The Liar Paradox - Introduction

Readings

• [Required] Sainsbury 1995. Paradoxes. 2nd edition. Cambridge University Press. Sections 5.2-5.8. hkulib:b1480077, hkulib:b3048828 intranet:sainsbury-paradoxes-ch5.pdf

History

• 'para' = beyond, and 'doxa' = opinion, belief.
• Aristotle's Metaphysics:

@To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or of what is not that it is not, is true.@

• According to Athenaeus, a gravestone on the Greek Island of Cos contains this poem:

@O Stranger: Philetas of Cos am I,
'Twas the Liar who made me die,
And the bad nights caused thereby.@

• Eubulides of Miletus. Eubulides reportedly said: A man says that he is lying. Is what he says true or false?
• Epimenides Paradox

@Even one of their own Prophets has said, "Cretans are always liars, evil brutes, lazy gluttons." This testimony is true. Therefore, rebuke them sharply, so that they will be sound in the faith and will pay no attention to Jewish myths or to the commands of those who reject the truth. (Titus 1:12-14)@

• Buridan (late medieval):

@God exists.
None of the sentences in this pair is true.@

The Paradox

Liar sentence #1

@(L1) L1 is false.@

Problem:

• L1 is true → "L1 is false" is true → L1 is false → L1 is both true and false.
• L1 is false → "L1 is false" is true → L1 is true → L1 is both true and false.
• So L1 is both true and false.

@Bivalence: L1 is either true or false.@ @Non-contradiction: It is not the case that L1 is both true and false.@

Relevance of the paradox: the concept of truth, logical laws, consistency of language.

Some responses

• Give up bivalence - L1 is neither true nor false. [tv gap]
• Give up non-contradiction - L1 is both true and false. [tv glut]
• Reform - Paradoxes inevitably arise in natural language, but it is possible to avoid the paradox by changing the language.