The symbol ⊨, also written as ⊧, is called the double turnstile in logic.
Understanding the Double Turnstile
The double turnstile (⊨) is a crucial symbol in mathematical logic, particularly in semantics. Here's a breakdown of its meaning and usage:
- Meaning: The double turnstile (⊨) represents a semantic consequence or entailment. It indicates that the truth of one statement is a consequence of the truth of another statement or set of statements.
- Pronunciation: It's commonly read as "entails," "models," "is a semantic consequence of," or "is stronger than."
- Relation to the Single Turnstile: The double turnstile (⊨) is closely related to the single turnstile symbol (⊢). While ⊨ denotes semantic consequence, ⊢ denotes syntactic consequence. In simple terms, ⊨ deals with the meaning and truth of statements, while ⊢ deals with the rules of inference and logical deduction.
Semantic Consequence Explained
The double turnstile (⊨) is typically used in the following format:
-
A ⊨ B
This is read as "A semantically entails B" or "B is a semantic consequence of A."
It means that if A is true, then B must also be true.
It is related to the truth of the statements, rather than their deduction. -
Γ ⊨ φ
This indicates that the formula (or statement) φ is a semantic consequence of the set of formulas (or statements) Γ. In other words, for every possible model (or interpretation) that satisfies all the formulas in Γ, φ is also satisfied in the same model.
Examples
Here are some illustrative examples of how to use the double turnstile:
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Propositional Logic:
- Let A represent "It is raining" and B represent "The ground is wet."
- The assertion (A ⊨ B) could only be true if the semantics of A and B made it so. If we defined 'it is raining' as 'water is falling from the sky' and 'the ground is wet' as 'there is water on the ground', and water on the ground can only come from the sky, this would hold. If however, water on the ground could also come from an underground spring, the implication would not be true.
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Predicate Logic:
- Let P(x) mean "x is a dog" and Q(x) mean "x is an animal."
- The assertion "∀x (P(x) → Q(x)) ⊨ ∀x (P(x) → Q(x)) " (i.e. All dogs are animals) shows a tautological semantic implication.
Key Differences
Here’s a table highlighting the differences between the double turnstile and the single turnstile:
| Feature | Double Turnstile (⊨) | Single Turnstile (⊢) |
|---|---|---|
| Symbol | ⊨ | ⊢ |
| Concept | Semantic Consequence | Syntactic Consequence |
| Focus | Truth & Meaning | Rules of Inference |
| Interpretation | "Entails," "Models" | "Derives," "Proves" |
Importance in Logic
The double turnstile plays a critical role in:
- Model Theory: It provides a means for relating formulas and models, helping us understand the meaning of sentences in different contexts.
- Soundness and Completeness: The notions of soundness and completeness are defined in relation to these two symbols.
- Formal Verification: Ensuring the validity of logical arguments and the correctness of software and hardware designs.
In conclusion, the double turnstile (⊨) is a vital symbol in logic that signifies semantic consequence, focusing on truth and meaning, and is essential in model theory and formal logic.
