A paradox is based on premises that are obviously bet365 Play online games，By apparently valid reasoning，The process or phenomenon of drawing obviously contradictory inferences。Semantic paradox involves truth、False、Reference、Satisfied、The paradox of semantic concepts such as definition。As the oldest semantic paradox，The "Liar Paradox" can be traced back to the ancient Greek philosopher Epimenides in the 6th century BC。Original form: Epimenides, a Cretan, claimed that "all Cretans lie"。Later，This paradox is simplified to: a person claims "I am lying"。The "liar" paradox is the most basic、The most typical semantic paradox，Because its form is the simplest and clearest，And most semantic paradoxes have a similar structure to the "liar" paradox。Resolved issues related to the "Liar" paradox，This also solves most of the problems about semantic paradoxes。

In the history of logic，The semantic paradox represented by the "liar" paradox is full of controversy，Also valued by logicians。To solve it，Logicians based on their respective positions，Proposing many different solutions。However, to bet365 Play online games day，We still haven’t found a universally accepted solution。Fortunately，Late 19th to early 20th century，Major progress has been made in the basic theoretical research of modern logic and mathematics，bet365 Play online games allows people to be stricter、Discuss the issue of semantic paradox more scientifically。Important results based on modern logic，Especially the "Diagonal Lemma" used by Gödel in the process of proving his "First Incompleteness Theorem"，And Tarski’s “Theorem of Indefinability (of Arithmetic Truths)”，We have the ability to find out the root cause of the semantic paradox。Based on bet365 Play online games，We think，Syntactic richness、The combination of naive truth theory and classical logic will lead to inconsistencies。bet365 Play online games is to say，At least one of these three is problematic，It must be abandoned or modified。Any solution that attempts to truly solve the semantic paradox，Everyone needs to make a choice in the face of bet365 Play online games conclusion，And give the corresponding answer。

The classic liar sentence is a self-referential sentence of the following form: "L: L is false"，That is, statement L claims that it is false。If you want to consider whether L is bet365 Play online games or false，You will be in a dilemma: if L is bet365 Play online games，Then the sentence "L is false" is bet365 Play online games，Thus L is false，Contradiction；If L is false，Then the sentence "L is false" is bet365 Play online games，Therefore L is bet365 Play online games，Contradiction。We deduce that L is false from L being bet365 Play online games；From L being false, L is bet365 Play online games。In classical logic，According to the law of excluded middle，Every statement is bet365 Play online games or false；According to the principle of explosiveness (contradictions lead to everything)，Every statement cannot be both bet365 Play online games and false，Because "Contradiction leads to everything"。But according to the above reasoning，No matter whether the value of L is bet365 Play online games or false，We will all deduce that L is both bet365 Play online games and false，Obviously contradictory。

Except for direct self-reference，The "liar" paradox also has a form of circular reference。For example，Plato: “Aristotle’s assertion is bet365 Play online games。”Aristotle: “Plato’s assertion is false。”We assume that Plato’s assertion is bet365 Play online games，And Plato asserts that "Aristotle's assertion is bet365 Play online games"，Then，Aristotle’s assertion is bet365 Play online games；But Aristotle asserts that “Plato’s assertion is false”，Then，Plato’s assertion is false。We derived from “Plato’s assertion is bet365 Play online games” to “Plato’s assertion is false”，Contradiction。Vice versa，We deduce "Plato's assertion is bet365 Play online games" from "Plato's assertion is false"，Contradiction。Conversation similar to this，Often called the Liar Cycle，This chain of reference can be long enough，Finally forms a closed loop。

Since semantic paradoxes are usually related to self-reference and circular reference in form，If you have a point of view, just think，To avoid paradox，The use of self-reference and circular reference must be completely prohibited。We think，This idea is too simplistic，Cannot fundamentally solve the problem of semantic paradox。First，Self-reference and circular reference are very common phenomena in our language，Not all statements containing self-referential or circular references will lead to contradictions。For example，Someone promises someone else，"Everything I said is bet365 Play online games"；Or both sides of the debate blame each other，"Your point of view is wrong"。Also，Self-referential words like "this article" and "this book"，Nothing more normal in our language。If self-reference and circular reference are completely prohibited，It will cause our language to be too poor。Secondly，Some sentences may depend on some accidental empirical facts to become self-referential sentences。For example，The teacher asks everyone in the classroom to say a word，Then determine whether the statement is bet365 Play online games or false。One of the people said: "What the person with the lowest IQ in this classroom said is false。”Unfortunately，This speaker happens to be the person with the lowest IQ in the classroom。In other words，He asserted that what he said was false。For this type of statement，There seems to be no reason for us to ban it，Because this statement can be a normal statement in other contexts。Again，The formation process of some paradoxes does not involve self-reference or circular reference，The most typical example is Yablo’s paradox，The statements that constitute this paradox do not speak of themselves, directly or indirectly。Finally，Also the most important，Even if we completely ban self-reference and circular reference in language，But as long as our language is rich enough to talk about elementary arithmetic，By Gödel encoding，We can still construct sentences similar to the "liar" paradox。

Hypothesisis a formal bet365 Play online games rich enough to include Peano arithmetic，The direct discussion is bet365 Play online games and their related properties。Gödel discovered，If we are rightEach symbol is encoded (that is, a specific natural number is selected to correspond to it)，Then encode each symbol string and the sequence of symbol strings，bet365 Play online games is our commonly used grammatical category，Same as normal、variable、Term、Predicate、Formula、Formula sequence、Axiom、Theorems and proofs, etc.，All can be encoded。After encoding like bet365 Play online games，We can change the original statement about natural numbers，See as indirectly talking about the symbols encoded by these natural numbers、Symbol strings and symbol string sequences。The grammatical categories listed above，All in these encoded symbols、Among symbol strings and symbol string sequences。In other words，In language中，We can do this by talking about natural numbers and their associated properties，Indirectly talking about the grammar of the language itself。If a formal language is rich enough to include Peano arithmetic，Let’s just say the language has “grammatical richness”。In the process of proving the famous "First Incompleteness Theorem"，Gödel proved the "Diagonal Lemma"。We can intuitively understand this lemma as: given a property A，We can construct a statement D，Claims that it has property A，And D and A ([D]) are equivalent。Among them，[D] is "the name of D"，A ([D]) means "D has property A"。Shortly after Gödel proposed the Diagonal Lemma，Tarsky derived the indefinability theorem of arithmetic truth based on this lemma。This theorem shows，Any consistent formal language containing sufficiently rich arithmetic，bet365 Play online games predicates that cannot consistently contain themselves。

In the process of deriving this conclusion，A rule called the "Naive Truth Principle" needs to be used。This rule was also used in the above process of inferring contradictions from the "liar" statement。The principle of naive truth has three different expressions: the first expression is Tarski’s T-equivalence: (T) “[P] is bet365 Play online games” if and only if P。Among them，"P" is any statement，"[P]" is the name of statement P。The second way of expressing it is to regard it as the general name of the two inference rules "T-introduction rule" and "T-elimination rule"。"T-introduction rule" says: any statement P，We can deduce the statement "[P] is bet365 Play online games" from the statement P；The "T-elimination rule" says: from the statement "[P] is bet365 Play online games"，We can derive the statement P。The third way of expressing it is about the bet365 Play online games "equivalent substitution rule"。This rule says，The statement P and the statement "P is bet365 Play online games" are always interchangeable，But it will not affect the truth value of the entire statement。

According to the principle of simple truth，For any statement P，The statement "[P] is bet365 Play online games" and the statement P should be equivalent，Can be replaced or introduced from each other。And Tarski’s indefinability theorem tells us，We cannot consistently maintain the principle of naive truth under the above conditions。Gödel’s diagonal lemma and Tarski’s indefinability theorem (of arithmetic truths)，is an important result that we must face when discussing semantic paradoxes。According to the above theorem，We observed，Syntactic richness、Principle of naive truth and classical logic，These three factors combined can lead to inconsistency。The richness of grammar，allows us to construct the liar sentence L in language，Making L satisfy: L if and only if “[L] is not bet365 Play online games”。Among them，"[L]" is the name of statement L。This condition means，L is equivalent to "[L] is not bet365 Play online games"。Requirements of the principle of simple truth，For any statement P，It and the statement "[P] is bet365 Play online games" satisfy the T equivalence and can be replaced or derived from each other。Classical logic provides us with the inference rules needed to derive contradictions from liar statements，Like the law of excluded middle、Explosion principle、Disjunctive and additive principles。

Since the grammar is rich、If the principle of naive truth and classical logic are established at the same time, it will lead to inconsistency，Then at least one of them is problematic，Need to abandon or modify it。First，We seem to be unable to give up the richness of grammar no matter what，Because our language should be able to talk about at least the most basic arithmetic。Secondly，The Naive Truth Principle is consistent with our intuitions about the concept of “truth” in natural language，Intuitively we think it should be bet365 Play online games。It is difficult to imagine a situation in which “‘Snow is white’ is bet365 Play online games if and only if snow is white” is not bet365 Play online games。Moreover，Classical logic is a model of logic system，The most “orthodox”、The most "standard" logic system，If it is not based on some specific philosophical position，Or for some special purpose，It seems that we should not abandon classical logic。

Any solution that attempts to truly solve the semantic bet365 Play online games，We all need to face the above conclusion and answer: How rich is the grammar of the language we are discussing？Do you recognize the principle of simple truth？Whether to use classical logic？Different answers to these questions，These plans will be divided into different factions。Etarski、Logicians represented by Birch and Gupta advocate retaining classical logic，Modifications and restrictions on the principle of naive truth，The solution is called the classical logical solution。With Kripke、Logicians represented by Field and Priest advocate retaining the principle of naive truth，Modify classic logic，The solution is called a non-classical logic solution。

　　(bet365 Play online games article is a major project of the National Social Science Fund "The Historical Development of Generalized Logical Paradox、Theoretical Frontier and Interdisciplinary Application Research” (18ZDA031) phased results)

(Author’s affiliation: School of Philosophy, Renmin University bet365 Play online games)