Posts Tagged ‘Axioms of Belief’

Logical Analysis of Consciously Held Beliefs

Posted by allzermalmer on June 10, 2013

Axioms of Consciously Believing

Axiom: “I believe that p” if and only if “I believe that I believe that p”
EBpBBp = Bp <-> BBp

Axiom: “I don’t believe that p” if and only if “I believe that I don’t believe that p”
ENBpBNBp = ~Bp <-> B~Bp

Axiom: If “I believe that not p” then “I don’t believe that p”
CBNpNBp = B~p -> ~Bp

Axiom: If “I believe that p implies y” then “I believe that p implies I believe that y”
CBCpyCBpBy = B(p->y) -> (Bp->By)

Theorems of Consciously Believing

T1: “I don’t believe that both p and not p”
NBKpNp = ~B(p&~P)

T2: “I believe that p is equivalent to y” implies “I believe that p is equivalent to I believe that y”.
CBEpyEBpBy = B(p<->y) -> (Bp<->By)

T3: “I believe that p or I believe that y” implies “I believe that both p or y”
CABpByBApy = (BpvBy) -> B(pvy)

T4: “I believe that both p & y” if and only if “I believe that p & I believe that y”
EBKpyKBpBy = B(p&y) <-> (Bp&By)

T5: “I believe that p” implies “I don’t believe that not p”
CBpNBNp = Bp->~B~p

T6: “I believe that either p or y & I don’t believe that p” implies “I don’t believe not p”
CKBApyNBpNBNp = B(pvy)&~Bp -> ~B~p

T7: “I believe that either p or y & I believe that not p” implies “I believe that y”
CKBApyBNpBy = B(pvy)&B~p -> By

T8: If “I believe that p implies y” then “I believe that not y implies I believe that not p”
CBCpyCB~yB~p = B(p->y) -> (B~y->B~p)

T9: If “I don’t believe that p implies y” then “I don’t believe that not p & I don’t believe that y”.
CNBCpyKNBNpNBy = ~B(p->y) -> (~B~p&~By)

T10: If “I believe that I believe that p or I believe that y” then “I believe that either p or y”
CBABpByBApy = B(BpvBy) -> B(pvy)

T11: If “I believe that p implies y” then “I believe that I believe that p implies y”
CBCpyBCBpy = B(p->y) -> B(Bp->y)

T12: “I believe that I believe that p implies p”
BCBpp = B(Bp->p)