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Properties of ąuantifiers

The commutatiye properties of ąuantifiers

Vx Vy p(x, y) <==> Vy Vxp(x, y);

Bx Byp(x, y) <==>By Bxp(x, y).

But the order of universal and existential ąuantifiers

cannot be

changed.

Vx By p(x, y) <== By Vxp(x> y).

Example: x, y Vx By x 6 y <=By Vx x6 y . 1 *0 Logic

The Morgan’s Laws for ąuantifiers

~ Vx p{x) <==>Bx t?(x);

~Bx p{x) <=$ Vx ~p(x).

Example: x, y

A Vx Byx = y2) OBx Vyx6=y2 Change of the rangę of ąuantifiers

Vx (p(x) =łg(x)) <=$ Vp(x) g(x) - reduction of rangę of the

universal ąuantifier;

Bx (p(x) Ag(x)) &Bp(x) g(x) - reduction of rangę of the

existentialquantifier.

Logic

Distributiye Laws for ąuantifiers