Solution

Correct Answer: Option 1) α ⊨ β

Key Points

• Logical entailment refers to the relationship between two statements (formulas) where one logically follows from the other.
• Notation: α ⊨ β means: "In every model (or interpretation) in which α is true, β must also be true."
• This is a fundamental concept in formal logic and model theory, particularly in propositional and predicate logic.
• The symbol "⊨" is known as the double turnstile, used specifically to denote semantic entailment.

 Example:

• Let α = "It is raining", and β = "The ground is wet".
• If α ⊨ β, this means that whenever it is raining (α is true), the ground being wet (β) is also always true in that scenario.

 Why Other Options Are Incorrect:

• Option 2) α ⊆ β: This is set notation, meaning "α is a subset of β". It is used in set theory, not in logic for entailment.
• Option 3) β ⊨ α: This means "β entails α", which is the reverse direction of what is asked. This would imply α is a logical consequence of β, which is not the case here.
• Option 4) β ⊆ α: Again, subset notation, and reversed direction. Not relevant to logical entailment in this context.

 Summary:

• Entailment in logic is directional: α ⊨ β says β follows logically from α.
• Correct formal representation uses ⊨, not set inclusion ⊆.

✅ Final Answer: Option 1) α ⊨ β