validity of conditional proof

Question

In many cases, many mathematical proofs tacitly use conditional proof(CP).

Including logic books, I've never seen why we can use CP. can somebody suggest some proof for validity of CP in general? or just recommendation books will be fine.

Answer 1

A conditional proof is a proof where, having derived $B$ from assumption $A$, we conclude with :

$A \to B$.

This proof is valid because form the definition of logical consequence we have that :

if $\Gamma \cup \{ \alpha \} \vDash \beta$, then $\Gamma \vDash \alpha \to \beta$.

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If for simplicity we consider propositiona logic, we have that :

$\Gamma$ logically implies $\sigma$ (written : $\Gamma \vDash \sigma$) iff every truth assignment for the sentence symbols in $\Gamma$ and $\sigma$ that satisfies every member of $\Gamma$ also satisfies $\sigma$.

Thus, considering : $\Gamma \cup \{ \alpha \} \vDash \beta$, we have that every truth assignment for the sentence symbols in $\Gamma, \alpha$ and $\beta$ that satisfies every member of $\Gamma$ and $\alpha$ also satisfies $\beta$.

Consider now a truth assignement $v$ that satisfies every member of $\Gamma$. We have two cases :

(i) if $v$ does not satisfy $\alpha$, then it satisies $\alpha \to \beta$ [see truth table for $\to$];

(ii) if instead $v$ satisfies $\alpha$, then by the assumption it satisfies also $\beta$, and so $v$ satisfies $\alpha \to \beta$.

In any case, $v$ satisfies $\alpha \to \beta$, and thus we have $\Gamma \vDash \alpha \to \beta$.