Paradoxes of self-teaching

Does every rule have an exception? Is each claim true or false? Is every argument valid or invalid, without a third possibility? The possibility of self-teaching in the language raises many difficulties and requires us to examine it very carefully

Marius Cohen "Galileo" magazine

Does every rule have an exception? If so - then even this rule has an exception and this means that it is not true, that is, not every rule has an exception. Since this claim, that every rule has an exception, leads to a self-contradiction, it cannot be true. However, there is no paradox in this, because its negation (the claim that there is no exception to every rule) does not contradict itself, and therefore it is probably the truest of the two. On the other hand, it is possible, as we will see below, that both an assertion and its negation will contradict themselves, thus creating a paradox.

But before approaching the paradoxes themselves, we will try to examine how a claim can contradict itself. First, it is important to distinguish between the concept of a sentence (an indicative sentence, in this context), which is a linguistic expression that expresses a claim, and the claim itself, which constitutes the intention of the speaker of the sentence. For example, if at the end of a running competition two runners argue with each other and each of them says: "I came first", then even if they both say the same sentence, their arguments are different (otherwise there would be no argument between them). In the same way, if one of them says "I won", and his opponent answers him "you won", then the two sentences are different from each other, but they express the same claim. Because of this, the claim, and not the sentence, is the one that carries a truth value, that is, it is the one that can be said to be true or false (it is not the sentence "I arrived first" that is true or false, but the claim it expresses).

An important question in this context is what determines whether a particular claim is true or false. Well, according to a widespread philosophical position (and we will not deal with other positions here), a claim about the world (for example, "the sun rises") is perceived as true if it corresponds to reality (if the sun really shines), otherwise it is perceived as false. However, the richness of language allows us to make claims not only about the world but also about other claims. I can, for example, say: "You are lying," thus claiming that your claim is false. If indeed he is, then my claim is true, otherwise I am the one who is lying. But if a claim can claim not only about facts in the world but also about other claims, it seems that there is no obstacle for a claim to also refer to itself. For example, I can say: "This claim is made on Sunday", or: "This claim is expressed in Hebrew", and on the surface it seems that we can attribute truth values ​​to such claims as to any other claim. We say of such a claim, which refers to itself, that it includes self-instruction. So is the claim: "Every rule has an exception." Since this claim itself expresses a rule, it proves itself to be false, and thus a self-contradiction is obtained.

More in the series of paradoxes: The stacking paradox

The liar paradox

The most famous paradox of self-teaching is the "liar's paradox", attributed to the philosopher Eubulides of Miletus, a contemporary of Aristotle. The paradox has gone through various incarnations (such as, for example, the sentence "All Cretans are liars", supposedly said by someone who was a Cretan himself), and is known today in its popular version: "This sentence is a lie". But since, as we have seen, it is not the sentences that carry truth values ​​but the claims, we will use the more accurate version, which is:

"This claim is false"

When the expression "this claim" refers to the claim that the above sentence expresses. Since a claim can be either true or false, we expect this claim to also have a definite truth-value. But the claim cannot be true, since it shows itself to be false, and from its truth comes its falsehood. Likewise, it cannot be false either, since it follows that it is not false, that is, it is true. If so - is the claim true or false?

It can be argued that the paradox stems from our very requirement that every claim have a definite truth-value (true or false), but there are claims, such as this claim, that do not have a definite truth-value and therefore there is no paradox here - the claim is neither true nor false. But this attempt to solve the paradox involves several problems: first, there is an exception here from the two-valued classic logic, which assigns to each statement one of two truth values: true or false. This logic is so entrenched in our way of thinking that we will not easily agree to give it up. Furthermore, the main problem involved in this position is simpler: it is unable to resolve the paradox resulting from the claim below:

"This claim is false or has no definite truth value"

It is not possible for the claim to be true, because its truth implies that it is false. Nor is it possible that the claim is false, because it follows that it is true. But it is also not possible that it lacks a definite truth-value, because it follows that the claim is true, and therefore has a definite truth-value! If so, it seems that the attempt to solve the liar's paradox by renouncing classical logic is not justified, and we must look for another solution to the problem.

Cree paradox

In order to understand the Curry paradox, we must first familiarize ourselves with the concept of a conditional sentence. A conditional sentence is a sentence that expresses a conditional claim, which has the structure: "If A then B", where A and B are themselves claims (A is called the root of the conditional claim, and B is called the conclusion). For example: the sentence "If it rains, the game will be postponed" is a conditional sentence, and it claims that if the claim "it will rain" is true (that is, if it really rains), then the claim "the game will be postponed" is also true (that is, the game will indeed be postponed). The question arises in which situations such a claim, expressed in a conditional sentence, will be true, and in which situations it will be false.

Undoubtedly, if it rains, then the claim must be seen as true if indeed the game will be postponed and as false if the game will not be postponed. In other words, if statement A is true, then the conditional statement is true if and only if B is also true. But what truth value should be attributed to the conditional claim if it did not rain? Did we lie when we said that if it rained the game would be postponed? Of course not, because we only stated what would happen if it rained. Therefore, if statement A is false, the conditional statement "if A then B" is perceived as true whether statement B is true or false. This can be summarized using the truth table in front of you:

AB If A then B

true true true
true false false
lie true true
lie lie true

Another thing we need to know to understand the Kree paradox is the rule of logical inference known as modus ponens. The rule says that from the truth of the two statements "A" and "If A then B" also follows the truth of the statement "B". For example, if the statements "today is Sunday" and "if today is Sunday then tomorrow is Monday" are both true, then by necessity the statement "tomorrow is Monday" is also true.

And now for the Kerry paradox. We will examine this claim: "If this claim is true, Haifa is the capital of Israel"

We immediately see that this claim includes a self-instruction, since it refers to itself in the head (with the words "this claim"). Well, is the claim true or false? If it is real, then its origin is also real, and from the general inference modus ponens it follows that its sipa is also real, that is, that Haifa is the capital of Israel. Suppose, then, that the claim is false. But then its predicate ("this claim is true") is also false, and as we saw above, if the predicate of a conditional claim is false, then the entire claim is true. That is, from assuming the falsity of the claim, its truth follows. However, we have already seen that from the truth of the claim it follows that Haifa is the capital of Israel. In other words, no matter what truth-value we attribute to the claim, its conclusion is the same: Haifa is the capital of Israel! (And of course we could show in the same way the truth of any other claim, including the claim that Haifa is not the capital of Israel).

More in the series of paradoxes: the logic behind Zeno's paradoxes

Paradoxes of validity

An argument is a collection of claims, one of which is the conclusion, and the rest are assumptions that try to lead to it. If the structure of the argument is such that, since all its premises are true, its conclusion must also be true, we say that the argument is valid (otherwise it is not valid). For example, in the argument:

Assumption 1: Today is Sunday
Assumption 2: If today is Sunday then tomorrow is Monday
Conclusion: Tomorrow is Monday

If both premises are true, so is the conclusion (necessarily), and therefore the argument is valid (it is possible for an argument to have only one premise, as we will see in the validity paradox below).

We will now examine the argument below, which includes self-instruction:

Assumption: This argument is valid

Conclusion: Haifa is the capital of Israel

Is the argument valid? If so, then the assumption ("this argument is valid") is true, and therefore the conclusion is also true. That is, Haifa is the capital of Israel. Since this is the case, we would like to argue that the argument is invalid. But by definition this argument is invalid if and only if it is possible that its premise is true while its conclusion is false. If this argument is not valid, then its assumption ("this argument is valid") is necessarily false, and the condition of the argument being invalid cannot be fulfilled. If so, the argument must be valid, and as we have seen its assumption is true, and there is no escaping the conclusion that Haifa is the capital of Israel.
Another validity paradox is illustrated by the argument:
Assumption: Jerusalem is the capital of Israel
Conclusion: This argument is not valid

Is the argument valid? If so, then since his premise is true, his conclusion is also true, that is, the argument is invalid, and we get a contradiction. Therefore, we would like to argue that the argument is invalid, but by definition this argument is invalid if and only if the premise is true while the conclusion is false. And if, as we established, this argument is not valid, then its conclusion ("this argument is not valid") is necessarily true, and the condition for the argument being invalid cannot be fulfilled (and similar to bivalence regarding the truth values ​​of claims, every argument in classical logic is either valid or invalid, without a third option). And here too the paradox was created due to the use of self-teaching.

Suggest a solution

Much has been written about paradoxes of self-teaching, and especially about the paradox of the liar, when you could write an entire book on all the existing positions and opinions on the subject. Below I will present one proposed solution to the paradoxes of self-teaching, which, despite being controversial, may shed light on all the paradoxes presented above and others like them.

We like this series of claims:

1. The following statement is true
2. The following statement is false
3. The following statement is false
4. London is the capital of England

To determine the truth value of claim #1 we must first determine the truth value of claim #2. To do this we must first determine the truth value of claim #3, which is determined by the truth value of claim no. 4, the last on the list. Since this last claim is true, then the claim that preceded it, claim #3, is false. Therefore claim #2 is true, and thus we can determine that the claim with which we opened the list is also true. Such a list can be as long as we like, but in the end we must arrive at a claim, the truth value of which is not based on the truth value of another claim, but is anchored in some fact (the claim "London is the capital of England" is true because London is indeed the capital of England). But what if our list is infinite? In a list like this:

1. The following statement is false
2. The following statement is false
3. The following statement is true

The truth value of each of the claims is based on the truth value of another claim, and there is no claim whose truth value is anchored in any fact. As a result, it is impossible to determine the truth value of any of the claims. A similar situation can also be obtained with a finite number of claims when they create a circular teaching situation:

1. The following statement is true
2. The previous claim is false

And as we saw with only one claim: "This claim is false". Here, too, it is impossible to anchor the truth value of the claim in some fact, and therefore it is impossible to determine its truth value. Seemingly there is here a waiver of classical logic and a return to the proposition that the truth-value of such a claim is undefined. However, this can be avoided if it is established that a sentence that purports to make a claim that is not grounded in fact does not make a claim at all. A similar thing happens, for example, when the sentence "This chair is blue", which on the face of it expresses a very clear claim, is said by a person who is not pointing to any chair. In such a situation the sentence said does not express any claim (although it may do so in a different context), and therefore there is no place to ask whether that person spoke the truth or a lie.

This position also resolves the other paradoxes of self-teaching, since they are all based on sentences that do not express anchored claims. In the first paradox of attacks, for example, the sentence "This argument is valid" does not express an anchored claim, and since according to the concept presented here such a sentence does not express a claim at all, then in practice we do not have an argument, because according to the definition of the concept, both the premises of the argument and its conclusion are Claims.

Dessert

As we have seen, self-teaching gives rise to semantic difficulties (that is, those concerning the determination of the truth values ​​of claims), but it also has, for self-teaching, an amusing aspect. Below are some thought provoking (and smile) sentences that make use of self-teaching. Most of the sentences are based on examples from Douglas Hofstadter's book: Metamagical Themas.:

• In this sentence, the letter "S" appears in the third, twelfth, twenty-fifth, twenty-eighth, thirty-first, thirty-sixth, fortieth, forty-eighth...
* Although this claim begins with the word "because", it is false
* You will understand this sentence if you change one celebration in it
* This sentence was written without the appearance of the letter "
* This shet was written without the letters ' ', ' ' and ' '
* ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ' – ' '
* The end of this sentence is written in
* If we take it out every third will still be understood
* Ignore this prompt
* This claim is true