The Liar Paradox: Solutions

That certain sentences involving liars are paradoxical has been known for a long time. Perhaps the earliest written formulation of the liar paradox within Western philosophy dates back to the Ancient Greek period: it is due to Eubulides of Miletus, author of several other important logical paradoxes too. If a person says that she is a liar, is what she’s saying true or false? Suppose it’s true: then the person is not a liar; but then it’s actually false that she is a liar.

If instead what the person is saying is false, then she is a liar; but then what she is saying is true. Either way, we reach conclusions that seem – literally – incredible; that’s the mark of paradoxicality. In this article we will concentrate on the solutions to the paradox.

What To Fix?
In order to understand how the liar paradox can be addressed, we should look at its components. The key ingredients of a liar paradox are two: a truth predicate and a principle of truth. In classical logic, the truth predicate is a function that is either satisfied or it’s not. That is, we have two truth values, true and false: each and every sentence has to be assigned one or the other, and cannot be assigned both or none. The truth predicate, instead, is that predicate we use to express the truth of a sentence within a language: typically, logicians have endorsed a principle that was captured by Alfred Tarski during the last century, which goes under the name of disquotational principle.

According to it, the sentence "The water is cold" is true just when, indeed, the water is cold. That is, we ought to accept as true only those sentences that we would be willing to assert.

Now, the liar paradox pressures us to rethink the thesis that every sentence is either true or false, or how the truth predicate is applied to evaluate sentences, or both of those aspects of our conception of truth.

Both True and False
A first type of solution is the so-called paraconsistent one: the liar paradox is simply pointing out that, in some circumstances, we should accept that a sentence is both true and false. This is of course contradictory, but endorsing a contradiction at some point means not to accept contradictions throughout.

This position also goes under the name of dialetheism, and has been accounted for to a great extent over the past few years.

Neither True Nor False
Another option is the so-called para-complete view. According to it, not every sentence is either true or false; some sentences are neither. What we should do with liar-style assertions, hence, is simply to suspend judgment regarding truth or falsity.

Fixing Classical Logic
Plenty of authors have also attempted to fix classical logic by revising the definition of the truth predicate in a way that would accommodate the problematic cases, such as the liar paradox. The solutions in this herd are hard to grasp without entering into more details of the logic of truth predicates than what we can do here. This is one of the most fertile areas of research in logic of the past few decades, which has contributed developments that are significant for the field at large.

Embracing Inconsistency
Finally, some authors have tried to develop a semantics compatible with a view of language as inconsistent. Although intuitively this remark may seem quite right and promising, perhaps even obvious, on the other hand to have a credible theory of semantic inconsistency is laborious.

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