implies its negation. If a statement implies its own negation, then the implication is false. Or we can say, a statement cannot imply its negation.
Lesson: Only some language statements are logically valid propositions.
The Grelling-Nelson Paradox
There are two kinds of adjectives:
First, those that describe themselves, are called autological.
Second, those that do not describe themselves are called heterological.
The word readable is readable so it is autological.
The word English is an English language word. So it is autological.
The word unreadable is still readable, so it is heterological.
The word German is not in Deutsch so it is heterological.
Is the word autological itself autological? Yes.
Consider:
"The word heterological itself is heterological"
If true then it is autological. Hence not heterological.
If false then it is heterological, hence not autological.
This statement "The word heterological is itself heterological" implies its negation.
A implies not-A.
But whatever truth value you take for A:
If false then not-A is true, so A doesn't imply not-A.
If true, then not-A is false so A again doesn't imply not-A.
The implication is a contradiction, always false.
A self-referencing statement is an implication that is a contradiction, hence always false.
The resolution is in the fact that a self-referencing statement is not valid, it declares itself as the opposite of whatever it is.
The presence of something doesn't imply the absence of it.
The absence of something doesn't imply the presence of it.
Either it is present or absent.
So A cannot imply not-A.
Either it implies something else or implies itself.
Implying itself is a trivial tautology.
A statement hence can only meaningfully imply something else.
The Pinocchio Paradox
If Pinocchio said "my nose will grow"
If true, then the nose grows, then the statement is false.
If false, then the nose doesn't grow, then the statement is true.
What is the implication here?
"My nose will grow" is simply equivalent to:
"My statement implies that my nose will grow hence my statement will be false."
Here again, the statement implies its negation.
So the implication is always false.
His nose will hence not grow finally. Why? It is simple not because he is saying something true but because what he is saying is meaningless. Equivalent to making honking noises.
Also in the case of Pinocchio, it can be explained alternatively as the future state of his nose is known to Pinocchio himself ONLY when he knows that he is lying.
Let's say, he says, "Tomorrow will rain."
If it won't, then should his nose grow?
No, the fact of rain is in the future, it has not happened yet, what that is not happened yet is neither true nor false.
Lying is lying only when the liar is aware that he is lying.
Otherwise, it is just a contingency, a proposition for which no fixed truth value exists.
Some more
For necessary truths, a contingency is a meaningless statement.
For factual truths, a contingency is a conjecture.
Important: False implication doesn't mean that these self-referencing statements are themselves false, it means they are meaningless. It is equivalent to honking noises unless you are a horn.
The problem with other common attempts to describe these paradoxes is that they often analyse the paradox in terms of factual truths where it is possible to have an uncertain statement. Natural knowledge is always uncertain. You cannot do deductive reasoning there.
It is always inductive reasoning, the entirety of scientific knowledge is based on inductive reasoning.
Some arguments also mix the idea of contingency with the law of excluded middle and conclude that the law of excluded middle is wrong. If you take out the law of excluded middle, the entire system of logic collapses due to the principle of explosion.
The relevant classical laws of logic are:
Law of excluded middle - a proposition is either true or false. Not somewhere in the middle.
Law of contradiction - something is either A or not A. Where A can be anything.
Law of identity - a thing is identical or same to itself, A = A for all A.
Law of excluded middle is necessary for logic. If a statement "A" is both true and false then anything can be proven to be true. The statement "law of excluded middle is true" can also be proven to be true which is contradictory to the supposed assumption that "law of excluded middle is false". So either the law of excluded middle has to be true or all the propositions are true and false at the same time leading to the collapse of logic.
How does the principle of explosion work:
Let us say : “no pigs can fly” and “there are flying pigs”
"There are flying pigs or There exist unicorns." This statement is true in the above inconsistent system of axioms. The truth value of “there are flying pigs” is one and the truth value of “unicorns exists” is say x. So the combined truth value is one But also “no pigs can fly”, so the truth value of “there are flying pigs” is now zero. But since the combined Statement is true. Unicorns must exist. So using inconsistent axioms, we can prove anything, any nonsense. In the same manner we can simultaneously prove that unicorns don't exist. So if we start with inconsistent axioms, we will reach self-contradicting theorems and results. Principle of explosion is central to the "proof by contradiction" in mathematics.
The root problem is in the assumption that all possible sentences are valid propositions. No. Propositions are special formalized sentences that follow the laws of logic.
The logic "paradoxes" are like the "paradoxes in special relativity." They teach us about our wrong assumptions.