Catuskoti :
Paraconsistent, Paracomplete,
Both, or None ?
Fabien SCHANG
National Research University, HSE, Moscow
University of Istanbul
UNILOG, 25-30 June 2015
This work is an output of a research project implemented as part of the Basic Research Program at the National
Research University Higher School of Economics (HSE)
Content
1 Catuskoti, and its dual
2 Question-Answer Semantics
3 Dialectical negation
4 Conclusion
1
Catuskoti, a d its dual
1 Catuskoti, a d its dual
Nāgā ju a 5 -250), was said to express the ultimate view of denial by
rejecting each of four combined sentences:
(a) Does a being come out itself?
(b) Does a being come out the other?
(c) Does a being come out of both itself and the other?
(d) Does a being come out neither?
1 Catuskoti, a d its dual
Nāgā ju a 5 -250), was said to express the ultimate view of denial by
rejecting each of four combined sentences:
(a) Does a being come out itself?
(b) Does a being come out the other?
(c) Does a being come out of both itself and the other?
(d) Does a being come out neither?
1 Catuskoti, a d its dual
Nāgā ju a 5 -250), was said to express the ultimate view of denial by
rejecting each of four combined sentences:
(a) Does a being come out itself?
No.
(b) Does a being come out the other?
No.
(c) Does a being come out of both itself and the other?
No.
(d) Does a being come out neither?
No.
1 Catuskoti, a d its dual
Catuskoti (Tetralemma): a set of denied sentences
(a)
not: p
(b)
not: p
(c)
(d)
not: p p
not: (p p)
1 Catuskoti, a d its dual
Catuskoti (Tetralemma): denial as classical negation
(a)
(b)
(c)
(d)
(p)
(p)
(p p)
((p p))
1 Catuskoti, a d its dual
Catuskoti (Tetralemma): denial as classical negation
(a)
p
(b)
p
(c)
p p
(d)
p p
1 Catuskoti, a d its dual
Catuskoti (Tetralemma): denial as classical negation
(a)
p
(b)
p
(c)
p p
(d)
p p
(d) is inconsistent and redundant with (a)-(b)
1 Catuskoti, and its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis as also a po e ful ad o ate of such a position.
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , which aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis was also a powerful advocate of such a position.
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , which aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis was also a powerful advocate of such a position.
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , hi h aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis as also a po e ful ad o ate of su h a positio .
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , which aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, and its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis as also a po e ful ad o ate of such a position.
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , which aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis as also a po e ful ad o ate of su h a positio .
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , which aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis as also a po e ful ad o ate of such a position.
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , which aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis was also a powerful advocate of such a position.
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , which aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis as also a po e ful ad o ate of su h a positio .
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , which aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis was also a powerful advocate of such a position.
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , which aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis as also a po e ful ad o ate of su h a positio .
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , hi h ai s to apture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, and its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis was also a powerful advocate of such a position.
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , hi h ai s to aptu e the i dete i ate atu es of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis as also a po e ful ad o ate of su h a positio .
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , which aims to capture the indeterminate natures of
things, when we attempt to say anything about anything.
E t
“kepti is
Whi h
: Stanford Encyclopedia of Philosophy)
o pli ated
ode of spee h to
ake se se of the Catuskoti?
1 Catuskoti, and its dual
A Weste
ou te pa t: P
ho s ou mallon
It is necessary above all to consider our own knowledge; for if it is in our nature to know nothing, there is no
eed to i ui e a fu the i to othe thi gs. […] P ho of Elis was also a powerful advocate of such a position.
Timon says that Pyrrho reveals that things are equally indifferent and unstable and indeterminate; for this
reason, neither our perceptions nor our beliefs tell the truth or lie. For this reason, then, we should not trust
them, but should be without opinions and without inclinations and without wavering, saying about each single
thing that it no more is than is not, or both is and is not, or neither is nor is not (ou mallon estin ê ouk estin ê kai
esti kai ouk estin ê oute estin oute ouk estin). Timon says that the result for those who are so disposed will be first
speechlessness (aphasia), but then freedom from worry (ataraxia).
Pyrrho infers that our perceptions and beliefs are neither true nor false. They are not truth-evaluable,
presumably because there are no facts which could be correctly captured.
Pyrrho does not say that we should cease to speak. He suggests that we adopt a complicated mode of speech,
constructed around the expression ou mallon
o o e , hi h ai s to aptu e the i dete i ate atu es of
things, when we attempt to say anything about anything.
E t
“kepti is
: Stanford Encyclopedia of Philosophy)
Whi h complicated mode of speech to
ake se se of the Catuskoti?
1 Catuskoti, a d its dual
Catuskoti (Tetralemma): a set of denied truth-values of p
(a)
v(p) T
(b)
v(p) F
(c)
v(p) B
(d)
v(p) N
1 Catuskoti, and its dual
The most obvious way to proceed is now to take
this possibility as a fifth semantic value, and
construct a five-valued logic.
Thus, we add a new value, E, to our existing four
(T, B, F, and N).
Priest (2011: 15)
1 Catuskoti, and its dual
The most obvious way to proceed is now to take
this possibility as a fifth semantic value, and
construct a five-valued logic.
Thus, we add a new value, E, to our existing four
(T, B, F, and N).
Priest (2011: 15)
1 Catuskoti, a d its dual
Limits of Thought = Limits of compossible acceptance/rejection?
p is E : p is eithe T, nor F, nor B, nor N in V = {T,F,B,N}
Catuskoti: rejection limited by a Law of Excluded 5th in V = {T,F,B,N}?
Saptabhangi: acceptance limited by a Law of Excluded 4th in V = {T,F,A}?
(A fo the se a ti p edi ate a akta a
Law of Excluded (n+1)th:
there is no (n+1)th truth-value in a domain of n truth-values V = {X1, …, Xn}
Example
Law of Excluded 3rd Middle : the e is o + =
rd
truth-value in V
Why should the Tetralemma stop rejecting p at the n = 4th predication?
What of a generalized n-lemma, about n truth-values?
1 Catuskoti, a d its dual
Limits of Thought = Limits of compossible acceptance/rejection?
p is E : p is eithe T, nor F, nor B, nor N in V = {T,F,B,N}
Catuskoti: rejection limited by a Law of Excluded 5th in V = {T,F,B,N}?
Saptabhangi: acceptance limited by a Law of Excluded 4th in V = {T,F,A}?
(A fo the se a ti p edi ate a akta a
Law of Excluded (n+1)th:
there is no (n+1)th truth-value in a domain of n truth-values V = {X1, …, Xn}
Example
Law of Excluded 3rd Middle : the e is o + =
rd
truth-value in V
Why should the Tetralemma stop rejecting p at the n = 4th predication?
What of a generalized n-lemma, about n truth-values?
1 Catuskoti, a d its dual
Limits of Thought = Limits of compossible acceptance/rejection?
p is E : p is eithe T, nor F, nor B, nor N in V = {T,F,B,N}
Catuskoti: rejection limited by a Law of Excluded 5th in V = {T,F,B,N}?
Saptabhangi: acceptance limited by a Law of Excluded 4th in V = {T,F,A}?
(A fo the se a ti p edi ate a akta a
Law of Excluded (n+1)th:
there is no (n+1)th truth-value in a domain of n truth-values V = {X1, …, Xn}
Example
Law of Excluded 3rd Middle : the e is o + =
rd
truth-value in V
Why should the Tetralemma stop rejecting p at the n = 4th predication?
What of a generalized n-lemma, about n truth-values?
1 Catuskoti, a d its dual
Dual logics: Catuskoti (LC) vs Saptabhangi (LS)?
paraconsistent vs paracomplete?
co-intuitionistic vs intuitionistic logics?
See Bahm (1958):
Does “e e -Fold Predication equal Four-Co e ed Negatio ‘e e sed?
Logic L = L,╞
L is a theory (i.e. a set of formulas, including pL)
╞ is a relation of consequence, such that:
╞ Δ iff
v(p)D v(qΔ)D
D is a set of designated values (where TD)
1 Catuskoti, a d its dual
Dual logics: Catuskoti (LC) vs Saptabhangi (LS)?
paraconsistent vs paracomplete?
co-intuitionistic vs intuitionistic logics?
See Bahm (1958):
Does “e e -Fold Predication equal Four-Cornered Negation Reversed?
Logic L = L,╞
L is a theory (i.e. a set of formulas, including pL)
╞ is a relation of consequence, such that:
╞ Δ iff
v(p)D v(qΔ)D
D is a set of designated values (where TD)
1 Catuskoti, a d its dual
The principle of four- o e ed egatio , stated as
is neither a, nor non-a, nor both a and non-a, nor
neither a nor non-a o as joi t de ial of is a , is
non-a , is oth a a d o -a , a d is eithe a o
non-a
he e a a d o -a are interpreted as
opposites , if e e sed, ould e stated as
is a,
non-a, both a and non-a, and neither a nor non-a o
as the joi t affi atio of
is a ,
is o -a ,
is
both a and non-a a d
is eithe a o o -a
(where a and non-a are interpreted as opposites). This
reversed statement consists of the first four of the
seven syads e ept that is eithe a o o -a is
epla ed
is i des i a le .
Bahm (1958): 128
1 Catuskoti, a d its dual
The principle of four-cornered negation, stated as
is neither a, nor non-a, nor both a and non-a, nor
neither a nor non-a or as joint denial of is a , is
non-a , is oth a a d o -a , a d is eithe a o
non-a
he e a a d o -a are interpreted as
opposites), if reversed, ould e stated as
is a,
non-a, both a and non-a, and neither a nor non-a o
as the joi t affi atio of
is a ,
is o -a ,
is
both a and non-a a d
is eithe a o o -a
(where a and non-a are interpreted as opposites). This
reversed statement consists of the first four of the
seven syads e ept that is eithe a o o -a is
epla ed
is i des i a le .
Bahm (1958): 128
1 Catuskoti, a d its dual
The principle of four- o e ed egatio , stated as
is neither a, nor non-a, nor both a and non-a, nor
neither a nor non-a o as joi t de ial of is a , is
non-a , is oth a a d o -a , a d is eithe a o
non-a
he e a a d o -a are interpreted as
opposites), if reversed, ould e stated as
is a,
non-a, both a and non-a, and neither a nor non-a o
is a ,
is o -a ,
is
as the joint affirmation of
both a and non-a a d
is eithe a o o -a
(where a and non-a are interpreted as opposites). This
reversed statement consists of the first four of the
seven syads e ept that is eithe a o o -a is
epla ed
is i des i a le .
Bahm (1958): 128
1 Catuskoti, a d its dual
A positi e
e sio of the Catuskoti:
(a) Does a being come out itself?
(b) Does a being come out the other?
(c) Does a being come out of both itself and the other?
(d) Does a being come out neither?
1 Catuskoti, a d its dual
A positi e
e sio of the Catuskoti:
(a) Does a being come out itself?
Yes.
(b) Does a being come out the other?
Yes.
(c) Does a being come out of both itself and the other?
Yes.
(d) Does a being come out neither?
Yes.
1 Catuskoti, a d its dual
A positi e
e sio of the Catuskoti:
(a)
p
(b)
p
(c)
p p
(d)
(p p)
1 Catuskoti, a d its dual
A positi e
e sio of the Catuskoti:
(a)
v(p) = T
(b)
v(p) = F
(c)
v(p) = B
(d)
v(p) = N
How can a sentence p be true, false, both true and false, and neither true
nor false at once?
1 Catuskoti, a d its dual
(1)
(2)
(3)
(4)
(5)
(6)
(7)
bhaṅgī
syād asty eva
syad nāsty eva
syād asty eva
syad nāsty eva
syād asty avaktavyam
eva
syād asty eva
syād avaktavyam eva
syād nāsty eva
syād avaktavyam eva
syād asty eva
syād nāsty eva
syād avaktavyam eva
English translation
arguably, it exists
arguably, it does not
exist
arguably, it exists;
arguably, it does not
exist
arguably, it is
unspeakable
arguably, it exists;
arguably, it is
unspeakable
arguably, it does not
exist;
arguably, it is
unspeakable
arguably, it exists;
arguably, it does not
exist; arguably, it is
unspeakable
speech-act
assertion
denial
successive assertion
and denial
simultaneous assertion
and denial
assertion and
simultaneous assertion
and denial
denial and simultaneous
assertion and denial
assertion and denial and
simultaneous assertion
and denial
1 Catuskoti, a d its dual
Saptabhangi: a classical formalization
(1)
p
(2)
p
(3)
(4)
(5)
(6)
(7)
p p
p p
p (p p)
p (p p)
p p (p p)
1 Catuskoti, a d its dual
Saptabhangi: a classical formalization
(1)
p
(2)
p
(3)
(4)
(5)
(6)
(7)
p p
p p
p (p p)
p (p p)
p p (p p)
(3) is inconsistent
(3) and (4) are indistinguishable
from each other
(5)-(7) collapse into (3)-(4), by
simplification
s ad efe s to sta dpoi ts:
a combination of various models
1 Catuskoti, a d its dual
Saptabhangi: truth-values in distinctive models a akta a as B)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
w v(w,p) = T
w v(w,p) = F
w v(w,p) = T,
w v(w,p) = F
w v(w,p) = T,
w v(w,p) = B
w v(w,p) = T,
w v(w,p) = F,
w v(w,p) = B
w v(w,p) = F,
w v(w,p) = B
w v(w,p) = B
1 Catuskoti, a d its dual
Saptabhangi: truth- alues i disti ti e
(1)
(2)
(3)
(4)
(5)
(6)
(7)
odels a akta a as B)
w v(w,p) = T
w v(w,p) = F
w v(w,p) = T,
w v(w,p) = F
w v(w,p) = T,
w v(w,p) = B
w v(w,p) = T,
w v(w,p) = F,
w v(w,p) = B
w v(w,p) = F,
w v(w,p) = B
w v(w,p) = B
1 Catuskoti, a d its dual
Saptabhangi: truth-values in distinctive models a akta a as N
(1)
(2)
(3)
(4)
(5)
(6)
(7)
w v(w,p) = T
w v(w,p) = F
w v(w,p) = T,
w v(w,p) = F
w v(w,p) = T,
w v(w,p) = N
w v(w,p) = T,
w v(w,p) = F,
w v(w,p) = N
w v(w,p) = F,
w v(w,p) = N
w v(w,p) = N
1 Catuskoti, a d its dual
Saptabhangi: truth- alues i disti ti e
(1)
(2)
(3)
(4)
(5)
(6)
(7)
odels a akta a as N)
w v(w,p) = T
w v(w,p) = F
w v(w,p) = T,
w v(w,p) = F
w v(w,p) = T,
w v(w,p) = N
w v(w,p) = T,
w v(w,p) = F,
w v(w,p) = N
w v(w,p) = F,
w v(w,p) = N
w v(w,p) = N
1 Catuskoti, a d its dual
Duality (Marcos & Molick 2013)
Duality:
╞ Δ iff Δd╞d d
If X{,} s.t. X = (p1, …, pn), then Xd = (p1, …, pn)
Examples
Let = (p p)
Then d = (p p) = p p = p p = p p
Let = (p p)
Then d = (p p) = p p = p p = p p
1 Catuskoti, a d its dual
Duality (Marcos & Molick 2013)
Duality:
╞ Δ iff Δd╞d d
If X{,} s.t. X = (p1, …, pn), then Xd = (p1, …, pn)
Examples
Let = (p p)
Then d = (p p) = p p = p p = p p
Let = (p p)
Then d = (p p) = p p = p p = p p
1 Catuskoti, a d its dual
Are LC and LS dual logics?
╞ Δ
iff
Δd╞d d
!
1 Catuskoti, a d its dual
Are LC and LS dual logics?
p p╞ q
iff
q╞d p p
!
1 Catuskoti, a d its dual
Are LC and LS dual logics?
p p╞/ q
iff
q╞/d p p
?
P o le s a out duals :
- defined between models (preservation relation of values)
LC and LS are defined by counter-models
- defined in terms of connectives
there are no sentential connectives in the original statements
- even if and are cancelled, remains
1 Catuskoti, a d its dual
Are LC and LS dual logics?
p p╞/ q
iff
q╞/d p p
?
P o le s a out duals :
- defined between models (preservation relation of values)
LC and LS are defined by counter-models
- defined in terms of connectives
there are no sentential connectives in the original statements
- even if and are cancelled, remains
1 Catuskoti, a d its dual
Are LC and LS dual logics?
p p╞/ q
iff
q╞/d p p
?
P o le s a out duals :
- defined between models (preservation relation of values)
LC and LS are defined by counter-models
- defined in terms of connectives
there are no sentential connectives in the original statements
- even if and are cancelled, remains
1 Catuskoti, a d its dual
My reference to the non-bivalence or paraconsistent logic,
in connection with Jainism, should not be overemphasized. I have already noted that Jaina logicians did
not develop, unlike the modern logicians, truth matrices
for Negation, Conjunction, and so on. It would be difficult,
if not impossible, to find intuitive interpretations of such
matrices, if one were to develop them in any case.
Matilal (1998): 139
1 Catuskoti, a d its dual
My reference to the non-bivalence or paraconsistent logic,
in connection with Jainism, should not be overemphasized. I have already noted that Jaina logicians did
not develop, unlike the modern logicians, truth matrices
for Negation, Conjunction, and so on. It would be difficult,
if not impossible, to find intuitive interpretations of such
matrices, if one were to develop them in any case.
Matilal (1998): 139
1 Catuskoti, a d its dual
Are LC and LS dual logics?
p p╞/ q
iff
q╞/d p p
?
P o le s a out duals :
- defined between models (preservation relation of values)
LC and LS are defined by counter-models
- defined in terms of connectives
there are no sentential connectives in the original statements
- even if and are cancelled, remains
1 Catuskoti, a d its dual
What sort of logic is LC?
paraconsistent?
paracomplete?
both paraconsistent and paracomplete?
neither paraconsistent nor paracomplete?
1 Catuskoti, a d its dual
LC is paraconsistent iff
p, p╞/ q
{p, p}D / qD
(Non-Explosion)
1 Catuskoti, a d its dual
What sort of logic is LC?
paraconsistent?
paracomplete?
both paraconsistent and paracomplete?
neither paraconsistent nor paracomplete?
1 Catuskoti, a d its dual
LC is paracomplete iff
p╞/ q, q
pD / {q, q}D
(Non-Implosion)
1 Catuskoti, a d its dual
What sort of logic is LC?
paraconsistent?
paracomplete?
both paraconsistent and paracomplete?
neither paraconsistent nor paracomplete?
1 Catuskoti, a d its dual
LC is both paraconsistent and paracomplete iff
p, p╞/ q
{p, p}D / qD
p╞/ q, q
pD / {q, q}D
(Non-Explosion)
(Non-Implosion)
1 Catuskoti, a d its dual
LC is both paraconsistent and paracomplete iff
p, p╞/ q
{p, p}D / qD
p╞/ q, q
pD / {q, q}D
(Non-Explosion)
(Non-Implosion)
1 Catuskoti, a d its dual
LC is both paraconsistent and paracomplete iff
p, p╞/ q
{p, p}D / qD
p╞/ q, q
pD / {q, q}D
(Non-Explosion)
(Non-Implosion)
1 Catuskoti, a d its dual
What sort of logic is LC?
paraconsistent?
paracomplete?
both paraconsistent and paracomplete?
neither paraconsistent nor paracomplete?
1 Catuskoti, a d its dual
LC is neither paraconsistent nor paracomplete iff
Not: p, p╞/ q
{p, p}D / qD
Not: p╞/ q, q
pD / {q, q}D
(Non-Explosion)
(Non-Implosion)
Neither paraconsistent nor paracomplete
=
Both consistent and complete
?
Is metalinguistic negation an involutive operator
?
1 Catuskoti, a d its dual
LC is neither paraconsistent nor paracomplete iff
Not: p, p╞/ q
{p, p}D / qD
Not: p╞/ q, q
pD / {q, q}D
(Non-Explosion)
(Non-Implosion)
Neither paraconsistent nor paracomplete
=
Both consistent and complete
?
Is metalinguistic negation an involutive operator
?
1 Catuskoti, a d its dual
LC is neither paraconsistent nor paracomplete iff
p, p╞ q
{p, p}D / qD
p╞ q, q
pD / {q, q}D
(Non-Explosion)
(Non-Implosion)
Neither paraconsistent nor paracomplete
=
Both consistent and complete
?
Is metalinguistic negation an involutive operator
?
1 Catuskoti, a d its dual
LC is neither paraconsistent nor paracomplete iff
p, p╞ q
{p, p}D / qD
p╞ q, q
pD / {q, q}D
(Non-Explosion)
(Non-Implosion)
Neither paraconsistent nor paracomplete
=
Both consistent and complete
?
Is metalinguistic negation an involutive operator
?
1 Catuskoti, a d its dual
LC is neither paraconsistent nor paracomplete iff
p, p╞ q
{p, p}D / qD
p╞ q, q
pD / {q, q}D
(Non-Explosion)
(Non-Implosion)
Neither paraconsistent nor paracomplete
=
Both consistent and complete
?
Is metalinguistic negation an involutive operator
?
1 Catuskoti, a d its dual
How can catuskoti and saptabhangi be dual logics ?
Duals?
Both theories (sets of sentences) are asymmetric (4 vs 7 sentences)
Logics?
The formalization of these theories is dubious
- syntactic version: increasingly complex sentences
p, p, p p, …
ea s of {,,} in L
- semantic version: increasingly complex truth-values
p, togethe ith T, F, B, N, … in D
Duality: between answers to questions about truth-values
1 Catuskoti, a d its dual
The difference between Buddhism and Jainism in
this respect lies in the fact that the former avoids
by rejecting the extremes altogether, while the
latter does it by accepting both with qualifications
and also by reconciling them.
Matilal (1998): 129
1 Catuskoti, a d its dual
The difference between Buddhism and Jainism in
this respect lies in the fact that the former avoids
by rejecting the extremes altogether, while the
latter does it by accepting both with qualifications
and also by reconciling them.
Matilal (1998): 129
(a) Is x a?
(b) Is x non-a?
(c) Is x a and non-a?
(d) Is x neither x nor non-a?
No.
No.
No.
No.
1 Catuskoti, a d its dual
The difference between Buddhism and Jainism in
this respect lies in the fact that the former avoids
by rejecting the extremes altogether, while the
latter does it by accepting both with qualifications
and also by reconciling them.
Matilal (1998): 129
(a) Is x arguably a?
(b) Is x arguably non-a?
(c) Is x arguably a, arguably non-a?
(d) Is x arguably unspeakable?
(e) Is x arguably a, arguably unspeakable?
(f) Is x arguably a, arguably unspeakable?
(g) Is x arguably a, arguably non-a, arguably unspeakable?
Yes.
Yes.
Yes.
Yes.
Yes.
Yes.
Yes.
1 Catuskoti, a d its dual
(1)
(2)
(3)
(4)
(5)
(6)
(7)
bhaṅgī
syād asty eva
syad nāsty eva
syād asty eva
syad nāsty eva
syād asty avaktavyam
eva
syād asty eva
syād avaktavyam eva
syād nāsty eva
syād avaktavyam eva
syād asty eva
syād nāsty eva
syād avaktavyam eva
English translation
arguably, it exists
arguably, it does not
exist
arguably, it exists;
arguably, it does not
exist
arguably, it is
unspeakable
arguably, it exists;
arguably, it is
unspeakable
arguably, it does not
exist;
arguably, it is
unspeakable
arguably, it exists;
arguably, it does not
exist; arguably, it is
unspeakable
speech-act
acceptance
acceptance
acceptance
acceptance
acceptance
acceptance
acceptance
1 Catuskoti, a d its dual
At least one duality prevails between theories:
- catuskoti relies on systematic rejection
- saptabhangi relies on systematic acceptance
What is the logical status of acceptance and rejection?
- logical connectives:
affirmation vs negation?
- truth-values:
truth vs falsity?
- none:
answers to questions about sentences!
LS and LC include higher-order sentences, viz. statements
A common semantics requires sentences, truth-values, and speech-acts
Question-Answer Semantics
(dialogical feature of ancient theories)
1 Catuskoti, a d its dual
At least one duality prevails between theories:
- catuskoti relies on systematic rejection
- saptabhangi relies on systematic acceptance
What is the logical status of acceptance and rejection?
- logical connectives:
affirmation vs negation?
- truth-values:
truth vs falsity?
- none:
answers to questions about sentences!
LS and LC include higher-order sentences, viz. statements
A common semantics requires sentences, truth-values, and speech-acts
Question-Answer Semantics
(dialogical feature of ancient theories)
2
Question-Answer Semantics
2 Question-Answer Semantics
What does
is a sta d fo ?
A statement of the form Xp = p is X
X: semantic predicate
E a ple: p is t ue
Xp
�p
Xp
Tp
Fp
Bp
Np
told alue
complementation
a ked alue
p has the value X in D
p has not the value X in D
p has only the value X in D
p is true and not false
p is not true and false
p is true and false
p is not true and not false
Tp आ �p
�p आ Fp
Tp आ Fp
�p आ �p
2 Question-Answer Semantics
What is a t uth- alue ?
a class of sentences (see Frege (1892))
Mono-valence: each sentence is in the True, or not
Bivalence: each sentence is either true or not, i.e. false
an information about a sentence
Ontology: about being and not-being (how things are)
Epistemology: about affirming and not-affirming (how things are held)
p is t ue :
p is false :
it is the case that p
it is asserted that p
acceptance of p
it is the case that not-p
it is asserted that not-p
rejection of p
2 Question-Answer Semantics
What is a t uth- alue ?
a class of sentences (see Frege (1892))
Mono-valence: each sentence is in the True, or not
Bivalence: each sentence is either true or not, i.e. false
an information about a sentence
Ontology: about being and not-being (how things are)
Epistemology: about affirming and not-affirming (how things are held)
p is t ue :
p is false :
it is the case that p
it is asserted that p
acceptance of p
it is the case that not-p
it is asserted that not-p
rejection of p
2 Question-Answer Semantics
What is a t uth- alue ?
a class of sentences (see Frege (1892))
Mono-valence: each sentence is in the True, or not
Bivalence: each sentence is either true or not, i.e. false
an information about a sentence
Ontology: about being and not-being (how things are)
Epistemology: about affirming and not-affirming (how things are held)
p is t ue :
p is false :
it is the case that p
it is asserted that p
acceptance of p
it is the case that not-p
it is asserted that not-p
rejection of p
2 Question-Answer Semantics
What is a t uth- alue ?
A generalization of truth-values, beyond monovalence and bivalence
See Zaitsev & Shramko (2013)
Referential truth- alues: p is t ue/false
Inferential truth- alues: p is held t ue/false
(T/F)
(1/0)
A parallel with ontological vs epistemic truth-values
In the following:
- truth-values are treated as abstract objects
- no special interpretation is associated to these objects
(ontological, epistemic; referential, inferential)
Ho to deal ith t uth- alues i catuskoti and saptabhangi?
2 Question-Answer Semantics
What is a t uth- alue ?
A generalization of truth-values, beyond monovalence and bivalence
See Zaitsev & Shramko (2013)
Referential truth- alues: p is t ue/false
Inferential truth- alues: p is held t ue/false
(T/F)
(1/0)
A parallel with ontological vs epistemic truth-values
In the following:
- truth-values are treated as abstract objects
- no special interpretation is associated to these objects
(ontological, epistemic; referential, inferential)
Ho to deal ith t uth- alues i catuskoti and saptabhangi?
2 Question-Answer Semantics
What is a t uth- alue ?
A generalization of truth-values, beyond monovalence and bivalence
See Zaitsev & Shramko (2013)
Referential truth- alues: p is t ue/false
Inferential truth- alues: p is held t ue/false
(T/F)
(1/0)
A parallel with ontological vs epistemic truth-values
In the following:
- truth-values are treated as abstract objects
- no special interpretation is associated to these objects
(ontological, epistemic; referential, inferential)
Ho to deal ith t uth- alues i catuskoti and saptabhangi?
2 Question-Answer Semantics
How many truth-values are there in the catuskoti and saptabhangi?
A common interpretation: 7 in the saptabhangi, 4 in the catuskoti
A common objection: Indian schools assumed bivalence
Paribhāṣā: general criteria of logical rationality
Consistency: no sentence p can be accepted and rejected
Solution:
- distinction told vs marked values
- truth-values are elements in increasingly complex sets
n=1
{T} = {T}
n=2
{T,{ }} = {T,F}
n=3
{{T},{F},{ }} = {T,F,N}
n=4
{{T},{F},{T,F},{ }} = {T,F,B,N}
…
2 Question-Answer Semantics
How many truth-values are there in the catuskoti and saptabhangi?
A common interpretation: 7 in the saptabhangi, 4 in the catuskoti
A common objection: Indian schools assumed bivalence
Paribhāṣā: general criteria of logical rationality
Consistency: no sentence p can be accepted and rejected
Solution:
- distinction told vs marked values
- truth-values are elements in increasingly complex sets
n=1
{T} = {T}
n=2
{T,{ }} = {T,F}
n=3
{{T},{F},{ }} = {T,F,N}
n=4
{{T},{F},{T,F},{ }} = {T,F,B,N}
…
2 Question-Answer Semantics
How many truth-values are there in the catuskoti and saptabhangi?
A common interpretation: 7 in the saptabhangi, 4 in the catuskoti
A common objection: Indian schools assumed bivalence
Paribhāṣā: general criteria of logical rationality
Consistency: no sentence p can be accepted and rejected
Solution:
- distinction told vs marked values
- truth-values are elements in increasingly complex sets
n=1
{T} = {T}
n=2
{T,{ }} = {T,F}
n=3
{{T},{F},{ }} = {T,F,N}
n=4
{{T},{F},{T,F},{ }} = {T,F,B,N}
…
2 Question-Answer Semantics
Saptabhangi:
set of 7 marked values expressing standpoints (naya)
each told value expresses one kind of standpoint
there may be several standpoints in a single model
3 basic predications (mūlabhaṅga): told values in {T,F,A}
Semantic predicates (in boldface) are assigned to sentences in the form of
statements (see Priest (2011))
1st bhanga:
p is true
p{T}
2nd bhanga:
p is false
p{F}
3rd bhanga: p is avaktavyam
1st int.: true and false simultaneously
2nd int.: neither true nor false
p{A}
p{B}
p{N}
2 Question-Answer Semantics
Saptabhangi:
set of 7 marked values expressing standpoints (naya)
each told value expresses one kind of standpoint
there may be several standpoints in a single model
3 basic predications (mūlabhaṅga): told values in {T,F,A}
Semantic predicates (in boldface) are assigned to sentences in the form of
statements (see Priest (2011))
(a)
p is true
Tp
(b)
p is false
Fp
(c)
p is avaktavyam (?)
1st int.: true and false simultaneously
2nd int.: neither true nor false
Bp
Np
2 Question-Answer Semantics
What do t uth a d falsit
Non-falsity = truth
Non-falsity truth
in V = {T,F}
in V = {T,F,N}
ean in various domains of values V?
�p = Fp
�p = Fp or Np
in V = {T,F}
in V = {T,F,N}
How many truth-values are there in the saptabhangi?
Truth-value: told values (paribhasa) and/or marked values (bhangi)?
subset of the set of basic, told values
Domain of values: the set of the subsets of sets of basic, told values
Example: TFp = Bp
2 Question-Answer Semantics
What do t uth a d falsit
Non-falsity = truth
Non-falsity truth
in V = {T,F}
in V = {T,F,N}
ea i
a ious do ai s of alues D?
�p = Fp
�p = Fp or Np
in V = {T,F}
in V = {T,F,N}
How many truth-values are there in the saptabhangi?
Truth-value: told values (paribhasa) and/or marked values (bhangi)?
subset of the set of basic, told values
Domain of values: the set of the subsets of sets of basic, told values
Example: TFp = Bp
2 Question-Answer Semantics
Option #1: a combination of 7 marked values among {T,F,A}
(i.e. a combination of 7 combinations of told values)
1
2
3
4
5
6
7
Tp
Fp
Tp, Fp
Ap
Tp, Ap
Fp, Ap
Tp, Fp, Ap
Tp
�p
Tp
�p
Tp
�p
Tp
आ
आ
आ
आ
आ
आ
आ
�p
Fp
Fp
�p
�p
Fp
Fp
आ
आ
आ
आ
आ
आ
आ
p
p
p
Ap
Ap
Ap
Ap
2 Question-Answer Semantics
Option #1: a combination of 7 marked values among {T,F,A}
(i.e. a combination of 7 combinations of told values)
1
2
3
4
5
6
7
Tp
Fp
Tp, Fp
Ap
Tp, Ap
Fp, Ap
Tp, Fp, Ap
Tp
�p
Tp
�p
Tp
�p
Tp
आ
आ
आ
आ
आ
आ
आ
�p
Fp
Fp
�p
�p
Fp
Fp
आ
आ
आ
आ
आ
आ
आ
p
p
p
Ap
Ap
Ap
Ap
2 Question-Answer Semantics
Option #1: a combination of 7 subsets of elements of the set {a,b,c}
(i.e. a combination of 7 combinations of told values)
1
2
3
4
5
6
7
Tp
Fp
Tp, Fp
Ap
Tp, Ap
Fp, Ap
Tp, Fp, Ap
7 = P(3) – 1
(1 for the empty set: { })
a
a
b
b
a
a
b
b
c
c
c
c
2 Question-Answer Semantics
Option #1: a combination of 7 subsets of elements of the set {a,b,c}
(i.e. a combination of 7 combinations of told values)
1
2
3
4
5
6
7
Tp
Fp
Tp, Fp
Ap
Tp, Ap
Fp, Ap
Tp, Fp, Ap
7 = P(3) – 1
(1 for the empty set: { })
a
a
b
b
a
a
b
b
c
c
c
c
2 Question-Answer Semantics
Option #1: a combination of 7 subsets of elements of the set {a,b,c}
(i.e. a combination of 7 combinations of told values)
1
2
3
4
5
6
7
Tp
Fp
Tp, Fp
Ap
Tp, Ap
Fp, Ap
Tp, Fp, Ap
7 = P(3) – 1
(1 for the empty set: 000)
1
0
1
0
1
0
1
0
1
1
0
0
1
1
0
0
0
1
1
1
1
2 Question-Answer Semantics
Option #2: a combination of 15 marked values among {T,F,B,N}
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Tp
Fp
Bp
Np
Tp, Fp
Tp, Bp
Tp, Np
Fp, Bp
Fp, Np
Bp, Np
Tp, Fp, Bp
Tp, Fp, Np
Tp, Bp, Np
Fp, Bp, Np
Tp, Fp, Bp, Np
Tp
�p
�p
�p
Tp
Tp
Tp
�p
�p
�p
Tp
Tp
Tp
�p
Tp
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
�p
Fp
�p
�p
Fp
�p
�p
Fp
Fp
�p
Fp
Fp
�p
Fp
Fp
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
p
p
Bp
p
p
Bp
p
Bp
p
Bp
Bp
p
Bp
Bp
Bp
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
आ
�p
�p
�p
Np
�p
�p
Np
�p
Np
Np
�p
Np
Np
Np
Np
2 Question-Answer Semantics
Option #2: a combination of 15 subsets of elements of the set {a,b,c,d}
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Tp
Fp
Bp
Np
Tp, Fp
Tp, Bp
Tp, Np
Fp, Bp
Fp, Np
Bp, Np
Tp, Fp, Bp
Tp, Fp, Np
Tp, Bp, Np
Fp, Bp, Np
Tp, Fp, Bp, Np
15 = P(4) – 1
(1 for the empty set: { })
a
b
c
d
a
a
a
b
c
d
b
b
a
a
a
a
b
b
b
b
c
c
c
c
c
c
d
d
d
d
d
d
2 Question-Answer Semantics
Option #2: 15 marked values among {a,b,c,d} (where c = Bp, d = Np)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Tp
Fp
Bp
Np
Tp, Fp
Tp, Bp
Tp, Np
Fp, Bp
Fp, Np
Bp, Np
Tp, Fp, Bp
Tp, Fp, Np
Tp, Bp, Np
Fp, Bp, Np
Tp, Fp, Bp, Np
15 = P(4) – 1
(1 for the empty set: 0000)
1
0
0
0
1
1
1
0
0
0
1
1
1
0
1
0
1
0
0
1
0
0
1
1
0
1
1
0
1
1
0
0
1
0
0
1
0
1
0
1
1
0
1
1
1
0
0
0
1
0
0
1
0
1
1
0
1
1
1
1
2 Question-Answer Semantics
Option #3: 1 marked value among {T} = {T}
1
Tp
Tp
2 Question-Answer Semantics
Option #3: 1 singleton of the set {a}
1
Tp
1 = P(1) – 1
(1 for the empty set: 0)
a
2 Question-Answer Semantics
Option #3: 1 singleton of the set {a}
1
Tp
1 = P(1) – 1
(1 for the empty set: 0)
1
2 Question-Answer Semantics
Generalized truth values
Zaitsev & Shramko (2013: 1300)
Definition 1.1.
Let X be a (basic) set of initial truth values, and let P(X) be the power-set of
X. Then the elements of P(X) are called generalized truth values defined on
the basis of X.
Definition 1.2.
Let X be a (basic) set of initial truth values, P(X) the set of generalized truth
values defined on the basis of X, and L a given language. Then a generalized
truth value function (defined on the basis of X) is a function from the set of
sentences of L into P(X).
2 Question-Answer Semantics
Generalized truth values
Zaitsev & Shramko (2013: 1300)
Definition 1.1.
Let X be a (basic) set of initial truth values, and let P(X) be the power-set of
X. Then the elements of P(X) are called generalized truth values defined on
the basis of X.
Definition 1.2.
Let X be a (basic) set of initial truth values, P(X) the set of generalized truth
values defined on the basis of X, and L a given language. Then a generalized
truth value function (defined on the basis of X) is a function from the set of
sentences of L into P(X).
2 Question-Answer Semantics
Generalized truth values
Zaitsev & Shramko (2013: 1300)
Definition 1.1.
Let X be a (basic) set of initial truth values, and let P(X) be the power-set of
X. Then the elements of P(X) are called generalized truth values defined on
the basis of X.
Definition 1.2.
Let X be a (basic) set of initial truth values, P(X) the set of generalized truth
values defined on the basis of X, and L a given language. Then a generalized
truth value function (defined on the basis of X) is a function from the set of
sentences of L into P(X).
2 Question-Answer Semantics
A generalization of generalized truth values
Question-Answer Semantics
Definition 1.1.
Let n be a (basic) set of initial questions, and let m be the corresponding set
of answers to n. Then the elements of mn are called generalized truth
values defined on the basis of n.
Definition 1.2.
Let n be a (basic) set of initial questions, m the corresponding set of
answers to n, and L a given language. Then a generalized truth value
function (defined on the basis of n) is a function from the set of sentences
of L into mn.
2 Question-Answer Semantics
A generalization of generalized truth values
Question-Answer Semantics
Definition 1.1.
Let n be a (basic) set of initial questions, and let m be the corresponding set
of answers to n. Then the elements of mn are called generalized truth
values defined on the basis of n.
Definition 1.2.
Let n be a (basic) set of initial questions, m the corresponding set of
answers to n, and L a given language. Then a generalized truth value
function (defined on the basis of n) is a function from the set of sentences
of L into mn.
2 Question-Answer Semantics
A generalization of generalized truth values
Question-Answer Semantics
Definition 1.1.
Let n be a (basic) set of initial questions, and let m be the corresponding set
of answers to n. Then the elements of mn are called generalized truth
values defined on the basis of n.
Definition 1.2.
Let n be a (basic) set of initial questions, m the corresponding set of
answers to n, and L a given language. Then a generalized truth value
function (defined on the basis of n) is a function from the set of sentences
of L into mn.
2 Question-Answer Semantics
Algebras of Acceptance and Rejection: ARmn
A common framework for arbitrary semantics
A Acceptance
ai(p) = 1
R Rejection
ai(p) = 0
m number of answers
Ai(p) = a1 p ,…, an(p)
n number of questions
Qi(p) = q1 p ,…, qn(p)
Not every truth value is an element of a power-set, in ARmn
n
does
not
equate
with
P(n)
=
2
m
m = , i “h a ko & Wa si g s f a e o k
n
2 Question-Answer Semantics
•
marked vs told values in ARmn
For any logical value A(p) = a1 p , …, an(p) in ARmn:
- each element ai(p) of ARmn is a told value
- marked values A(p) are meets of elements: A(p) = ai(p) ⊓ aj(p)
- told values Xp in ARmn correspond to marked values Xp in ARmn+1
In AR21 = AR2, Xp = Xp (the difference marked/told values is redundant)
Tp = Tp, so that A(p) = a1(p) = 1
Fp = Fp, so that A(p) = a1(p) = 0
In AR22 = AR4, Xp = Xp (the difference marked/told values is not redundant)
Tp = Tp ⊓ P �p, so that A(p) = a1(p),a2(p) = 10
Fp = �p ⊓ Fp, so that A(p) = a1(p),a2(p) = 01
Xp Xp whenever n >
o e tha state e t a out p s t uth-value)
2 Question-Answer Semantics
•
marked vs told values in ARmn
For any logical value A(p) = a1 p , …, an(p) in ARmn:
- each element ai(p) of ARmn is a told value
- marked values A(p) are meets of elements: A(p) = ai(p) ⊓ aj(p)
- told values Xp in ARmn correspond to marked values Xp in ARmn+1
In AR21 = AR2, Xp = Xp (the difference marked/told values is redundant)
Tp = Tp, so that A(p) = a1(p) = 1
Fp = Fp, so that A(p) = a1(p) = 0
In AR22 = AR4, Xp = Xp (the difference marked/told values is not redundant)
Tp = Tp ⊓ P �p, so that A(p) = a1(p),a2(p) = 10
Fp = �p ⊓ Fp, so that A(p) = a1(p),a2(p) = 01
Xp Xp whenever n >
o e tha state e t a out p s t uth-value)
2 Question-Answer Semantics
•
marked vs told values in ARmn
For any logical value A(p) = a1 p , …, an(p) in ARmn:
- each element ai(p) of ARmn is a told value
- marked values A(p) are meets of elements: A(p) = ai(p) ⊓ aj(p)
- told values Xp in ARmn correspond to marked values Xp in ARmn+1
In AR21 = AR2, Xp = Xp (the difference marked/told values is redundant)
Tp = Tp, so that A(p) = a1(p) = 1
Fp = Fp, so that A(p) = a1(p) = 0
In AR22 = AR4, Xp = Xp (the difference marked/told values is not redundant)
Tp = Tp ⊓ P �p, so that A(p) = a1(p),a2(p) = 10
Fp = �p ⊓ Fp, so that A(p) = a1(p),a2(p) = 01
Xp Xp whenever n >
o e tha state e t a out p s t uth-value)
2 Question-Answer Semantics
An example of 4-valuedness: bilateralist logic AR22 = AR4
AR4 = L, A,V4,आ,इ
L
set of fo ulae {p, , …}
set of logical functions {,,,}
A(p)
valuation function, mapping from L to V4
A(p) = a1(p), a2(p)
V4
{11, 10, 00, 01}
1आ0 = 0आ1 = 0आ0 = 0, 1आ1 = 1
1इ1 = 1इ0 = 0इ1 = 1, 0इ0 = 0
2 Question-Answer Semantics
Negation
A(p) = a1(p), a2(p) = a2(p), a1(p)
a1(p) = 1
a2(p) = 1
iff
iff
a2(p) = 1
a1(p) = 1
p
11
10
01
00
p
11
01
10
00
2 Question-Answer Semantics
Negation
A(p) = a1(p), a2(p) = a2(p), a1(p)
a1(p) = 1
a2(p) = 1
iff
iff
a2(p) = 1
a1(p) = 1
p
11
10
01
00
p
11
01
10
00
2 Question-Answer Semantics
Conjunction
A(p q) = a1(p q), a2(p q) = a1(p)आa2(q), a1(p)इa2(q)
a1(p q) = 1
a2(p q) = 1
iff
iff
a1(p) = 1 and a2(p) = 1
a1(p) = 0 or a2(p) = 0
11
10
01
00
11
11
11
01
01
10
11
10
01
00
01
01
01
01
01
00
01
00
01
00
2 Question-Answer Semantics
Conjunction
A(p q) = a1(p q), a2(p q) = a1(p)आa2(q), a1(p)इa2(q)
a1(p q) = 1
a2(p q) = 1
iff
iff
a1(p) = 1 and a2(p) = 1
a1(p) = 0 or a2(p) = 0
11
10
01
00
11
11
11
01
01
10
11
10
01
00
01
01
01
01
01
00
01
00
01
00
2 Question-Answer Semantics
Disjunction
A(pq) = a1(p q), a2(p q) = a1(p)इa2(q), a1(p)आa2(q)
a1(p q) = 1
a2(p q) = 1
iff
iff
a1(p) = 1 or a2(p) = 1
a1(p) = 0 and a2(p) = 0
11
10
01
00
11
11
10
11
10
10
10
10
10
10
01
11
10
01
00
00
10
10
00
00
2 Question-Answer Semantics
Disjunction
A(pq) = a1(p q), a2(p q) = a1(p)इa2(q), a1(p)आa2(q)
a1(p q) = 1
a2(p q) = 1
iff
iff
a1(p) = 1 or a2(p) = 1
a1(p) = 0 and a2(p) = 0
11
10
01
00
11
11
10
11
10
10
10
10
10
10
01
11
10
01
00
00
10
10
00
00
2 Question-Answer Semantics
Conditional st e ghe ed
A(p q) = a1(p q), a2(p q) = a1(p)आa1(q), a1(p)आa2(q)
a1(p q) = 1
a2(p q) = 1
iff
iff
a1(p) = 1 and a2(q) = 1
a1(p) = 1 and a2(q) = 0
11
10
01
00
11
11
11
00
00
10
10
10
00
00
01
01
01
00
00
00
01
00
00
00
2 Question-Answer Semantics
Conditional st e ghe ed
A(p q) = a1(p q), a2(p q) = a1(p)आa1(q), a1(p)आa2(q)
a1(p q) = 1
a2(p q) = 1
iff
iff
a1(p) = 1 and a2(q) = 1
a1(p) = 1 and a2(q) = 0
11
10
01
00
11
11
11
00
00
10
10
10
00
00
01
01
01
00
00
00
01
00
00
00
2 Question-Answer Semantics
Conditional st e ghe ed
A(p q) = a1(p q), a2(p q) = a1(p)आa1(q), a1(p)आa2(q)
a1(p q) = 1
a2(p q) = 1
iff
iff
a1(p) = 1 and a2(q) = 1
a1(p) = 1 and a2(q) = 0
11
10
01
00
11
11
11
00
00
10
10
10
00
00
01
01
01
00
00
00
01
00
00
00
See Schang & Trafford (201X):
Is o a fo e-i di ato ? Yes, soo e o late ! to e su
itted
2 Question-Answer Semantics
0-valuedness?
AR0n = ARm0 = AR0
m ?
or a1(p) { }
n ?
or { }
A(p) { }
P iest s sile e ?
Not a value, but a lack of value (compare with ½ in AR21)!
2 Question-Answer Semantics
1-valuedness?
AR1n = AR1
m ai(p){1}
n ?
A(p) {1}
fo
es
Saptabhangi (Balcerowicz 2011)
2 Question-Answer Semantics
1-valuedness?
AR1n = AR1
m ai(p){0}
n ?
A(p) {0}
fo
o
Catuskoti (Schang 2013)
2 Question-Answer Semantics
2-valuedness?
AR21 = AR2
m ai(p){1,0}
n q1(p) = Tp?
A(p) {1,0}
1 for
fo
es
o
2 Question-Answer Semantics
3-valuedness?
AR31 = AR3
m ai(p) {1,1/2,0}
n q1(p) = Tp ?
A(p) {1,1/2,0}
fo yes
½ fo both yes and no , if qi(p) = Bp
fo absolutely o
Glutty logics
2 Question-Answer Semantics
3-valuedness?
AR31 = AR3
m ai(p) {1,1/2,0}
n q1(p) = Tp ?
A(p) {1,1/2,0}
fo es
½ fo oth es a d o , if qi(p) = Bp
fo a solutel o
Glutty logics
2 Question-Answer Semantics
3-valuedness?
AR31 = AR3
m ai(p) {1,1/2,0}
n q1(p) = Tp ?
A(p) {1,1/2,0}
fo es
½ fo neither yes nor o , if qi(p) = Np
fo o
Gappy logics
2 Question-Answer Semantics
3-valuedness?
AR31 = AR3
m ai(p) {1,1/2,0}
n q1(p) = Tp ?
A(p) {1,1/2,0}
fo
½ fo
fo
es
eithe
o
Gappy logics
es o
o , if qi(p) = Np
2 Question-Answer Semantics
3-valuedness?
AR22 – 1 = AR3
m ai(p) {1,0}
n q1(p) = Tp ?
q2(p) = Fp ?
A(p) {1,1,1,0,0,1,0,0} – {1,1}
A(p) {1,0,0,1,0,0}
fo
fo
es
o
Gappy logics
2 Question-Answer Semantics
3-valuedness?
AR22 – 1 = AR3
m ai(p) {1,0}
n q1(p) = Tp ?
q2(p) = Fp ?
A(p) {1,1,1,0,0,1,0,0} – {1,1}
A(p) {1,0,0,1,0,0}
fo
fo
es
o
Gappy logics
2 Question-Answer Semantics
3-valuedness?
AR22 – 1 = AR3
m ai(p) {1,0}
n q1(p) = Tp ?
q2(p) = Fp ?
A(p) {1,1,1,0,0,1,0,0} – {0,0}
A(p) {1,1,1,0,0,1}
fo
fo
es
o
Glutty logics
2 Question-Answer Semantics
3-valuedness?
AR22 – 1 = AR3
m ai(p) {1,0}
n q1(p) = Tp ?
q2(p) = Fp ?
A(p) {1,1,1,0,0,1,0,0} – {0,0}
A(p) {1,1,1,0,0,1}
fo
fo
es
o
Glutty logics
2 Question-Answer Semantics
4-valuedness?
AR41 = AR4
m ai(p) { ,⅔,⅓,0}
n q1(p) = Tp ?
A(p) {1,2/3,1/3,0}
fo es
2/
yes and no
3 fo
1/
neither yes nor no
3 fo
fo no
2 Question-Answer Semantics
4-valuedness?
AR22 = AR4
m ai(p) {1,0}
n q1(p) = Tp ?
q2(p) = Fp ?
A(p) {1,1,1,0,0,1,0,0}
fo
fo
es
o
2 Question-Answer Semantics
7-valuedness?
AR23 – 1 = AR8-1 = AR7
m ai(p) {1,0}
n q1(p) = Tp ?
q2(p) = Np ?
q3(p) = Fp ?
A(p) {1,1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0,0,1}
fo
fo
es
o
Gappy logics
2 Question-Answer Semantics
7-valuedness?
AR23 – 1 = AR8-1 = AR7
m ai(p) {1,0}
n q1(p) = Tp ?
q2(p) = Np ?
q3(p) = Fp ?
A(p) {1,1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0,0,1}
fo
fo
es
o
Gappy logics
2 Question-Answer Semantics
7-valuedness?
AR23 – 1 = AR8-1 = AR7
m ai(p) {1,0}
n q1(p) = Tp ?
q2(p) = Bp ?
q3(p) = Fp ?
A(p) {1,1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0,0,1}
fo
fo
es
o
Glutty logics
2 Question-Answer Semantics
7-valuedness?
AR23 – 1 = AR8-1 = AR7
m ai(p) {1,0}
n q1(p) = Tp ?
q2(p) = Bp ?
q3(p) = Fp ?
A(p) {1,1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0,0,1}
fo
fo
es
o
Glutty logics
2 Question-Answer Semantics
16-valuedness?
AR42 = AR16
m ai(p) {1,2/3,1/3,0}
n q1(p) = Tp ?
q2(p) = Fp ?
A(p) {1,1,1,2/3,1,1/3,1,0,2/3,1,2/3,2/3,2/3,1/3,2/3,0,
1/3,1,1/3,1,1/3,1,1/3,1,0,1,0,1,0,1/3,0,0}
fo es
2/
yes and no
3 fo
1/
neither yes nor no
3 fo
0 fo o
o o l es
o es, but not only )
o o, but not only )
o o l o
2 Question-Answer Semantics
16-valuedness?
AR24 = AR16
m a1(p) {1,0}
n q1(p) = Tp ?
q2(p) = Bp ?
q2(p) = Np ?
q2(p) = Fp ?
A(p)
{1,1,1,1,1,1,1,0,1,1,0,1,1,0,1,1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,1,
0,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0}
1 fo
0 fo
es
o
2 Question-Answer Semantics
Catuskoti in ARmn:
AR22 + 1 = AR5
(Priest (2011))
AR14 = AR1
(Schang (2013))
or else?
Saptabhangi in ARmn:
AR23 – 1 = AR7
(Ganeri (2002), Priest (2008))
AR42 – 1 = AR15 (Sylvan (1987))
AR14 = AR1
(Schang (2013))
AR17 = AR1
(Balcerowicz (2011))
or else?
How can one-valued theories AR1 e p ope logi s ?
- designated values in ARn (with n > 2): marked values including T
- which sentence p does not include T, in the saptabhangi?
- which sentence p does include T, in the catuskoti?
3
Dialectical negation
3 Dialectical negation
What is negation in the catuskoti and saptabhangi?
(1) locutionary negation ( pa udāsap atiṣedha : operator
In ARmn:
- involutive (Boolean and De Morgan) operator in AR21, such that
ai(p) = 1 iff ai(p) = 0
- negations in AR2n (with n > 1)
epistemic, Boolean negation b: switching operator, such that
b(a1(p), ... ,an(p)) = ((a1(p)ꞌ, ... ,an(p)ꞌ )
ontological, De Morgan negation d: permuting operator, such that
d(a1(p), ... ,an(p)) = ((an(p), ... ,a1(p))
3 Dialectical negation
What is negation in the catuskoti and saptabhangi?
(1) locutionary negation pa udāsapratiṣedha : ope ato
In ARmn:
- involutive (Boolean and De Morgan) operator in AR21, such that
ai(p) = 1 iff ai(p) = 0
- negations in AR2n (with n > 1)
epistemic, Boolean negation b: switching operator, such that
b(a1(p), ... ,an(p)) = ((a1(p)ꞌ, ... ,an(p)ꞌ )
ontological, De Morgan negation d: permuting operator, such that
d(a1(p), ... ,an(p)) = ((an(p), ... ,a1(p))
3 Dialectical negation
What is negation in the catuskoti and saptabhangi?
(2) locutionary egatio
pa udāsap atiṣedha : ope ato
In ARmn:
- involutive (Boolean and De Morgan) operator in AR21, such that
ai(p) = 1 iff ai(p) = 0
- negations in AR2n (with n > 1)
epistemic, Boolean negation b: switching operator, such that
b(a1(p), ... ,an(p)) = ((a1(p)ꞌ, ... ,an(p)ꞌ )
ontological, De Morgan negation d: permuting operator, such that
d(a1(p), ... ,an(p)) = ((an(p), ... ,a1(p))
3 Dialectical negation
What is negation in the catuskoti and saptabhangi?
(3) locutionary egatio
pa udāsap atiṣedha : ope ato
In ARmn:
- involutive (Boolean and De Morgan) operator in AR21, such that
ai(p) = 1 iff ai(p) = 0
- negations in AR2n (with n > 1)
epistemic, Boolean negation b: switching operator, such that
b(a1(p), ... ,an(p)) = ((a1(p)ꞌ, ... ,an(p)ꞌ )
ontological, De Morgan negation d: permuting operator, such that
d(a1(p), ... ,an(p)) = ((an(p), ... ,a1(p))
3 Dialectical negation
What is negation in the catuskoti and saptabhangi?
(4) illocutionary negation ( prasajyapratiṣedha : operand
In ARmn: negation as a speech-act of rejection (no-answer), such that
ai(p) = 0
- rejection is the same as negative assertion in AR21, only
o, p is ot T = es, p is � = F = es, p is T
o, p is ot F = es, p is � = T = es, p is T
- rejection is complementation in AR2n
o, p is ot X = es, p is �
3 Dialectical negation
What is negation in the catuskoti and saptabhangi?
(2) illocutionary negation prasajyapratiṣedha : ope a d
In ARmn: negation as a speech-act of rejection (no-answer), such that
ai(p) = 0
- rejection is the same as negative assertion in AR21, only
o, p is ot T = es, p is � = F = es, p is T
o, p is ot F = es, p is � = T = es, p is T
- rejection is complementation in AR2n
o, p is ot X = es, p is �
3 Dialectical negation
A third reading of negation
(3) dialectical negation
A metalinguistic operator d mapping on algebras, such that:
d(ARmn) = ARmn + 1 = AR1n+1
How can negation be applied to a whole set of values V, rather than a
single value A(p)V?
2 interpretations of dialectical negation on truth-values:
- epistemological: society semantics (formal epistemology)
- ontological truth-values: ontological monism (formal ontology)
Catuskoti:
d(AR1n)
Saptabhangi: d(AR1n)
= AR1n+1 (where a(p) = 1)
= AR1n+1 (where a(p) = 0)
3 Dialectical negation
A Hegelia extension of the saptabhangi: L = {p}
- everything is (one unique thing exhausts the world: the Absolute)
Every predication is true of the Absolute; thus, for every p, A(p) = 1
A Hegelia extension of the catuskoti: L = { }
- nothing is (the world is empty: Buddhist nothingness)
No predication is true of the Absolute; thus, for every p, A(p) = 1
Hegelia diale ti s: thesis-anthesis-synthesis
- dialectics leads to Aufhebung (overcome the negative): X, �, ��
- the True = p, conserved through negation without being rejected
Truth-values as proper names:
each sentence p refers to a single truth-value X, s.t. X = p
3 Dialectical negation
In L:
p
p
p p
(p p)
(p p) (p p)
((p p) (p p))
(p p) (p p) ((p p) (p p))
…
3 Dialectical negation
In L: everything is p
p
p
p p
(p p)
(p p) (p p)
((p p) (p p))
(p p) (p p) ((p p) (p p))
…
3 Dialectical negation
In L
p
q
r
s
…
3 Dialectical negation
In V: everything is T
T
T
TT
TT
TTTT
TTTT
TTTTTTTT
…
3 Dialectical negation
T
thesis in L1 = {p}
T
antithesis in L1
TT
synthesis in L2 = thesis in L2 = {p,q}
TT
TTTT
TTTT
TTTTTTTT
antithesis in L2
synthesis in L2 = thesis in L3 = {p,q,r}
antithesis in L3
synthesis in L3 = thesis in L4 = {p,q,r,s}
…
3 Dialectical negation
T
thesis in L1 = {p}
T
antithesis in L1
TT
synthesis in L2 = thesis in L2 = {p,q}
TT
TTTT
TTTT
antithesis in L2
synthesis in L2 = thesis in L3 = {p,q,r}
P iest s 5th value?
TTTTTTTT
antithesis in L3
synthesis in L3 = thesis in L4 = {p,q,r,s}
…
3 Dialectical negation
Base-2 arithmetics
Base-10 arithmetics
1
1
10
2
11
3
100
4
101
5
110
6
111
7
…
3 Dialectical negation
3 Dialectical negation
Dialectical negation as a successor operator S(n) = n +1
A difference with dichotomy (see Priest (2011): 31)
- dichotomy corresponds to AR2n+1 (powersetting)
- dialectical negation corresponds to addition AR1n+1
3 Dialectical negation
T
AR11 = AR1
3 Dialectical negation
1
AR11 = AR1
3 Dialectical negation
T
AR21 = AR2
T
3 Dialectical negation
1
AR21 = AR2
0
3 Dialectical negation
XD
AR21 = AR2
1
0
XD
3 Dialectical negation
AR22 = AR4
TT
TT
TT
TT
3 Dialectical negation
AR22 = AR4
10
00
11
01
3 Dialectical negation
1st Boolean semi-negation:
10
00
11
01
b1/2(a1(p),a2(p)) = (a1(p))ꞌ,a2(p)
AR22 = AR4
3 Dialectical negation
2nd Boolean semi-negation:
10
00
11
01
b2/2(a1(p),a2(p)) = a1(p),(a2(p))ꞌ
AR22 = AR4
3 Dialectical negation
10
00
XD
XD
11
AR22 = AR4
01
3 Dialectical negation
AR23 = AR8
TTTT
TTTT
TTTT
TTTT
TTTT
TTTT
TTTT
TTTT
3 Dialectical negation
AR23 = AR8
101
100
001
000
111
110
011
010
3 Dialectical negation
101
100
001
000
XD
XD
111
AR23 = AR8
110
011
010
3 Dialectical negation
AR24 = AR16
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
TTTTTTTT
3 Dialectical negation
1010 1000 0010 0000
1011 1001 0011 0001
1110 1100 0110 0100
1111 1101 0111 0101
AR24 = AR16
3 Dialectical negation
1010 1000 0010 0000
1011 1001 0011 0001
XD
XD
1110 1100 0110 0100
1111 1101 0111 0101
AR24 = AR16
3 Dialectical negation
Dialectical negation as a successor operator S(n) = n +1
A difference with dichotomy (see Priest (2011): 31)
- dichotomy corresponds to AR2n+1 (powersetting): partition of V
- dialectical negation corresponds to addition AR1n+1
Extension of the saptbhangi
AR11 A(p) = 1
AR12 A(p) = 11
AR13 A(p) = 111
…
AR112 A(p) = 111111111111
…
AR1n A p =
…
n times
3 Dialectical negation
Dialectical negation as a successor operator S(n) = n +1
A difference with dichotomy (see Priest (2011): 31)
- dichotomy corresponds to AR2n+1 (powersetting): partition of V
- dialectical negation corresponds to addition AR1n+1
Extension of the saptbhangi
AR11 A(p) = 1
AR12 A(p) = 11
AR13 A(p) = 111
…
AR112 A(p) = 111111111111
…
AR1n A(p) = 111111111111 … 1
n times
3 Dialectical negation
Dialectical negation as a successor operator S(n) = n +1
A difference with dichotomy (see Priest (2011): 31)
- dichotomy corresponds to AR2n+1 (powersetting): partition of V
- dialectical negation corresponds to addition AR1n+1
Extension of the catuskoti
AR11 A(p) = 0
AR12 A(p) = 00
AR13 A(p) = 000
…
AR112 A(p) = 000000000000
…
AR1n A p =
…
n times
3 Dialectical negation
Dialectical negation as a successor operator S(n) = n +1
A difference with dichotomy (see Priest (2011): 31)
- dichotomy corresponds to AR2n+1 (powersetting): partition of V
- dialectical negation corresponds to addition AR1n+1
Extension of the catuskoti
AR11 A(p) = 0
AR12 A(p) = 00
AR13 A(p) = 000
…
AR112 A(p) = 000000000000
…
AR1n A(p) = 000000000000 … 0
n times
3 Dialectical negation
A dialectical
(a)
(b)
(c)
(d)
e sio of the Catuskoti:
v(p) T
v(p) T
v(p) TT
v(p) TT
3 Dialectical negation
Law of n-th negation: cyclic negation modulo n, s.t. X = � in ARmn
⁞ ⁞
n times
Designated values = accepted values
For every value X in V, XpD
iff
p is accepted
XpD
iff
p is not accepted, i.e. rejected
Meaning of truth-values (see Suszko (1977)): t uth ? falsit ?
- logical truth/falsity: the class of truth-values X including T
p is logically true iff A(p)D
p is logically false iff A(p)D
- algebraic truth/falsity: told values T and F = �
For every truth-value X,
X
is a designated value
X
is a non-designated value
iff
iff
TX
TX
3 Dialectical negation
Law of n-th negation: cyclic negation modulo n, s.t. X = � in ARmn
⁞ ⁞
n times
Designated values = accepted values
For every value X in V, XpD
iff
p is accepted
XpD
iff
p is not accepted, i.e. rejected
Meaning of truth-values (see Suszko (1977)): t uth ? falsit ?
- logical truth/falsity: the class of truth-values X including T
p is logically true iff A(p)D
p is logically false iff A(p)D
- algebraic truth/falsity: told values T and F = �
For every truth-value X,
X
is a designated value
X
is a non-designated value
iff
iff
TX
TX
4.
Conclusion (and Prospects)
4 Conclusion
Are LC and LS dual logics?
if there is no designated value XD in LC,
LC is both paraconsistent and paracomplete
For every truth-value Xp:
Xp╞/ Xq
Xp╞/ Xq, Xq
Tarskian trivial logic: every sentence follows from every other one
if there is no non-designated value XD in LC,
LS is neither paraconsistent nor paracomplete: Tarskian trivial logic
For every truth-value X:
Xp╞ Xq
Xp╞ Xq, Xq
Tarskian trivial logic: no sentence follows from no other one
dualized by Boolean negation: p in LC = b(p) in LS
b
… =
…
4 Conclusion
Are LC and LS dual logics?
if there is no designated value XD in LC,
LC is both paraconsistent and paracomplete
For every truth-value Xp:
Xp╞/ Xq
Xp╞/ Xq, Xq
Tarskian trivial logic: every sentence follows from every other one
if there is no non-designated value XD in LC,
LS is neither paraconsistent nor paracomplete: Tarskian trivial logic
For every truth-value X:
Xp╞ Xq
Xp╞ Xq, Xq
Tarskian trivial logic: no sentence follows from no other one
dualized by Boolean negation: p in LC = b(p) in LS
b
… =
…
4 Conclusion
Are LC and LS dual logics?
if there is no designated value XD in LC,
LC is both paraconsistent and paracomplete
For every truth-value Xp:
Xp╞/ Xq
Xp╞/ Xq, Xq
Tarskian trivial logic: every sentence follows from every other one
if there is no non-designated value XD in LC,
LS is neither paraconsistent nor paracomplete: Tarskian trivial logic
For every truth-value X:
Xp╞ Xq
Xp╞ Xq, Xq
Tarskian trivial logic: no sentence follows from no other one
dualized by Boolean negation: p in LC = b(p) in LS
b
… =
…
4 Conclusion
Are LC and LS dual logics?
if there is no designated value XD in LC,
LC is both paraconsistent and paracomplete
For every truth-value Xp:
Xp╞/ Xq
Xp╞/ Xq, Xq
Tarskian trivial logic: every sentence follows from every other one
if there is no non-designated value XD in LC,
LS is neither paraconsistent nor paracomplete: Tarskian trivial logic
For every truth-value X:
Xp╞ Xq
Xp╞ Xq, Xq
Tarskian trivial logic: no sentence follows from no other one
dualized by Boolean negation: p in LC = b(p) in LS
b
… =
…
4 Conclusion
Are LC and LS dual logics?
if there is no designated value XD in LC,
LC is both paraconsistent and paracomplete
For every truth-value Xp:
Xp╞/ Xq
Xp╞/ Xq, Xq
Tarskian trivial logic: every sentence follows from every other one
if there is no non-designated value XD in LC,
LS is neither paraconsistent nor paracomplete: Tarskian trivial logic
For every truth-value X:
Xp╞ Xq
Xp╞ Xq, Xq
Tarskian trivial logic: no sentence follows from no other one
dualized by Boolean negation: p in LS = b(p) in LC
b
… =
…
4 Conclusion
Are LC and LS dual logics?
if there is no designated value XD in LC,
LC is both paraconsistent and paracomplete
For every truth-value Xp:
Xp╞/ Xq
Xp╞/ Xq, Xq
Tarskian trivial logic: every sentence follows from every other one
if there is no non-designated value XD in LC,
LS is neither paraconsistent nor paracomplete: Tarskian trivial logic
For every truth-value X:
Xp╞ Xq
Xp╞ Xq, Xq
Tarskian trivial logic: no sentence follows from no other one
dualized by Boolean negation: p in LS = b(p) in LC
b
… =
…
4 Prospects
Limits of rationality in dialogue
Are there impossible answers in every dialogical situation?
A indefinite range of logics within e t e e a s e s: all yes
Non-classical answers
What is the logical ea i g of
ARmn (with m > 2)
es a d o , o
eithe
Future works
Inclusive algebras in ARmn: ARmn ARmn+1, ARmn ARm+1n
Equivalent algebras in ARmn: ARmn = AR
Non-classical answers and many-valued modal logics
Many-valuedness: set of questions n in ARmn
Modalities: modes of answers m in ARmn
s all
es o
?
?
o
o ?
4 Prospects
Limits of rationality in dialogue
Are there impossible answers in every dialogical situation?
A i defi ite a ge of logi s ithi e t e e a s e s: all es
Non-classical answers
What is the logi al ea i g of
ARmn (with m > 2)
es a d o , o
eithe
Future works
Inclusive algebras in ARmn: ARmn ARmn+1, ARmn ARm+1n
Equivalent algebras in ARmn: ARmn = AR
Non-classical answers and many-valued modal logics
Many-valuedness: set of questions n in ARmn
Modalities: modes of answers m in ARmn
s all
es o
?
?
o
o ?
4 Prospects
Limits of rationality in dialogue
Are there impossible answers in every dialogical situation?
A i defi ite a ge of logi s ithi e t e e a s e s: all es
Non-classical answers
What is the logical ea i g of
ARmn (with m > 2)
es a d o , o
eithe
Future works
Inclusive algebras in ARmn: ARmn ARmn+1, ARmn ARm+1n
Equivalent algebras in ARmn: ARmn = AR
Non-classical answers and many-valued modal logics
Many-valuedness: set of questions n in ARmn
Modalities: modes of answers m in ARmn
s all
es o
?
?
o
o ?
4 Prospects
Limits of rationality in dialogue
Are there impossible answers in every dialogical situation?
A indefinite range of logics ithi e t e e a s e s: all es
Non-classical answers
What is the logi al ea i g of
ARmn (with m > 2)
es a d o , o
eithe
Future works
Inclusive algebras in ARmn: ARmn ARmn+1, ARmn ARm+1n
Equivalent algebras in ARmn: ARmn = AR
Non-classical answers and many-valued modal logics
Many-valuedness: set of questions n in ARmn
Modalities: modes of answers m in ARmn
s all
es o
?
?
o
o ?
References
Bah , A.J.
5 . Does “e e -Fold Predications equal Four-Cornered Negation
‘e e sed? , Philosophy East and West, Vol. 7: 127-130
Bal e o i z, P
. Do atte pts to fo alize the syād-vāda ake se se , pape
presented at the 11th Jaina Studies Workshop: Jaina Scriptures and Philosophy, SOAS, 151. London.
Ganeri, J. (2002). Jai a Logi a d the Philosoph
Philosophy of Logic, Vol. 23: 267-281
Basis of Plu alis
. History and
Marcos, J. & Molick, S. (2013). The mystery of duality unraveled: dualizing rules,
operators and logics . Talk given at GeTFuN Workshop 1.0, IV World Congress and School
on Universal Logic
Matilal, B.K. (1998). The Jai a o t i utio to logi . I Ga e i, J. & Ti a i, H. eds ., The
Character of Logic in India. State University of New Press, 1998: 127-139.
P iest, G.
. The Logic of the Catuskoti . Comparative Philosophy, Vol. 1: 24-54
“ ha g, F.
. A No -One Sided Logic for Non-One-“ided ess . International Journal
of Jaina Studies (Online) Vol. 9: 1-25
Suszko, R. (1977). The Fregean A io
Logica, Vol. 36: 87–90
a d Polish
athe ati al logi i the
s , Studia
“ l a , ‘.
. A Ge e ous Jai ist I te p etatio of Co e ‘ele a t Logi s , Bulletin of
the Section of Logic, Vol. 16: 58-66
)aitse , D. & “h a ko, Y.
. Bi-facial truth: a case for generalized truth- alues ,
Studia Logica, Vol. 101: 1299-1318
Extra:
Two cases of deaf dialogues
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
you just accepted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is true?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is true.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p and p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p are true?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
you just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
you just accepted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
Deaf dialogue #1: Aristotle (A), Heraclites (H)
A: Is it the ase that p, i.e. p is t ue?
H: Yes, p is t ue.
A: He e p is false, ight?
H: No, it is ot.
A: Do ou ea that oth p a d p a e t ue?
H: Yes, I do.
A: Well, let us assu e that p a d p can be true together. Then your
position is indefensible, because you should reject the negation of what
ou just a epted.
H: That is?
A: If ou a ept p a d p at once, then you accept (p p). And if you do
so, then you cannot but reject (p p). Therefore, you eventually
endorse now what you just refuted a couple of minutes ago. I am right, and
ou a e o g.
H: Wh o ea th?!
A: Be ause this is ho la guage a d thought a e ade, a d ou a ot
reply everything to this necessity. Therefore, you cannot accept and reject
one and the same proposition at once. Consequently, you eventually
assume PNC whenever you recognize that a proposition like (p p)
cannot be accepted and rejected at once.
H: I ag ee ith the fi st pa t of ou o lusio . Not the se o d,
ho e e .
A: You a ot p o eed i su h a a !
H: Yes, I do a d p o e it as follo s. I told ou that p a d p are true
together: and I do not see any difference between this statement and the
claim that (p p) is equally true, by virtue of what conjunction means.
Now you are mistaken by assuming that I should reject (p p) for the
2 Question-Answer Semantics
endorse now what you just refuted a couple of minutes ago. I am right, and
ou a e o g.
H: Wh o ea th?!
A: Be ause this is ho la guage a d thought a e ade, a d ou a ot
reply everything to this necessity. Therefore, you cannot accept and reject
one and the same proposition at once. Consequently, you eventually
assume PNC whenever you recognize that a proposition like (p p)
cannot be accepted and rejected at once.
H: I ag ee ith the fi st pa t of ou o lusio . Not the se o d,
ho e e .
A: You a ot p o eed i su h a a !
H: Yes, I do a d p o e it as follo s. I told ou that p a d p are true
together: and I do not see any difference between this statement and the
claim that (p p) is equally true, by virtue of what conjunction means.
endorse now what you just refuted a couple of minutes ago. I am right, and
ou a e o g.
H: Wh o ea th?!
A: Be ause this is ho la guage a d thought a e ade, a d ou a ot
reply everything to this necessity. Therefore, you cannot accept and reject
one and the same proposition at once. Consequently, you eventually
assume PNC whenever you recognize that a proposition like (p p)
cannot be accepted and rejected at once.
H: I ag ee ith the fi st pa t of ou o lusio . Not the se o d,
ho e e .
A: You a ot p o eed i su h a a !
H: Yes, I do a d p o e it as follo s. I told ou that p a d p are true
together: and I do not see any difference between this statement and the
claim that (p p) is equally true, by virtue of what conjunction means.
endorse now what you just refuted a couple of minutes ago. I am right, and
ou a e o g.
H: Wh o ea th?!
A: Be ause this is ho la guage a d thought a e ade, a d ou a ot
reply everything to this necessity. Therefore, you cannot accept and reject
one and the same proposition at once. Consequently, you eventually
assume PNC whenever you recognize that a proposition like (p p)
cannot be accepted and rejected at once.
H: I ag ee ith the fi st pa t of ou o lusio . Not the se o d,
ho e e .
A: You a ot p o eed i su h a a !
H: Yes, I do a d p o e it as follo s. I told ou that p a d p are true
together: and I do not see any difference between this statement and the
claim that (p p) is equally true, by virtue of what conjunction
meansNow you are mistaken by assuming that I should reject (p p) for
endorse now what you just refuted a couple of minutes ago. I am right, and
ou a e o g.
H: Wh o ea th?!
A: Be ause this is how language and thought are made, and you cannot
reply everything to this necessity. Therefore, you cannot accept and reject
one and the same proposition at once. Consequently, you eventually
assume PNC whenever you recognize that a proposition like (p p)
cannot be accepted and rejected at once.
H: I ag ee ith the fi st pa t of ou o lusio . Not the se o d,
ho e e .
A: You a ot p o eed i su h a a !
H: Yes, I do a d p o e it as follo s. I told ou that p a d p are true
together: and I do not see any difference between this statement and the
claim that (p p) is equally true, by virtue of what conjunction
meansNow you are mistaken by assuming that I should reject (p p) for
endorse now what you just refuted a couple of minutes ago. I am right, and
ou a e o g.
H: Wh o ea th?!
A: Be ause this is ho la guage a d thought a e ade, a d ou a ot
reply everything to this necessity. Therefore, you cannot accept and reject
one and the same proposition at once. Consequently, you eventually
assume PNC whenever you recognize that a proposition like (p p)
cannot be accepted and rejected at once.
H: I ag ee ith the fi st pa t of ou o lusio . Not the se o d,
ho e e .
A: You a ot p o eed i su h a a !
H: Yes, I do a d p o e it as follo s. I told ou that p a d p are true
together: and I do not see any difference between this statement and the
claim that (p p) is equally true, by virtue of what conjunction
meansNow you are mistaken by assuming that I should reject (p p) for
endorse now what you just refuted a couple of minutes ago. I am right, and
ou a e o g.
H: Wh o ea th?!
A: Be ause this is ho la guage a d thought a e ade, a d ou a ot
reply everything to this necessity. Therefore, you cannot accept and reject
one and the same proposition at once. Consequently, you eventually
assume PNC whenever you recognize that a proposition like (p p)
cannot be accepted and rejected at once.
H: I ag ee ith the fi st pa t of ou o lusio . Not the se o d,
ho e e .
A: You a ot p o eed i su h a a !
H: Yes, I do a d p o e it as follo s. I told ou that p a d p are true
together: and I do not see any difference between this statement and the
claim that (p p) is equally true, by virtue of what conjunction
meansNow you are mistaken by assuming that I should reject (p p) for
endorse now what you just refuted a couple of minutes ago. I am right, and
ou a e o g.
H: Wh o ea th?!
A: Be ause this is ho la guage a d thought a e ade, a d ou a ot
reply everything to this necessity. Therefore, you cannot accept and reject
one and the same proposition at once. Consequently, you eventually
assume PNC whenever you recognize that a proposition like (p p)
cannot be accepted and rejected at once.
H: I ag ee ith the fi st pa t of ou o lusio . Not the se o d,
ho e e .
A: You a ot p o eed i su h a a !
H: Yes, I do a d p o e it as follo s. I told ou that p a d p are true
together: and I do not see any difference between this statement and the
claim that (p p) is equally true, by virtue of what conjunction means.
Now you are mistaken by assuming that I should reject (p p) for the
endorse now what you just refuted a couple of minutes ago. I am right, and
ou a e o g.
H: Wh o ea th?!
A: Be ause this is ho la guage a d thought a e ade, a d ou a ot
reply everything to this necessity. Therefore, you cannot accept and reject
one and the same proposition at once. Consequently, you eventually
assume PNC whenever you recognize that a proposition like (p p)
cannot be accepted and rejected at once.
H: I ag ee ith the fi st pa t of ou o lusio . Not the se o d,
ho e e .
A: You a ot p o eed i su h a a !
H: Yes, I do a d p o e it as follo s. I told ou that p a d p are true
together: and I do not see any difference between this statement and the
claim that (p p) is equally true, by virtue of what conjunction means.
Now you are mistaken by assuming that I should reject (p p) for the
endorse now what you just refuted a couple of minutes ago. I am right, and
ou a e o g.
H: Wh o ea th?!
A: Be ause this is ho la guage a d thought a e ade, a d ou a ot
reply everything to this necessity. Therefore, you cannot accept and reject
one and the same proposition at once. Consequently, you eventually
assume PNC whenever you recognize that a proposition like (p p)
cannot be accepted and rejected at once.
H: I ag ee ith the fi st pa t of ou o lusio . Not the se o d,
ho e e .
A: You a ot p o eed i su h a a !
H: Yes, I do a d p o e it as follo s. I told ou that p a d p are true
together: and I do not see any difference between this statement and the
claim that (p p) is equally true, by virtue of what conjunction means.
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, to accept p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g with words, and there is no point to go further with you
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, to accept p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith ou
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, to accept p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith ou
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, to accept p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith ou
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, to accept p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith you
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, to accept p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith you
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, accepting p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does not say anything relevant. It is just noise, nothi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith ou
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, accepting p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self-confidently.
A: What ou a e sa i g does ot ake se se. You a ot accept any
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith ou
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, accepting p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith you
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, accepting p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith you
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, accepting p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot accept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith you
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, accepting p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept contradictions , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith you
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, accepting p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept contradictions , o e agai . These a e your
contradictions, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith you
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, accepting p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does ot sa a thi g ele a t. It is just oise, othi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith you
doi g so.
H: As ou please. I do ot a t to o t adi t ou, a
a .
Now you are mistaken by assuming that I should reject (p p) for the
very reason that I just accepted (p p). It is natural to do so, so long as
you assume that every proposition and its negation are contradictories.
You are free to do that, but nothing compels me to do so. I do not, actually,
and that is why I also accept both (p p) and its negation (p p). In a
nutshell, accepting p is not the same as rejecting p. Or not for everybody,
as you seem to claim it so self- o fide tl .
A: What ou a e sa i g does ot ake se se. You a ot a ept a
contradiction as you do here, because whoever proceeds in such a way
does not say anything relevant. It is just noise, nothi g ea i gful he e.
H: I do ot a ept o t adi tio s , o e agai . These a e your
o t adi tio s, ot i e.
A: You a e pla i g ith o ds, a d the e is o poi t to go fu the ith ou
doi g so.
H: As ou please. I do ot a t to contradict you, anyway.
A: … .
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
p: Not a
o e.
A: I ll e entually know what you think, anyway! And this is the following,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is true?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
p: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e entually know what you think, anyway! And this is the following,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do not, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just accept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only true, al eit t ue togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only true, albeit true togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
P: Not a
o e.
A: I ll e entually know what you think, anyway! And this is the following,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
p: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
p: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
p: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
p: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
true o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
p: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
true nor false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
p: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
p: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again.
Deaf dialogue #2: Aristotle (A), Pyrrho (P)
A: Do ou thi k that p is t ue?
P: No, I do t.
A: Al ight. “o ou thi k that p is false?
P: I do ot, eithe .
A: Agai , afte He a lites this o i g? He lai ed he did ot just a ept
p and p, but both. Do you think the same by rejecting p and p at once, I
mean: do you insinuate that these are not only t ue, al eit t ue togethe ?
p: Not a
o e.
A: I ll e e tuall k o
hat ou thi k, a
a ! A d this is the follo i g,
namely: that neither p nor p are true or false, because they are neither
t ue o false. ‘ight?
P: No.
A: I a fed up. I gi e up. Again …
Fabien Schang