2

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."

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3

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
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