# A question about statements and conclusion

Consider the following statements:

All actors are dancers

No dancer is a singer

Some singers are musicians

Conclusion -

• 1.Some actors being musicians is a possibility
• 2.Some singers being actors is a possibility.

The options are:

• only 1 follows
• only 2 follows
• either 1 or 2 follows
• neither follows
• both follow

The answer is 1, just "1 follows."

It's possible for actors to be musicians because there's no rule constraining musicians in any way. However the second can be proven to be not possible.

1. If someone were a singer, that would rule out being a dancer.
2. Since that person couldn't be a dancer, being an actor would be ruled out.
3. Therefore, nobody can be both a singer and an actor.

This can be proven formally as follows:

```{1}         1.   ∀x[Ax → Dx]                     Prem.
{2}         2.   ~Ǝx[Dx & Sx]                    Prem.
{2}         3.   ∀x[~(Dx & Sx)]                  2 QI
{2}         4.   ∀x[~Dx ∨ ~Sx]                   3 DM
{2}         5.   ∀x[Dx → ~Sx]                    4 MI
{6}         6.   Sa                              Assum.
{1}         7.   Aa → Da                         1 UE
{2}         8.   Da → ~Sa                        5 UE
{2,6}       9.   ~Da                             6,8 MT
{1,2,6}     10.  ~Aa                             7,8 MT
{1,2}       11.  Sa → ~Aa                        6,10 CP
{1,2}       12.  ∀x[Sa → ~Aa]                    11 UI
{1,2}       13.  ∀x[~Sa ∨ ~Aa]                   12 MI
{1,2}       14.  ∀x[~(Sa & Aa)]                  13 DM
{1,2}       15.  ~Ǝx[Sa & Aa]                    14 QI
```