Strengthening the Consequent: From A implies B, infer A implies (B ^ C).

Question

Strengthening the Consequent: From A implies B, infer A implies (B ^ C).

How do I construct a Fitch style proof to prove this?

Answer 1

Not true:

If $A$ true, $B$ true and $C$ false, then $A\Longrightarrow B$ true, $B\wedge C$ false, and hence $A\Longrightarrow (B\wedge C)$ false.

Answer 2

You are probably thinking of proving the following theorem:

$\vdash (A \rightarrow B) \rightarrow (A \rightarrow B \vee C)$

If this is the case, you can just

(1) assume $A \rightarrow B$;

(2) assume A;

(3) derive B by modus-ponens;

(4) derive $B \vee C$ by $\vee I$;

(5) derive $A \rightarrow (B \vee C$) by $\rightarrow I$ (assumption 2);

(6) derive $(A \rightarrow B) \rightarrow (A \rightarrow (B \vee C$)) by $\rightarrow I$ (assumption 1).

However, if you do really want to prove $(A \rightarrow B) \rightarrow (A \rightarrow (B \wedge C$)) it is no use, since $\nvdash (A \rightarrow B) \rightarrow (A \rightarrow (B \wedge C$)), i.e. it's not a theorem in the propositional calculus (if you want, you can confirm it by a truth table calculus).