More Query Logic

So last night, I talked about the 4 chat rooms. Two being equivalent to the atomic concepts of True and False, the other two being equivalent to the unary functions NOT and IDENTITY. I gave the examples of Russell's paradox being equivalent to NOT, (it negates your assumptions) and I talked about the existence of lonely hearts paradoxes, which agree with your assumption. Off the top of my head one like Russell's paradox: "The set of all sets which contain themselves. I assume this set contains itself" True, it does, by definition. "I assume this set does not contain itself" True, it does not, by definition in assumption. This is a lonely hearts paradox, though these get less attention, poor things. People pay far more attention to snobs.

Anyway, some formalization. Lets get rid of the atomic true and false, and replace them with 4 unary functions. We have Contradiction (=F= atomic concept of False) Tautology (=T= atomic concept of True) Not(=N), and Identity(=I). Each of these is queried with an assumption of their values. The first two act like T and false, so if the second argument of q is the query, then q(T, T)=T, q(T,F)=F, q(F,F)=T, q(F,T) = F. For the snobs, q(N, T) = F, q(N, F)=T and lonely hearts q(I, T) = T q(I, F) = F. So since we are treating these 4 functions like atomic values, then they need to be defined for the others. T and F are both "honest" functions, we can trust them to answer that they are not N or I if they were queried. But can we trust the liars? Will the lonely hearts tell us that they are lonely? Will the snobs admit to being snobs? I think both the snobs and lonely hearts would respond with meta-honesty, which is to say, they would respond be being snobs and lonely hearts respectively. So the query Q(N, N) = N, Q(I, I) = I. In english:

"Is it the case that you are snobs?"

"well..."

"C'mon, its true isn't it?"

"No!"

or

"C'mon its a lie, you aren't snobs are you?"

"Actually we ARE!"

Which is to say, Q(Q(N,N), T) = F, Q(Q(N,N), F) = T Our liars are consistent in their lies.

So you can draw out a 4*4 truth table for TFNI, I haven't yet defined Q(T, I) for instance, maybe its more interesting than T and F being honest, but it really doesn't matter. The key idea of query logic is that while V() is final in its answer, Q() is not. if V(e) = T, then V(V(e)) = T. However, Q(N, Q(N, T)) != Q(N, T). The first is NOT NOT T, the second is NOT T.

So back to pragmatism, what's the big idea? De-construct a person thinking about a paradox. The define it "everything I say is a lie" then they observe it. "OMG! If its true, then its false, and if its false, then its true!" De-construct that last part of what they said; two if-then implications, of the form condition-implies-outcome, input output, a one way direction. The input is an ASSUMPTION (in caps because its that important) and the outcome is a logical conclusion of the assumption, the result of the evaluation or observation of the thing defined. Who holds the assumption? The observer. What is the result of the observation? It is dependent on the assumption, or the state of the observer. So the state of the observer and the system observed are entangled, that's one way to say it. A simple metaphor is that a paradox is like a mirror, what you see in it depends on the state you are in, what you get out depends on what you put in.

So wrapping this up for now, V(e) = Q(e, T) where e is any logical expression made up of T, F, or any of the traditional logical operators. (The principle of the excluded middle holds for v logic, so we never have to ask Q(e, F)). So v-logic is in query logic. Outside of that though, the wasteland.