Truth and the Liar Paradox

by casebash4 min read2nd Sep 201445 comments


Truth, Semantics, & Meaning
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Related: The map is not the territory, Unresolved questions in philosophy part 1: The Liar paradox

A well-known brainteaser asks about the truth of the statement "this statement is false". If the statement is true, then the sentence must be false, but if it false then the sentence must be true. This paradox, far from being just a game, illustrates a question fundamental to understanding the nature of truth itself.

A number of different solutions have been proposed to this paradox (and the closely related Epimenides paradoxPinocchio paradox). One approach is to reject the principal of bivalence - that every proposition must be true or false - and argue that this statement is neither true nor false. Unfortunately, this approach fails to resolve the truth of "this statement is not true". A second approach called Dialetheism is to argue that it should be both true and false, but this fails on "this statement is only false".

Arthur Prior's resolution it to claim that each statement implicitly asserts its own truth, so that "this statement is false" becomes "this statement is false and this statement is true". This later statement is clearly false. There do appear to be some advantages to constructing a system where each statement asserts its own truth, but the normative claim that truth should always be constructed in this manner seems to be hard to justify.

Another solution (non-cognitivism) is to deny that these statement have any truth content at all, similar to meaningless statements ("Are you a?") or non-propositional statements like commands ("Get me some milk?"). If we take this approach, then a natural question is "Which statements are meaningless?" One answer is to exclude all statements that are self referential. However, there are a few paradoxes that complicate this. One is the Card paradox where the front says that the sentence on the back is true and the back says that the sentence on the front is false. Another is Quine's paradox - ""Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation". One other common example is: "The statement on the blackboard in Carslaw Room 201 is false". The Card paradox and blackboard paradox are interesting in that if we declare the Liar paradox to be meaningless, these paradoxes are meaningless or meaningful depending on the state of the world.

This problem has been previously discussed on Less Wrong, but I think that there is more that is worth being said on this topic. Cousin_it noted that the formalist school of philosophy (in maths) believes that "meaningful questions have to be phrased in terms of finite computational processes". Yvain took a similar approach arguing that "you can't use a truth-function to evaluate the truth of a noun until you unpack the noun into a sentence" and that it would require infinite unpacking to evaluate, while "This sentence is in English" would only require a single unpacking.

I'll take a similar approach, but I'll be exploring the notion of truth as a constructed concept. First I'll note that there are at least two different kinds of truth - truth of statements about the world and truth of mathematical concepts. These two kinds of truth are about completely different kinds of objects. The first are true if part of world is in a particular configuration and satisfy bivalence because the world is either in that configuration or not in that configuration.

The second is a constructed system where certain basic axioms start off in the class of true formulas and we have rules of deduction to allow us to add more formulas into this class or to determine that formulas aren't in the class. One particularly interesting class of axiomatic systems has the following deductive rules:

if x is in the true class, then not x is in the false class
if x is in the false class, then not x is in the true class
if not x is in the true class, then x is in the false class
if not x is in the false class, then x is in the true class

If we start with certain primitive propositions defined as true or false and start adding operations like "AND", "OR", "NOT", ect. then we get propositional logic. If we define variables and predicates (functions from variables to boolean values) and "FOR EACH", "THERE EXISTS", ect, then we get first-order predicate logic and later higher order predicate logics. These logics work with the two given deductive rules and avoid a situation where both x and not x are in the true class which would for any non-trivial classical logic lead to all formulas being in the true class, which would not be a useful system.

The system has a binary notion of truth which satisfies the law of excluded model because it was constructed in this manner. Mathematical truth does not exist in its own right, in only exists within a system of logic. Geometry, arithmetic and set theory can all be modelled within the same set-theoretic logic which has the same rules related to truth. But this doesn't mean that truth is a set-theoretic concept - set-theory is only one possible way of modelling these systems which then lets us combine objects from these different domains into the one proposition. Set-theory simply shows us being within the true or false class has similar effects across multiple systems. This explains why we believe that mathematical truth exists - leaving us with no reason to suppose that this kind of "truth" has an inherent meaning. These aren't models of the truth, "truth" is really just a set of useful models with similar properties.

Once we realise this, these paradoxes completely dissolve. What is the truth value of "This statement is false"? Is it Arthur Prior's solution where he infers that the statement asserts its own truth? Is it invalid because of infinite recursion? Is it both true and false? These questions all miss the point. We define a system that puts statements into the true class, false class or whatever other classes that we want. There is no reason to assume that there is one necessarily best way of determining the truth of the statement. The value of this solution is that this dissolves the paradox without philosophically committing ourselves to formalism or Arthur Prior's notion of truth or Dialetheism or any other such system that would be difficult to justify as being "the true solution". Instead we simply have a choice of which system we wish to construct. 

I have also seen a few mentions of Tarski's type hierarchies and Kripke's fixed point theory of truth as resolving the paradox. I can't comment too much because I haven't had time to learn these yet. However, the point of this post is to resolve the paradox without committing us to a specific model of truth, as opposed to the general notion of truth as a construct.

Edit: I removed the discussion of "This statement is true" as it was incorrect (thanks to Manfred). The proper example was, "This statement is either true or false". If it is true, then that works. If it is false, then there is a contradiction. So is it true or is it meaningless given that it doesn't seem to refer to anything? This depends on how we define truth. We can either define truth only for statements that can be unpacked or we can define it for statements that have a single stable value allocation. Either version of truth could work.

Update: Perhaps I should have just argued that truth is constructed at least to some extent. Maybe there is actually a basic fundamental notion of truth for simple statements, but when we get to anything complicated, such as "this statement is false" any system for assigning truth values is simply a construction.