31 January 2022

THE LIAR'S PARADOX (Epimenides's too!)

The Epimenides paradox goes something like this:

“Epimenides the Cretan says, ‘that all the Cretans are liars,’ but Epimenides is himself a Cretan; therefore he is himself a liar. But if he be a liar, what he says is untrue, and consequently the Cretans are veracious; but Epimenides is a Cretan, and therefore what he says is true; saying the Cretans are liars, Epimenides is himself a liar, and what he says is untrue. Thus we may go on alternately proving that Epimenides and the Cretans are truthful and untruthful.” Thomas Fowler, The Elements of Deductive Logic (1869)


Another formulation, the so-called Liar’s Paradox goes: ‘Everything I say is false.’ ‘I am lying.’ 

Do not get lost in the truth-functional contradictions implied by these statements. For when you set truth and falsity aside, these statements convey a surprising amount of information. For example, we establish the assumption that:


            (0.1) There is such a thing as a statement of the language.

(0.2) This is a well-formed statement of the language. {function; copula; predicate}


This is obvious. It is the basis of the game we are playing. But moreover, simply by attempting to decide its ambiguity, we affirm that:


            (1.1) Some statements have truth.

            (1.2) Some statements have falsity.


Then, looking at the paradox and acknowledging its essential contradiction, we conclude that:


            (1.3) Some statements have neither truth nor falsity, and thus

            (1.4) are undecidable to our linguistic understanding.


In a world where truth functions determine meaning:


            (2.1) There is more to information than mere MEANING.


What other information can we glean from this logical paradox (other than attempting to solve it by noting that just because the statement “everything I say is false” is false does not imply that everything else I say is true, or, as is the case with most philosophers, explaining it away by saying that we are applying truth values ambivalently in the language and the metalanguage)? We can ascertain data about the speaker Epimenides, or the so-called Liar (L), who makes these statements:


            (3.1) L can make certain well-formed statements of the language about himself.


Whether they are true or false matters not at this point to us. Thus,


            (3.2) L is not necessarily a reliable witness about himself.


And while we can make no inferences about L’s self-consciousness of the truth or falsity of his statements, we can certainly assert that:


            (3.3) L’s statement sows confusion.


For example, if we imagine a contradiction machine, a machine that can calculate statements logically, then such a paradoxical input statement will disable the machine.

Without any further information about his intentions, we cannot determine whether L actually is a liar or or is merely mistaken or whether he’s intentionally sowing confusing or merely playing a game or whether he’s bullshitting us or is merely confused.


Generally, though, our thinking and thus our understanding of reality and, what’s more, our understanding of who we are is necessarily limited by the language we use. And Epimenides’s paradox here points us to merely one facet of this limitation.


To ask the question of meaning, to ask what it all means, is to ask the wrong question. It is to voluntarily stop at the gates of the prison that constrains us: the prison of language.

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