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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? If we take this approach, then a natural question is "Which statements are meaningless? However, there are a few paradoxes that complicate this. 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. 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:. 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?
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that he or she is lying: for instance, declaring. The Liar Paradox is an argument that arrives at a contradiction by reasoning about a Liar Sentence. The Classical Liar Sentence is the self-referential sentence.
Is it invalid because of infinite recursion? Is it both true and false? These questions all miss the point. What has gone wrong?
Must we revise our notion of truth and our logic? Or can we dispel the common conviction that there are such self-referential sentences? The present study explores the second path.
Detailed semantico-metaphysical arguments show that in this dynamic setting, an object can be referred to only after it has started to exist. Hence the circular reference needed in the Liar paradox cannot occur, after all.
As this solution is contextualist, it evades the expressibility problems of other proposals. More Options Prices excl.
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