Question

Give a formal proof for the following tautology by using the CP rule.

(A →(B →C)) ^ B →(A →C)

Answer 1

Solution for the problem is provided below, please comment if any doubts:

Note: Since the solution contains equations, to avoid format loss, U added the screenshot of the solution, raw data is also included at the end

Raw data:

The tautology proof using (CP) rules is following the logic to prove , start From R and derive S using rules, then we can say that R→S, R is the left hand side term and S is the right hand side term.

Here LHS=> (A →(B →C)) ^ B

And RHS => (A →C)

So to prove the tautology, derive “(A →C)” from “(A →(B →C)) ^ B” to prove the tautology

1. B →C                       :Given premise(P)
2. B                              : Given premise(P)
3. C                              :1, 2, Modus Ponens(MP) rule [B →C, B => C]
4. ¬ A v C                    :3, addition
5. A →C                      :4, Implication

QED                            :1-5 CP rule

Hence prove the tautology, “(A →(B →C)) ^ →B(A →C) using CP rule

Note: QED is used to represent follow the proof