When people say "Statement A follows logically from statement set B"，or "Statement A is bet365 bonus code louisiana of statement set B"，Or "The reasoning based on statement set B as the premise and statement A as the conclusion is valid"，What do they mean？How to characterize the "logical succession" relationship、The concept of “logically derived” relations or “valid reasoning” is undoubtedly one of the core tasks of the philosophy of logic。Although the development of logic has a history of more than two thousand years，But not until 1936，Tarski in the article "On the Concept of Logical Succession"，Only gave a far-reaching and clear model theory description of the intuitive logical successor relationship。

An attempt to characterize bet365 bonus code louisiana relationship is proof-theoretic in nature: statement A is bet365 bonus code louisiana of statement set B only if there are some simple and fidelity inference rules， A can be deduced from B by repeatedly using these inference rules。In Tarski’s opinion，This portrayal is unsuccessful。First，According to Gödel’s theorem，Given an appropriate system of arithmetic axioms，There is a true but unprovable successor to this axiom system。Secondly，Second-order logic is incomplete relative to standard semantics，There must be a true successor to the second-order theory but it cannot be proved，It is too narrow to use the concepts of "provable" and "deducible" to describe the concept of "(intuitive) logical successor"。

Carnap gave the following description of the "logical successor" relationship in his book "The Logical Structure of Language" published in 1934: Statement A is bet365 bonus code louisiana of statement set B and only if any containing The statement set B and the statement set other than A are all contradictory。Tarsky improved Carnap's characterization，Point out that a sufficient definition of "statement A is bet365 bonus code louisiana of statement set B" must satisfy two conditions: first，Impossibility condition or necessity condition-it is impossible that all the statements in B are true and A is false。“Necessary fidelity” should be the core of any logical sequel description。Second，Formal condition - it is only because of the logical form of the statements in statement A and B that it is impossible for the statements in B to be true and A to be false，The empirical content of the statement is the specific object that the non-logical constants in the statement such as names and predicates refer to、It doesn’t matter which specific property or relationship。

For example，The statement "There is an object that is a logician" is bet365 bonus code louisiana of the statement "Tarski is a logician"。This is because no matter which individual object we interpret the name "Tarski" to refer to，May as well be interpreted as Plato；No matter which property the predicate "is a logician" is interpreted as，It may be explained as the nature of a philosopher，It is impossible under this explanation，The statement "There is an object that is a logician" is false，And the statement "Tarski was a logician" is true。Because under this explanation，"Tarski is a logician" means Plato is a philosopher，And "there is a logician" means there is a philosopher，It is impossible that “Plato is a philosopher” is true and “there is a philosopher” is false。The statement "All birds can fly" is bet365 bonus code louisiana of the statement "Everything that cannot fly is not a bird"，Because no matter what nature the non-logical symbols "bird" and "flying" are interpreted as，If interpreted as the two properties of "odd number" and "not divisible by two" respectively，Not "All odd numbers are not divisible by two" is true and "All odd numbers that are divisible by two are not odd" is false。After a logical successor relationship or non-logical symbols in effective reasoning are uniformly interpreted with objects of the corresponding type，The premise is true and the conclusion is false。The reason why formal logic is “formal” logic，It’s because of abandoning “content”，Just like the arithmetic equation (a2-b2) = (a+b) × (a-b) is a mathematical truth，No matter what specific numbers "a" and "b" are interpreted as，Any special case of this arithmetic form or law is true。Similarly，All A is B is logically equivalent to all non-B is non-A，No matter what properties "A" and "B" refer to，There will be no situation where one side is true and the other side is false。

In order to describe the above necessary conditions and formal conditions，Tarski used the concept of mathematical models to describe bet365 bonus code louisiana: statement A is bet365 bonus code louisiana of statement set B and only if the given model is given the domain of discourse，Any non-logical constant (proper name、Function symbol、Explanation of predicate symbol)，Not all statements in statement set B are true under this interpretation and statement A is false under this interpretation。In other words，B’s models are all A’s models。A model can be regarded as a possible world，Possible worlds where the premise is true，The conclusion is also true。This is the meaning of "inevitable fidelity"。

Tarski found that the above characterization of bet365 bonus code louisiana successors relies on the division of bet365 bonus code louisiana constants and non-bet365 bonus code louisiana constants，This division determines the bet365 bonus code louisiana form of the statement，bet365 bonus code louisiana constants such as "not", "or", "and", "equal to", "if and only if", "all", "exists" and other words cannot be interpreted at will，The names or predicates of "Plato", "is a logician", "is a bird" and "can fly" are non-bet365 bonus code louisiana constants，can be freely interpreted using objects of the appropriate type。How to distinguish bet365 bonus code louisiana constants from non-bet365 bonus code louisiana constants，Tarski gave an interesting answer in his 1966 lecture "On bet365 bonus code louisiana Constants": objects that remain unchanged under any transformation are bet365 bonus code louisiana constants or bet365 bonus code louisiana objects。Geometers call bijective functions transformations，The domain and value range of functions are points on a plane or space graph，They usually study invariants under some kind of transformation，such as translation、Rotate、Orthogonal transformations such as axial reflection or flipping keep the distance between two points on the plane unchanged。What objects remain unchanged under any transformation？

Consider the following standard model of simple type theory: the lowest level of the model is a class of individual objects，This type of individual object is the most basic type；The second layer of the model is all possible classes composed of individual objects in the first layer；The third layer is the class composed of all individual object classes，And so on。Consider any bijective transformation from the lowest class of the standard model of simple type theory to itself，Such transformations naturally induce or extend to transformations of all types，It can be examined under such bijective transformation，Which objects in which layer remain unchanged under any transformation。It’s not hard to see，If there are more than one individual objects at the lowest level，No invariant，Because one object can always be mapped to another object；On the second floor，The empty set and the complete set are the only invariants，Because the empty set is bijectively mapped to the empty set，Full set mapped to full set；On the third floor，A class consisting of exactly n elements is an invariant，Because the image of a set of n elements is also a set of n elements。If ordered pairs are defined according to Kuratowski's method，Then define the binary relationship as a set of ordered pairs，At the fourth level, you can consider the set of ordered pairs，In it，Equality relationship、Full relationship、Empty relationship、The inequality relationship is an invariant，For example, equal objects are mapped to equal objects。On the fourth floor，Inclusion relationship between two classes、Intersect is not empty、Disjoint、Complementary relationships are all bet365 bonus code louisiana invariants。Tarski suggested that objects that do not change under any transformation are bet365 bonus code louisiana constants。Interesting thing，If the above Russellian type theory model is used as the background model of transformation，Belonging relationship is a bet365 bonus code louisiana invariant，Due to the belonging relationship between objects at adjacent levels, their images will maintain the belonging relationship under any transformation。In the context of the “tiled” set-theoretic universe model，Belongs to a relationship and is not a bet365 bonus code louisiana constant。

Tarski’s theory of bet365 bonus code louisiana succession has become a basic tool for modern bet365 bonus code louisiana research，Widely accepted。In recent years，McGee、Logicians such as Sher and MacFarlane are discussing how to exploit or modify Tarski’s invariant ideas，To describe common truth function connectives、bet365 bonus code louisiana constants represented by quantifiers and various modal words，And explore the impact of corresponding characterization on the concept of bet365 bonus code louisiana succession。

(Author’s unit: Department of bet365 bonus code louisiana, Wuhan University of Technology)