# The Liar Paradox and the Measurement Problem

Just a quick observation.  I think that a telling analogy can be drawn between the measurement problem in quantum mechanics (QM) and the liar paradox.  This aside is just meant to draw out those intuitions.

The liar paradox amounts to more or less the following statement.   ‘This sentence is false.’  You know the drill.  If the sentence is true, then it must be false (for it asserts its own falsity).  And if the sentence is false, then it must be true (for the negation of its falsity is its truth).  In light of this (and to avoid infinitely ‘looping’ through the truth-values of the sentence), we say that we cannot ascribe any truth-value to the sentence at all, dub it a paradox, and call it a day.

Another account of the measurement problem can be found here.  Nevertheless, here’s the gist.  The dynamic equations of motion (DEM) (the third axiom of QM) are thought to certainly determine the states and motions of all particles (all states and motions are calculable [via the Schroedinger equation]).  By DEM, if we measure the color of a hard electron, the measurement outcome should be in a superposition of being both black and white.  But this isn’t actually what happens (and is where the fifth axiom of QM comes in).  The measurement outcome is always either definitely black or definitely white (with each result have a probability of exactly .5).  (Somehow measurement ‘disrupts’ the outcome, collapsing the superpositional state into just one of its terms.)

Say our goal is to identify what ‘the liar paradox’ would look like in a physical, rather than linguistic, system.  Superposition seems like a good candidate.  When the hard electron is going through the color box, it is in a superposition of being both black and white.  But we only really understand what superposition means in a negative sense.  An electron in a superposition of being black and white is not black, nor is it white, and it is not definitely both black and white, but nor can it be neither — and what that means, we don’t really know (so we introduce the term, ‘superposition’).  In one sense, it seems that, prior to the measurement outcome, a color-property simply cannot even be predicated of the electron.  With regard to its color, literally nothing can be said.  (Until it emerges from the device, but this isn’t as relevant.)

And this starts to look like a liar paradox.  We refuse to ascribe a truth-value to ‘this sentence is false,’ in the same way we refuse to ascribe a color-property to the hard electron going through the color box.

DEM says that the result of a measurement is superposition, but the collapse postulate predicts a probalistic outcome of .5 for black.  (And how could you ever even see a superposition?)  Suppose the outcome is black.  When you measure a second hard electron, the outcome will necessarily be white.  And when you measure a third, the outcome will be black.1  This sounds like saying: suppose ‘this sentence is false’ is true.  Evaluate the truth-value of the sentence again; it must be false.  And on the next evaluation, it must be true.

The difference between the two is that, empirically, measurements must have outcomes, while the liar paradox doesn’t demand a truth-value in the same way — we can reserve our judgment.

1. We’re fudging a bit here, but bear with me.