Is the next statement correct: (A∧B)⊨(A⇔B) ?
Formal definition of entailment is this: α⊨β if and only if, in every model in which α is true, β is also true.
I used a truth table to show that there is only one model {A=True, B=True} where (A∧B)=True. (A⇔B) is also True in this model. It means (A∧B)⊨(A⇔B) is correct.
There exist model2 = {A=False, B=False}, (A⇔B) is True in model2. But (A∧B) is not True in model2.
Can you please provide intuition on what does it mean? Does it extend our knowledge base (A∧B) if we add a new statement (A⇔B)?
Best Answer
The statement is correct and you have shown it via truth table. Knowing that $A \land B$ implies that both $A$ and $B$ are true, thus $A \iff B$ follows, i.e. it does not extend our knowledge base if $A \iff B$ is added as a new statement, since we already know both $A$ and $B$ are true.
There is a different way to derive $A\iff B$ from $A\land B$, that is, via natural deduction:
\begin{align} 1. &\quad A\land B&\text{Premise}\\ 2. &\quad A &\text{Simplification}\\ 3. &\quad B &\text{Simplification}\\ 4. &\quad \neg B \lor A &\text{Conjunction}\\ 5. &\quad \neg A \lor B &\text{Conjunction}\\ 6. &\quad B \implies A &\text{Material Implication}\\ 7. &\quad A \implies B &\text{Material Implication}\\ 8. &\quad A \iff B &\text{Material Equivalence} \end{align}
or, if your definition of biconditional includes the alternate definition $(A \land B)\lor (\neg A \land \neg B)$ (instead of only the standard $(A \implies B) \land (B \implies A)$), then the conclusion can be drawn from one simple use of the Conjunction rule.