# Do all contradictory statements entail something self-referential?

"The house is all blue(B), and the house is all white(W)"

Those statements are, ostensibly, not of the form p ⋀ ¬p. However, they do seem to entail p ⋀ ¬p. For example "The house is all blue" entails that the house is not any other colour, including white. So the example entails B ⋀ ¬W ⋀ W ⋀ ¬B.

I can't think of any contrary statement that does not entail something self-referential. Is a self-reference necessary for a contradiction?

• I think you mean contradictory statement and not contrary statement. I don't see how your example is self-referential. – Git Gud Apr 18 '14 at 1:18
• To say that properties blue (B) and white (W) are contraries is simply to say that $B \rightarrow \lnot W$. – Hunan Rostomyan Apr 18 '14 at 1:20
• I may have misused "self-reference". But (B -> ¬W) and (W -> ¬B), so given B ⋀ W, I have B ⋀ ¬W ⋀ W ⋀ ¬B, which is logically equivalent to (B ⋀ ¬B) ⋀ (¬W ⋀ W) - both pairs in the parenthesis seem self-referential. – Hal Apr 18 '14 at 1:39
• @Hal - it is not so; $(B \rightarrow \lnot W)$ is $(\lnot B \lor \lnot W)$ which in turn is $\lnot (B \land W)$. If you apply the transformation to $(W \rightarrow \lnot B)$ you will have the same result : $\lnot (W \land B)$. Thus the two "says" the same thing : $B$ and $W$ are "incompatible". There is no self-reference. – Mauro ALLEGRANZA Apr 18 '14 at 7:11
• @MauroALLEGRANZA Thank you... again =) – Hal Apr 18 '14 at 12:09

$\forall P, x(Px \rightarrow \forall Q(Contrary(Q,P) \rightarrow \lnot Qx)$,
which says, when applied to your example, that for any color $P$ and object $x$ if $x$ is (uniformly) of color $P$, then it's not of any other color that is contrary to $P$ (where contrary is defined as follows:
$Contrary(P,Q) =_{df} \forall x(Px \rightarrow \lnot Qx)$),