Deleted lines 86-87:
See Terrence Parsons (1990) True Contradictions. ''Canadian Journal of Philosophy''. 20, 335-54.
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* 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).
Added lines 85-90:
!!Objection #5 - Dialetheism cannot express disagreement with someone
See Terrence Parsons (1990) True Contradictions. ''Canadian Journal of Philosophy''. 20, 335-54.
@@@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 ...@@@
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* Sainsbury (1995) ''Paradoxes'' 2nd edition. Cambridge University Press. Chapter 6.
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Reply: Rational belief is based on many considerations. Consistency need not be paramount.
!!Objection #3 - Dialetheism does not solve the truth-teller problem
@@@(L7) L7 is true.@@@
to:
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?
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* It is consistent to regard L7 as true, and equally consistent to regard it as false. There does not seem to be any reason to choose either side.
* Conclusion: L7 is neither true nor false.
to:
# OK. So dialetheism cannot deal with this paradox. But it does not mean that there are no true contradictions.
# [Priest] "utrue" and "true" are both exclusive predicates, AND they are also non-exclusive predicates.
# [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.@@@
Reply:
* 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.
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* Trivialism: All contradictions are true.\\
Problematic because P&¬P ⇒ P,¬P
to:
* Trivialism: All contradictions are true. Problematic because P&¬P ⇒ P,¬P (so all statements are true.)
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!!The truth-teller
What about:
to:
!!Objection #3 - Dialetheism does not solve the truth-teller problem
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What if we say that (L7) is neither true nor false?
to:
Reply:
* It is consistent to regard L7 as true, and equally consistent to regard it as false. There does not seem to be any reason to choose either side.
* Conclusion: L7 is neither true nor false.
* If so, what follows?
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to:
* Edwin Mares. 2004. ''Relevant Logic: a philosophical interpretation''. Cambridge University Press.
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# P (conjunction elimination)
# P∨Q (disjunction introduction)
# ¬P (conjunction elimination)
to:
# P (1 conjunction elimination)
# P∨Q (2 disjunction introduction)
# ¬P (1 conjunction elimination)
Changed lines 38-41 from:
# P&¬P\\
# P (conjunction elimination)\\
# P∨Q (disjunction introduction)\\
# ¬P (conjunction elimination)\\
to:
# P&¬P
# P (conjunction elimination)
# P∨Q (disjunction introduction)
# ¬P (conjunction elimination)
Changed lines 38-43 from:
P&¬P\\
P (conjunctin elimination)\\
P∨Q (disjunction introduction)\\
¬P\\
Q
to:
# P&¬P\\
# P (conjunction elimination)\\
# P∨Q (disjunction introduction)\\
# ¬P (conjunction elimination)\\
# Q (3,4 DS)
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to:
P (conjunctin elimination)\\
P∨Q (disjunction introduction)\\
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** Dialetheism rejects disjunctive syllogism.
to:
** Dialetheism rejects disjunctive syllogism. (e.g. paraconsistent logics)
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to:
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to:
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* Dialetheism: Some contradictions are true.
* Trivialism: All contradictions are true.\\
Problematic because P&¬P ⇒ P,¬P
@@@(L1) L1 is false.@@@
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* Dialetheism: Some contradictions are true.
* Trivialism: All contradictions are true. Problematic because P&¬P ⇒ P,¬P
to:
@@@Disjunctive syllogism: P or Q. ¬P. Therefore Q.@@@
P¬P\\
P\\
P∨Q\\
not;P\\
Q
* Dialetheism + DS lead to explosion
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* Disjunctive syllogism: P or Q. ¬P. Therefore Q.
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Paradoxes and cost-benefit analysis.
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to:
* Mortensen and Priest. 1981. The truth teller paradox. ''Logique et Analyse'', 95-6, 381-8.
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* What if we say that (L7) is neither true nor false?
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to:
@@@(L7) L7 is true.@@@
[[Category.LogicAndMaths]]
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* Trivialism: All contradictions are true.
** Problematic because P&Q⇒P, P&Q⇒Q
to:
* Trivialism: All contradictions are true. Problematic because P&¬P ⇒ P,¬P
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** Problematic because P&Q⇒P, P&Q⇒Q
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* It is never rational to believe a contradiction.
to:
* It is never rational to believe a contradiction.\\
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* Arguments based on purely logical considerations are question-begging.
* Arguments based on empirical evidence show at most that true contradictions are unlikely.
to:
* 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. Consistency need not be paramount.
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to:
* Graham Priest, JC Beall, and Bradley Armour-Garb (eds.) 2004. '' The Law of Non-Contradiction: New Philosophical Essays''. Oxford University Press.
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to:
!!Objection #1 Contradictions entail everything
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!!Objection #2 Contradictions cannot be true
* Arguments based on purely logical considerations are question-begging.
* Arguments based on empirical evidence show at most that true contradictions are unlikely.
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to:
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to:
!!The truth-teller
What about:
@@@(L5) L5 is true.@@@
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to:
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* Dialetheism: Some contradictions are true.
* Trivialism: All contradictions are true.
φ, ¬φ ⇒ ψ
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* Dialetheism: Some contradictions are true.
* Trivialism: All contradictions are true.
* Explosion: φ, ¬φ ⇒ ψ
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* Dialetheism rejects disjunctive syllogism.
to:
** Dialetheism rejects disjunctive syllogism.
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** Not applicable when P&#¬P is a true contradiction.
to:
** Not applicable when P&¬P is a true contradiction.
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** Not applicable when P&&#not;P is a true contradiction.
to:
** Not applicable when P&#¬P is a true contradiction.
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* Paraconsistent logicians often reject disjunctive syllogism.
to:
* Dialetheism rejects disjunctive syllogism.
** But we make use of disjunctive syllogism all the time.
** Not applicable when P&&#not;P is a true contradiction.
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* Heraclitus: "We step and do not step into the same rivers."
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* Dialetheism: Some contradictions are true.
* Trivialism: All contradictions are true.
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@@@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.@@@
to:
@@@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.@@@
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@@@佛告須菩提：「諸菩薩摩訶薩應如是降伏其心！所有一切眾生之類：若卵生、若胎生、若濕生、若化生；若有色、若無色；若有想、若無想、若非有想非無想，我皆令入無餘涅槃而滅度之。如是滅度無量無數無邊眾生，實無眾生得滅度者。何以故？須菩提！若菩薩有我相、人相、眾生相、壽者相，即非菩薩。@@@
to:
@@@佛告須菩提：「諸菩薩摩訶薩應如是降伏其心！所有一切眾生之類：若卵生、若胎生、若濕生、若化生；若有色、若無色；若有想、若無想、若非有想非無想，我皆令入無餘涅槃而滅度之。'''如是滅度無量無數無邊眾生，實無眾生得滅度者'''。何以故？須菩提！若菩薩有我相、人相、眾生相、壽者相，即非菩薩。@@@
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to:
* The Diamond Sutra 金剛般若波羅蜜經
@@@佛告須菩提：「諸菩薩摩訶薩應如是降伏其心！所有一切眾生之類：若卵生、若胎生、若濕生、若化生；若有色、若無色；若有想、若無想、若非有想非無想，我皆令入無餘涅槃而滅度之。如是滅度無量無數無邊眾生，實無眾生得滅度者。何以故？須菩提！若菩薩有我相、人相、眾生相、壽者相，即非菩薩。@@@
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@@@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.@@@
to:
@@@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.@@@
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to:
* Mao's essay [[http://www.marxists.org/reference/archive/mao/selected-works/volume-1/mswv1_17.htm|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.@@@
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Paraconsistent logicians often reject disjunctive syllogism.
Disjunctive syllogism: or Q. ¬P. Therefore Q.
to:
* Disjunctive syllogism: P or Q. ¬P. Therefore Q.
* Paraconsistent logicians often reject disjunctive syllogism.
Changed lines 20-22 from:
Paraconsistent logicians often reject disjunctive syllogism.
to:
Paraconsistent logicians often reject disjunctive syllogism.
Disjunctive syllogism: P or Q. ¬P. Therefore Q.
Changed lines 18-20 from:
“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.”@@@
to:
“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.”@@@
Paraconsistent logicians often reject disjunctive syllogism.
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φ, ¬φ ⇒ ψ
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1. Suppose 2+2=5.\\
2. Subtracting 2 from both sides we get 2=3.\\
to:
1. Suppose 2+2=5.\\
2. Subtracting 2 from both sides we get 2=3.\\
Changed lines 13-15 from:
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.\\
to:
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.\\
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to:
1. Suppose 2+2=5.\\
Changed lines 12-15 from:
->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.\\
to:
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.\\
Changed lines 12-15 from:
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.\\
to:
->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.\\
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!!Trivialism
@@@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.”@@@
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to:
!!True contradictions?
* "I am happy and I am not happy."
Added lines 1-6:
!Liar - truth-value glut
!!Readings
* stanford:dialetheism
!!