Quantifier Distribution Principles

  We will state these principles as implications and show corresponding Beta Graphs. Diagrams on the left were prepared in Adobe Illustrator; those on the right are done in .
   
1

(x)(Ax Bx) ((x)Ax (x)Bx)

 
 
2 (x)(Ax Bx) ((x)Ax (x)Bx)
 
 
3 ((x)Ax (x)Bx) (x)(Ax Bx)
 
 
4 (($x)Ax ($x)Bx) ($x)(Ax Bx)
 
 
5 ($x)(Ax Bx) (($x)Ax ($x)Bx)
 
 
6 ((x)Ax ($x)Bx) ($x)(Ax Bx)
 
 
7 ($x)(Ax Bx) ((x)Ax ($x)Bx)
 
 
8 (($x)Ax (x)Bx) (x)(Ax Bx)
 
 
   
  Some Useful Alpha-provable Graphs
   
9 ((P Q) (P R ) P ) (Q R)
 
 
10 ((Q P) (R   P)   (Q R)) P