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 .

(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