The symbol ⊢, often called a "turnstile," is a fundamental notation in mathematical logic and computer science, and it signifies a relationship of derivation or consequence. It can be read in several ways, with the most common interpretations being "yields," "proves," "satisfies," or "entails." Its name comes from its visual similarity to a turnstile when viewed from above.
Understanding the Turnstile Symbol
The turnstile symbol (⊢) is essential in formal systems and is used to represent inference or deduction. It helps describe how one statement or set of statements leads to another.
Meanings of the Turnstile
| Meaning | Description |
|---|---|
| Yields | Indicates that a statement or formula can be derived as a consequence of given premises. |
| Proves | States that a logical deduction exists to show a statement is true based on the assumptions. |
| Satisfies | Used in model theory, where it means that a formula is true in a certain structure or interpretation. |
| Entails | Expresses a relation between sets of statements where one set necessarily leads to the truth of the other. |
Usage in Logical Systems
-
Inference Rules: The turnstile is used in the context of formal logical systems to express inference rules. For instance,
A, B ⊢ Cmight mean "from statements A and B, we can derive C." -
Logical Validity: The notation
⊢ Pcan mean that the statementPis a theorem, that is, it can be derived from the axioms of the logical system. -
Model Theory: In model theory, we use
M ⊨ Pwhere⊨is the semantic entailment symbol to indicate that in modelMthe formulaPis true; This is related to the turnstile ⊢ symbol when it is used to represent satisfaction.
Example
Consider the following scenario:
- Premise 1: All men are mortal (∀x(Man(x) → Mortal(x))).
- Premise 2: Socrates is a man (Man(Socrates)).
Using the turnstile, we can represent the conclusion as:
∀x(Man(x) → Mortal(x)), Man(Socrates) ⊢ Mortal(Socrates)
This can be read as: “Given that all men are mortal and Socrates is a man, it yields that Socrates is mortal.”
Practical Insights
- The turnstile is not a connective like “and” or “or,” but rather a meta-logical symbol.
- It describes a relationship between statements, not the truth or falsity of those statements.
- Understanding the turnstile is crucial for working with formal systems, automated reasoning, and logic programming.
Summary
The symbol ⊢, the turnstile, represents a relation of derivation or consequence, often read as "yields," "proves," "satisfies" or "entails." It plays a crucial role in formal logic, showing how conclusions are drawn from premises.
