Liar - truth-value glut


  • Sainsbury (1995) Paradoxes 2nd edition. Cambridge University Press. Chapter 6.
  • stanford:dialetheism
  • Graham Priest, JC Beall, and Bradley Armour-Garb (eds.) 2004. The Law of Non-Contradiction: New Philosophical Essays. Oxford University Press.
  • Mortensen and Priest. 1981. The truth teller paradox. Logique et Analyse, 95-6, 381-8.
  • Edwin Mares. 2004. Relevant Logic: a philosophical interpretation. Cambridge University Press.


  • Dialetheism: Some contradictions are true.
  • Trivialism: All contradictions are true. Problematic because P&¬P ⇒ P,¬P (so all statements are true.)

@(L1) L1 is false.@

True contradictions?

  • "I am happy and I am not happy."
  • Heraclitus: "We step and do not step into the same rivers."
  • Mao's essay On Contradiction

@The universality or absoluteness of contradiction has a twofold meaning. One is that contradiction exists in the process of development of all things, and the other is that in the process of development of each thing a movement of opposites exists from beginning to end. ... The interdependence of the contradictory aspects present in all things and the struggle between these aspects determine the life of all things and push their development forward. There is nothing that does not contain contradiction; without contradiction nothing would exist.@

  • The Diamond Sutra 金剛般若波羅蜜經


Objection #1 Contradictions entail everything

@This is a story about the famous philosopher / logician Bertrand Russell. He was asked the question, “You mean from the statement 2+2=5 it follows that you are the Pope? Can you prove it?” Russell said “yes” and then came up with this argument:
   1. Suppose 2+2=5.
   2. Subtracting 2 from both sides we get 2=3.
   3. Transposing, we have 3=2.
   4. Subtracting 1 from both sides, we get 2=1.
“Now”, Russell continues, “the Pope and I are two. Since two equals one, then the Pope and I are one. Hence I am the Pope.”

@Disjunctive syllogism: P or Q. ¬P. Therefore Q.@

  1. P&¬P
  2. P (1 conjunction elimination)
  3. P∨Q (2 disjunction introduction)
  4. ¬P (1 conjunction elimination)
  5. Q (3,4 DS)
  • Dialetheism + DS lead to explosion
  • Explosion: φ, ¬φ ⇒ ψ
    • Dialetheism rejects disjunctive syllogism. (e.g. paraconsistent logics)
    • But we make use of disjunctive syllogism all the time.
    • Not applicable when P&¬P is a true contradiction.

Objection #2 Contradictions cannot be true

  • By definition, if a sentence is false then it is not true.
    Reply: Question-begging.
  • No empirical evidence for true contradictions.
    Reply: This shows at most that true contradictions are unlikely, not that there aren't any.
  • It is never rational to believe a contradiction.
    Reply: Rational belief is based on many considerations. Logical consistency need not be paramount.

Objection #3 - The liar paradox can be reformulated easily

  • So the liar sentence is both true and not true. So "true" and "not true" are not exclusive.
  • Let us define "utrue" to apply ONLY to sentences that are not true, and NEVER sentences that are true. Now consider:

@(L7) L7 is utrue.@

Is L7 true?


  1. OK. So dialetheism cannot deal with this paradox. But it does not mean that there are no true contradictions.
  2. [Priest] "utrue" and "true" are both exclusive predicates, AND they are also non-exclusive predicates.
  3. [Sainsbury] Perhaps there is no such predicate that we can define.

Objection #4 - Dialetheism does not solve the truth-teller problem

@(L8) L8 is true.@


  • It is consistent to regard L8 as true, and equally consistent to regard it as false. There does not seem to be any reason to choose either side.
  • Conclusion: L8 is neither true nor false.
  • If so, what follows?

Objection #5 - Dialetheism cannot express disagreement with someone

@Suppose you say 'β' and Priest replies '¬β'. Under ordinary circumstances you would think that he had disagreed with you. But you remember that Priest is a dialetheist, and it occurs to you that he might very well agree with you afterall - since he might think that β and ¬β are both true. How can he indicate that he geniunely disagrees with you? The natural choice is for him to say 'β is not true.' However, the truth of this assertion is also consistent with β's being true - for a dialetheist anyway ...@

  • Quote taken from Terrence Parsons (1990) True Contradictions. Canadian Journal of Philosophy. 20, 335-54.
  • See discussion in Shapiro's "Simple Truth, Contradiction, and Consistency" in Priest, Beall, and Armour-Garb (2004).


Paradoxes and cost-benefit analysis.