Question
Download Solution PDFThe mathematical notation to describe logical entailment of a scentence "α entails another sentence β" is
Answer (Detailed Solution Below)
Option 1 : α \(⊨\) β
Detailed Solution
Download Solution PDFCorrect 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