What is the difference between ⊢ and ⊨?

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In summary, the difference between ⊢ and ⊨ is that the former deals with provability and syntactic relationships between sentences, while the latter deals with truth and semantic relationships between sentences. ⊢ is used to indicate that one sentence can be deduced or proved from another, while ⊨ is used to indicate that one sentence is true in a particular model or interpretation.
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What is the difference between ⊢ and ⊨?

how to call them?
 
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steelcat said:
What is the difference between ⊢ and ⊨?

how to call them?

Maybe someone else can add more, but:

AFAIK A ⊢ B means that B can be deduced/derived from A, or there is a proof of B from A.

And M ⊨ N means M is a model for N , i.e., M is an interpretation in which all wffs in N are mapped into truths. Look up the meaning of interpretation.

The first '⊢' is syntactic, dealing with provability, so that B can be deduced/proved from A, i.e., there is a finite collection of sentences starting with A, where each is either a theorem, or follows from previous sentences by the application of some inference rule(s), the last sentence of which is B. 'A' on the left may be the empty set, in which case B is a tautology.
This deals only with formal relations between sentences, and not with their actual inner content.

The second , i.e., '⊨' deals with semantics, or the notion of truth. Informally, M is a possible world in which the wffs are all feasible/realizable. This deals with truth.
 

What is the difference between ⊢ and ⊨?

The symbols ⊢ and ⊨ are used in mathematical logic to represent different types of logical entailment. Logical entailment is a relationship between statements, where one statement follows logically from another. The main difference between these symbols is their level of strictness.

What does the symbol ⊢ mean?

The symbol ⊢ is read as "proves" or "derives". It is used to indicate that a statement can be deduced or derived from a set of premises using a particular logical system. This type of entailment is called syntactic entailment because it is based on the structure of the logical system.

What does the symbol ⊨ mean?

The symbol ⊨ is read as "entails". It is used to indicate that a statement is logically entailed by a set of premises. This type of entailment is called semantic entailment because it is based on the meaning or interpretation of the statements.

What is the difference between syntactic and semantic entailment?

Syntactic entailment is based on the structure and rules of a logical system, while semantic entailment is based on the meaning and interpretation of the statements. In other words, syntactic entailment is concerned with the form of an argument, while semantic entailment is concerned with the content.

Which is more strict, ⊢ or ⊨?

⊢ is more strict than ⊨ because it is based on the rules and structure of a logical system. If a statement is syntactically entailed, it can be derived or proven using the rules of the system. Semantic entailment is less strict because it is based on the interpretation of the statements, which can vary depending on the context or perspective.

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