To validate the $ \exists x [E(x) \land C(x)]$

Question

Show the validity of

(1) $\forall x [M(x) \implies C(x)]$

(2) $\exists x[M(x) \land H(x)]$

(3) $\forall x [E(x) \implies H(x)]$

so, (4) $ \exists x [E(x) \land C(x)]$

============================================

I try to simply them to

(5) $M(c) \implies C(c) $ ............. ui

(6) $M(c) \land H(c) $ ...............ei

(7) $E(c) \implies H(c)$ .........ui

(8) $C(c) \land H(c)$ ..........(4) & (5)

now I am stuck on step 8. Can anyone help?

Thanks in advanced!!

Answer 1

The argument would be valid if you had, for the third premise:

$$(3)\;\;\forall x\,(H(x) \rightarrow E(x))$$

Otherwise, one cannot validly derive the desired conclusion.

This is what things would look like with the third premise as posted here:

(1) $\forall x [M(x) \implies C(x)]$

(2) $\exists x[M(x) \land H(x)]$

(3) $\forall x [H(x) \implies E(x)]$

(5) $M(c) \implies C(c) $ ............. ui

(6) $M(c) \land H(c) $ ...............ei (c)

(7) $H(c) \implies E(c)$ .........ui

(8) $M(c)$ ........ (6) simplification ($\land$ - Elimination)

(9) $H(c)$ .........(6) simplification ($\land$ - Elimination)

(10) $C(c)$ .......(5, 8) modus ponens

(11) $E(c)$ .......(7, 9) modus ponens

(12) $C(c) \land H(c)$ ..........(10, 11) $\land$ - Introduction

Now we are justified in using existential introduction:

(13) $\exists x\,[C(x) \land H(x)]$.