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"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?

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  • $\begingroup$ I think you mean contradictory statement and not contrary statement. I don't see how your example is self-referential. $\endgroup$ – Git Gud Apr 18 '14 at 1:18
  • $\begingroup$ To say that properties blue (B) and white (W) are contraries is simply to say that $B \rightarrow \lnot W$. $\endgroup$ – Hunan Rostomyan Apr 18 '14 at 1:20
  • $\begingroup$ 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. $\endgroup$ – Hal Apr 18 '14 at 1:39
  • $\begingroup$ @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. $\endgroup$ – Mauro ALLEGRANZA Apr 18 '14 at 7:11
  • $\begingroup$ @MauroALLEGRANZA Thank you... again =) $\endgroup$ – Hal Apr 18 '14 at 12:09
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If you start with the following postulate:

$\forall P, x(Px \rightarrow \forall Q(Contrary(Q,P) \rightarrow \lnot Qx)$,

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:

$Contrary(P,Q) =_{df} \forall x(Px \rightarrow \lnot Qx)$),

then your example does lead to a contradiction: if the house is all white and all black, then since "all white" and "all black" are contraries, if the house is all white then it's not all black and vice versa, so we have a contradiction.

What I don't understand is what any of this has to do with self-reference.

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