Symbolize and use trees to test for validity

1.  All logic students work hard.  Some individuals who work hard are successful.  Therefore, some logic students are successful.

(x)(Lx -> Wx), (Ex)(Wx & Sx) |- (Ex)(Lx & Sx)   

(x)(Lx -> Wx)
(Ex)(Wx & Sx)
~(Ex)(Lx & Sx)
(x)(Lx -> ~Sx)
Wa
Sa
La -> Wa
La -> ~Sa
/      \
~La     Wa
/   \      
~La     ~Sa      
   x
invalid

2.  Some logic students work hard.  All individuals who work hard are successful.  Therefore, some logic students are successful.     

(Ex)(Lx & Wx), (x)(Wx -> Sx) |- (Ex)(Lx & Sx)

(Ex)(Lx & Wx)
(x)(Wx -> Sx)
~(Ex)(Lx & Sx)
(x)(Lx -> ~Sx)
La
Wa
Wa -> Sa
La -> ~Sa
/      \
~Wa    Sa
x      /   \
        ~La   ~Sa
         x       x
valid

3.  No students are lazy.  Anyone lazy will not do well.  Therefore, every student will do well.    

(x)(Sx -> ~Lx), (x)(Lx -> ~Wx)  |- (x)(Sx -> Wx)

(x)(Sx -> ~Lx)
(x)(Lx -> ~Wx)
~(x)(Sx -> Wx)
(Ex)(Sx & ~Wx)
Sa
~Wa
Sa -> ~La
La -> ~Wa
/    \
~Sa     ~La
 x       /    \
         ~La    ~Wa
               
invalid

4.  No students are lazy.  Anyone not doing well is lazy.  Therefore, every student does well.    

(x)(Sx -> ~Lx), (x)(~Wx -> Lx) |- (x)(Sx -> Wx)

(x)(Sx -> ~Lx)
(x)(~Wx -> Lx)
~(x)(Sx -> Wx)
(Ex)(Sx & ~Wx)
Sa
~Wa
Sa -> ~La
~Wa -> La
/     \
~Sa      ~La
 x          /   \
            Wa  La
             x     x
valid

5.  Every student is doing well.  Anyone lazy does not do well.  Therefore, no students are lazy. 

(x)(Sx -> Wx), (x)(Lx -> ~Wx) |- (x)(Sx -> ~Lx)

(x)(Sx -> Wx)
(x)(Lx -> ~Wx)
~(x)(Sx -> ~Lx)
(Ex)(Sx & Lx)
Sa
La
Sa -> Wa
La -> ~Wa
/    \
~Sa     Wa
 x         /   \
              ~La  ~Wa  
           x      x
valid


A suggested exercise for practice on your own:  provide derivations for the three valid argument forms above.