What is the difference between ⊢ and ⊨?

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The symbols ⊢ and ⊨ represent distinct concepts in formal logic. The expression A ⊢ B indicates that B can be syntactically derived from A, meaning there exists a proof of B based on A through a series of inference rules. Conversely, M ⊨ N signifies that M serves as a model for N, where M is an interpretation that makes all well-formed formulas (wffs) in N true. The former is concerned with provability, while the latter focuses on the semantics of truth.

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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.
 

Thread 'My basic understanding of set theory'

If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those...
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