Liar’s paradox

 Paradox is an interesting mechanism. The teacher asserts, "There is a mouse in this box. What I just said is false." The teacher continues, "But there is a mouse in the box."

 We were listening very attentively, but we were bemused to infer simply that it was false (since that is what the sentence said) or that it was true (since it spoke truly). There might be a mouse in the box, or there might not be a mouse in the box. Once the box is opened, we will be surprised either way, by a mouse or by no mouse. We were not so serious like Philitas, who was so worried over this kind of liar’s paradox that he died of insomnia, as reported by Athenaeus. The teacher was explaining with reference to the quantum superposition of Schrodinger’s cat, which is both dead and alive simultaneously.

 An ancient gravestone on the Greek island of Kos contained this poem, which might be about the difficulty of solving the paradox:

O Stranger! Philetas of Kos am I,

"It was the liar who made me die,

and the bad nights caused thereby."

 None of the sentences in this pair are true.

One often talks about truth with its academic hedges and caveats, but one cannot solve the curious twist of liar’s paradox. The oldest attribution of the liar’s paradox is to Eubulides of Miletus, who said, "A man says that he is lying". Is what he says true or false? The sentences cannot be consistently assigned a truth value, even though they are correct by grammar and semantic rules. If he says, "I am lying," then the liar is telling the truth; if he says the statement is false, then the so-called liar is telling the truth. Thus, the statement is true and false at the same time, consequently leads to contradiction. One way to escape this paradox is to deny the fact of a two-valued logic of true or false.

 Some logicians assert that there is nothing paradoxical about the liar paradox since every statement includes an implicit assertion of its own truth. The statement "It is true that two plus two equals four" contains no more information than the statement "two plus two equals four," because the phrase "It is true that..." is always implicit there. Working in classical logic, Tarski concluded that a language cannot define its truth predicate. The main avenue for resolving the paradox is the hierarchy of languages and meta-languages. The sentence within the quote is part of the "object language", whereas the language that determines its truth value is part of the "meta-language, which is semantically higher than the object language, which is semantically closed."Snow is white" is true if and only if snow is white. But one does not need to keep running through richer and richer meta-languages in order to chase our semantic tails of infinite regress.

 A simple example can formulate the paradox in terms of a speech act within social contexts. The barber’s paradox is a puzzle that shows an apparently plausible scenario of logical impossibility. A barber is defined as one "who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? Any answer to this question results in a contradiction. The barber cannot shave himself, as he only shaves those who do not shave themselves. Thus, if he shaves himself, he ceases to be the barber specified. Conversely, if the barber does not shave himself, then he fits into the group of people who would be shaved by the specified barber only, and thus, as that barber, he must shave himself. A loaded question  assumes the existence of a barber who both shaves and does not shave himself, which implies that no such barber exists, which is a vacuous proposition, and hence false.

The teacher declares: “learn from Bhartrhari, the Indian grammarian-philosopher.  Our interest is  not in strengthening the paradoxes by abstracting them from pragmatic context, but rather in exploring how a stubborn paradox may arise from unproblematic situations in daily communication”. 

Bhartrhari dealt with similar paradoxes (sarvam mithyā bravīmi "everything I am saying is false") and identified a hidden parameter that can defuse the stubborn paradox, if some form of the non-trivial effect of context dependence can be shown. The warring sides of the contradiction can consume themselves by allowing the reconciliation of the two perspectives of the world — the part before and after the function accomplishes its task. The capacity of time to prevent a confrontation of the two worlds is external to the "liar". Therefore, the unsolvable paradox lands one either in contradiction (virodha) or in infinite regress (anavasthā). Just as if someone says that the statement is unnameable (avācya), it becomes nameable (vyapadiśyate) precisely by calling it unnameable.  The "binary logic" allows one to operate the world into discrete units of meaning, but when one puts together the two separate worlds, one can explain and identify the two perspectives that are being deceptively sundry. 

 The liar paradox is occasionally used in fiction to shut down artificial intelligence. In the Doctor Who series, the Doctor temporarily stumps the insane computer BOSS by asking it, "If I were to tell you that the next thing I say would be true but that the last thing I say was a lie, would you believe me?" The boss tries to figure it out but cannot, and eventually decides the question is irrelevant and summons security.

 My friends! So, don’t worry about the politicians’ statements. Theirs’ are nothing but like liar-paradox, ready to be set aside as "I hear what he says; he says what I don't hear", where secondary meaning must be used to resolve the paradox.

 Again my Friends!! Don’t be disappointed! What we need to learn is the radical way out of the paradox, that is, how to live with the liar-paradox being both true and false. It offers a deeper understanding of how our semantic concepts and principles work to produce the paradox— a version of dialetheism that embraces the contradiction. One must remember that in ordinary thinking, truth is just one thing and falsity is just another thing. They do not come in hierarchical order. Heraclitus says: "We step and do not step into the same rivers; we are and we are not." The supra-proposition towers above the propositions and would stand like a monument (with a Janus head) over the propositions of logic.

 One gets the joke only if one sees through the deception of the mixed perspective of a Cretan calling all Cretans liars. In the paradox of the heap, where piling up grains of sand eventually becomes a heap at some point, the exact point of transformation cannot be identified. For example, to name the heap is to think of it from the universal perspective ("all grains of sand in the pile"), while to think of it as not yet a heap requires the perspective of the particulars ("each of the grains of sand"). Thus, all paradoxes are grounded in some conflation of perspectives, just as all humour is grounded in a thinly veiled attempt to deceive one’s listener into treating multiple perspectives as one. The claim that paradox is a problem is the ultimate logical lie, for it deceives us into believing that human reasoning is mono-perspectival. 

Similarly, Zeno’s most famous paradox (i.e., the race between Achilles and the tortoise, whereby the fastest runner never catches the slowest animal, because he can only ever reach half the distance between them) conflates two perspectives on time: mathematical and experiential. Does this awareness of perspectival differences actually resolve such time-honored paradoxes?

 If you disagree with me—especially if you believe that analytic logic is the only path to truth and meaning—you probably won’t learn to laugh! And so a Buddhist analysis of the liar sentence reveals that an answer to the liar paradox need not focus on the logical and semantic rules of reasoning. It ponders whether the real paradox resides in the interplay of possible values that seems to mirror the real-life use of language, essentially tied to intentions, social norms, etc.

 One important way to motivate non-classical solutions is to appeal to deflation of binary truth-value. Most strictly, the so-called transparency or ‘see-through’ teaches us that some sentences are neither true nor false. The Buddhist philosopher Nagarjuna refined the Catuskoti form of logic: "A", "not-A", "A and not-A'", and "not A and not not-A", plunging into an unexplored domain, called Shunya. 

Jainism made its own unique contribution by referring to the Saptabhangi naya, where our looking at things is perspectival. For example,  "A thing can be both chair and not-chair, not-mat …etc. so on" … which denies the whole world while at the same time including the whole world. But if there is a possibility of assigning a neutral third value like "indeterminate" between true and false, the value "gappy." can be given to the sentence over and above the true and false.

 The paradox of divine omnipotence is interesting to cite in this connection. For instance, "Can God make a stone too heavy for Him to lift?" God is all powerful. So God must have the power to undo what is done. He lives in the eternal present, so denying Him power over the past equates to denying Him power over current and future events, which is profane. Here also, God has all the properties of including contradictory things into one.

At the end let us celebrate nonsense poem from Through the Looking Glass of Lewis Carroll's Jabberwocky. Readers will find a meaning for it outside the literal:

 'Twas brillig1, and the slithy2 toves

Did gyre3 and gimble4 in the wabe5;

All mimsy6 were the borogoves7,

And the mome8 rath9 outgrabe10.

Thus, the main insight yields a very attractive definition of lying, namely, that you lie if and only if you say something that you do not believe and you intend to represent yourself as believing what you say.

Notes:

1. Brillig means— the time when you begin broiling things for dinner.’2

2. blend of slimy + lithe, Lithe" is the same as "active.e

3. gyre: twirl around like a gyre

4. gimble: bore holes like a gimlet

5. wabe: the wet side of a hill soaked by the rain

6. mimsy suggests miserable + flimsy.

7. borogoves: (fictional) type of bird

8. mome: grave or solemn

9. raths: (fictional) turtle with a mouth like a shark and a smooth green body; lives on swallows and oysters 

10. outgrabe: emitted a strange noise