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Most of my reading has lead me to the conclusion that, all statements are either true or false, but they cannot be both or neither. It's just sort of an on/off thing. If so, how do paradoxes exist. The liar paradox takes a statement, "this statement is false," and just turns it into a big mess. Wouldn't that mean there are three possible truth values: true, false, and a sort of null value?

N.D.H.
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    All statements are true or false, never neither nor both. That sentence is both both and neither true and false, therefore it cannot be a statement. Hence drawing conclusions about truth and falsity from it are impossible, because it can't be suborned to the logic necessary. – Nij Jan 08 '17 at 00:03
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    In "natural" languages not all statements are either true or false : "Shut up !" is not... In "formal languages" we usually restricts ourselves to sentences that have a unique truth value: true or false (usually called : propositions). – Mauro ALLEGRANZA Jan 08 '17 at 09:19
  • See the answer I just posted at the duplicate question, and in particular the linked post at http://math.stackexchange.com/a/1888389 which shows that it is feasible to analyze all the well-known paradoxes and conclude that they fall into a truth-value gap of some kind. Kripke's notion of groundedness is similar to mine and based on the same idea that we should distinguish statements about reality from other kinds of statements. Of course, the details differ but at least you know where to start looking. =) – user21820 Jan 08 '17 at 15:09

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