I have to proof the following statement.
Let $L$ be a language of propositional logic where formulas are built only from atomic formulas using the primitive connectives $¬$, $∧$, $∨$, $→$, and $↔$.
Let $A$ be a formula of $L$ containing no occurrence of $¬$ and $I$ an interpretation assigning to all atomic formulas of $A$ the truth value $1$.
Then, $A$ is true under $I$.
I wanted to archieve this using structural induction and I would like somebody to review it.
Let $P ⊆ L$ be a set containing propositional letters. The set $F$ of (propositional logic) formulas over $P$ without occurrence of $¬$ is defined recursively by:
(1) If $p ∈ P$, then $p$ is a formula over $F$.
(2) If $β$ and $γ$ are formulas over $F$, then so are:
- ($β ∧ γ$)
- ($β ∨ γ$)
- ($β → γ$)
- ($β ↔ γ$)
Suppose there is an interpretation $I$ such that for every propositional letter $α ∈ P$ that $I(α) = 1$.
Then the following holds:
(1) If $p ∈ P$, then $I(p) = 1$.
(2) Assume $β, γ ∈ F$ and $I(β) = I(γ) = 1$. Therefore the following applies for $β, γ ∈ F$.
- $I(β ∧ γ) = 1$
- $I(β ∨ γ) = 1$
- $I(β → γ) = 1$
- $I(β ↔ γ) = 1$
Therefore $A ∈ F$ is true under $I$.
Is this correct?