Catuskoti: paraconsistent, paracomplete, both, or none?

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, and its “dual”

1 Catuskoti, and its “dual”

Nāgārjuna (150-250), 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, and its “dual”

Nāgārjuna (150-250), 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, and its “dual”

Nāgārjuna (150-250), 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, and its “dual”

Catuskoti (Tetralemma): a set of denied sentences

(a)

not: p

(b)

not: p

(c)

not: p  p

(d)

not: (p  p)

1 Catuskoti, and its “dual”

Catuskoti (Tetralemma): denial as classical negation

(a)

(p)

(b)

(p)

(c)

(p  p)

(d)

((p  p))

1 Catuskoti, and its “dual”

Catuskoti (Tetralemma): denial as classical negation

(a)

p

(b)

p

(c)

p  p

(d)

p  p

1 Catuskoti, and 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 Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and its “dual”

A Western counterpart: Pyrrho’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 need to inquire any further into other things. […] Pyrrho 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 (“no more”), which aims to capture the indeterminate natures of things, when we attempt to say anything about anything.

(Entry “Skepticism”: Stanford Encyclopedia of Philosophy)

Which “complicated mode of speech” to make sense of the Catuskoti?

1 Catuskoti, and 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, and its “dual”

Limits of Thought = Limits of compossible acceptance/rejection? “p is E”: p is neither 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 for the semantic predicate “avaktavyam”) 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”): there is no 2+1 = 3rd 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, and its “dual”

Limits of Thought = Limits of compossible acceptance/rejection? “p is E”: p is neither 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 for the semantic predicate “avaktavyam”) 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”): there is no 2+1 = 3rd 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, and its “dual”

Limits of Thought = Limits of compossible acceptance/rejection? “p is E”: p is neither 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 for the semantic predicate “avaktavyam”) 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”): there is no 2+1 = 3rd 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, and its “dual”

Dual logics: Catuskoti (LC) vs Saptabhangi (LS)? paraconsistent vs paracomplete? co-intuitionistic vs intuitionistic logics? See Bahm (1958): “Does Seven-Fold Predication equal Four-Cornered Negation Reversed?” Logic L = L,╞ L is a theory (i.e. a set of formulas, including pL) ╞ is a relation of consequence, such that: ╞ Δ iff v(p)D  v(qΔ)D D is a set of designated values (where TD)

1 Catuskoti, and its “dual”

Dual logics: Catuskoti (LC) vs Saptabhangi (LS)? paraconsistent vs paracomplete? co-intuitionistic vs intuitionistic logics? See Bahm (1958): “Does Seven-Fold Predication equal Four-Cornered Negation Reversed?” Logic L = L,╞ L is a theory (i.e. a set of formulas, including pL) ╞ is a relation of consequence, such that: ╞ Δ iff v(p)D  v(qΔ)D D is a set of designated values (where TD)

1 Catuskoti, and its “dual”

The principle of four-cornered negation, stated as “x is neither a, nor non-a, nor both a and non-a, nor neither a nor non-a” or as joint denial of “x is a”, “x is non-a”, “x is both a and non-a”, and “x is neither a nor non-a” (where a and non-a are interpreted as opposites), if reversed, would be stated as “x is a, non-a, both a and non-a, and neither a nor non-a” or as the joint affirmation of “x is a”, “x is non-a”, “x is both a and non-a” and “x is neither a nor non-a” (where a and non-a are interpreted as opposites). This reversed statement consists of the first four of the seven syads except that “x is neither a nor non-a” is replaced by “x is indescribable”. Bahm (1958): 128

1 Catuskoti, and its “dual”

The principle of four-cornered negation, stated as “x is neither a, nor non-a, nor both a and non-a, nor neither a nor non-a” or as joint denial of “x is a”, “x is non-a”, “x is both a and non-a”, and “x is neither a nor non-a” (where a and non-a are interpreted as opposites), if reversed, would be stated as “x is a, non-a, both a and non-a, and neither a nor non-a” or as the joint affirmation of “x is a”, “x is non-a”, “x is both a and non-a” and “x is neither a nor non-a” (where a and non-a are interpreted as opposites). This reversed statement consists of the first four of the seven syads except that “x is neither a nor non-a” is replaced by “x is indescribable”. Bahm (1958): 128

1 Catuskoti, and its “dual”

The principle of four-cornered negation, stated as “x is neither a, nor non-a, nor both a and non-a, nor neither a nor non-a” or as joint denial of “x is a”, “x is non-a”, “x is both a and non-a”, and “x is neither a nor non-a” (where a and non-a are interpreted as opposites), if reversed, would be stated as “x is a, non-a, both a and non-a, and neither a nor non-a” or as the joint affirmation of “x is a”, “x is non-a”, “x is both a and non-a” and “x is neither a nor non-a” (where a and non-a are interpreted as opposites). This reversed statement consists of the first four of the seven syads except that “x is neither a nor non-a” is replaced by “x is indescribable”. Bahm (1958): 128

1 Catuskoti, and its “dual”

A “positive” version 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, and its “dual”

A “positive” version 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, and its “dual”

A “positive” version of the Catuskoti:

(a)

p

(b)

p

(c)

p  p

(d)

(p  p)

1 Catuskoti, and its “dual”

A “positive” version 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, and 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, and its “dual”

Saptabhangi: a classical formalization (1)

p

(2)

p

(3)

p  p

(4)

p  p

(5)

p  (p  p)

(6)

p  (p  p)

(7)

p  p  (p  p)

1 Catuskoti, and its “dual”

Saptabhangi: a classical formalization (1)

p

 (3) is inconsistent

(2)

p

(3)

p  p

 (3) and (4) are indistinguishable from each other

(4)

p  p

(5)

p  (p  p)

(6)

p  (p  p)

(7)

p  p  (p  p)

 (5)-(7) collapse into (3)-(4), by simplification  “syad” refers to standpoints: a combination of various models

1 Catuskoti, and its “dual”

Saptabhangi: truth-values in distinctive models (“avaktavya” as B) (1)

w v(w,p) = T

(2)

w v(w,p) = F

(3)

w v(w,p) = T,

(4)

w v(w,p) = B

(5)

w v(w,p) = T,

w v(w,p) = B

(6)

w v(w,p) = F,

w v(w,p) = B

(7)

w v(w,p) = T,

w v(w,p) = F,

w v(w,p) = F

w v(w,p) = B

1 Catuskoti, and its “dual”

Saptabhangi: truth-values in distinctive models (“avaktavya” as B) (1)

w v(w,p) = T

(2)

w v(w,p) = F

(3)

w v(w,p) = T,

(4)

w v(w,p) = B

(5)

w v(w,p) = T,

w v(w,p) = B

(6)

w v(w,p) = F,

w v(w,p) = B

(7)

w v(w,p) = T,

w v(w,p) = F,

w v(w,p) = F

w v(w,p) = B

1 Catuskoti, and its “dual”

Saptabhangi: truth-values in distinctive models (“avaktavya” as N) (1)

w v(w,p) = T

(2)

w v(w,p) = F

(3)

w v(w,p) = T,

(4)

w v(w,p) = N

(5)

w v(w,p) = T,

w v(w,p) = N

(6)

w v(w,p) = F,

w v(w,p) = N

(7)

w v(w,p) = T,

w v(w,p) = F,

w v(w,p) = F

w v(w,p) = N

1 Catuskoti, and its “dual”

Saptabhangi: truth-values in distinctive models (“avaktavya” as N) (1)

w v(w,p) = T

(2)

w v(w,p) = F

(3)

w v(w,p) = T,

(4)

w v(w,p) = N

(5)

w v(w,p) = T,

w v(w,p) = N

(6)

w v(w,p) = F,

w v(w,p) = N

(7)

w v(w,p) = T,

w v(w,p) = F,

w v(w,p) = F

w v(w,p) = N

1 Catuskoti, and 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, and 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, and its “dual”

Are LC and LS dual logics? ╞ Δ

iff

Δd╞d d

!

1 Catuskoti, and its “dual”

Are LC and LS dual logics? p  p╞ q

iff

q╞d p  p

!

1 Catuskoti, and its “dual”

Are LC and LS dual logics? p  p╞/ q

iff

q╞/d p  p

?

Problems about “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, and its “dual”

Are LC and LS dual logics? p  p╞/ q

iff

q╞/d p  p

?

Problems about “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, and its “dual”

Are LC and LS dual logics? p  p╞/ q

iff

q╞/d p  p

?

Problems about “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, and 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, and 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, and its “dual”

Are LC and LS dual logics? p  p╞/ q

iff

q╞/d p  p

?

Problems about “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, and its “dual”

What sort of logic is LC?

paraconsistent? paracomplete? both paraconsistent and paracomplete? neither paraconsistent nor paracomplete?

1 Catuskoti, and its “dual”

LC is paraconsistent iff p, p╞/ q {p, p}D / qD

(Non-Explosion)

1 Catuskoti, and its “dual”

What sort of logic is LC?

paraconsistent? paracomplete? both paraconsistent and paracomplete? neither paraconsistent nor paracomplete?

1 Catuskoti, and its “dual”

LC is paracomplete iff p╞/ q, q pD / {q, q}D

(Non-Implosion)

1 Catuskoti, and its “dual”

What sort of logic is LC?

paraconsistent? paracomplete? both paraconsistent and paracomplete? neither paraconsistent nor paracomplete?

1 Catuskoti, and its “dual”

LC is both paraconsistent and paracomplete iff p, p╞/ q {p, p}D / qD

(Non-Explosion)

p╞/ q, q pD / {q, q}D

(Non-Implosion)

1 Catuskoti, and its “dual”

LC is both paraconsistent and paracomplete iff p, p╞/ q {p, p}D / qD

(Non-Explosion)

p╞/ q, q pD / {q, q}D

(Non-Implosion)

1 Catuskoti, and its “dual”

LC is both paraconsistent and paracomplete iff p, p╞/ q {p, p}D / qD

(Non-Explosion)

p╞/ q, q pD / {q, q}D

(Non-Implosion)

1 Catuskoti, and its “dual”

What sort of logic is LC?

paraconsistent? paracomplete? both paraconsistent and paracomplete? neither paraconsistent nor paracomplete?

1 Catuskoti, and its “dual”

LC is neither paraconsistent nor paracomplete iff Not: p, p╞/ q {p, p}D / qD

(Non-Explosion)

Not: p╞/ q, q pD / {q, q}D

(Non-Implosion)

Neither paraconsistent nor paracomplete = Both consistent and complete ? Is metalinguistic negation an involutive operator ?

1 Catuskoti, and its “dual”

LC is neither paraconsistent nor paracomplete iff Not: p, p╞/ q {p, p}D / qD

(Non-Explosion)

Not: p╞/ q, q pD / {q, q}D

(Non-Implosion)

Neither paraconsistent nor paracomplete = Both consistent and complete ? Is metalinguistic negation an involutive operator ?

1 Catuskoti, and its “dual”

LC is neither paraconsistent nor paracomplete iff p, p╞ q {p, p}D / qD

(Non-Explosion)

p╞ q, q pD / {q, q}D

(Non-Implosion)

Neither paraconsistent nor paracomplete = Both consistent and complete ? Is metalinguistic negation an involutive operator ?

1 Catuskoti, and its “dual”

LC is neither paraconsistent nor paracomplete iff p, p╞ q {p, p}D / qD

(Non-Explosion)

p╞ q, q pD / {q, q}D

(Non-Implosion)

Neither paraconsistent nor paracomplete = Both consistent and complete ? Is metalinguistic negation an involutive operator ?

1 Catuskoti, and its “dual”

LC is neither paraconsistent nor paracomplete iff p, p╞ q {p, p}D / qD

(Non-Explosion)

p╞ q, q pD / {q, q}D

(Non-Implosion)

Neither paraconsistent nor paracomplete = Both consistent and complete ? Is metalinguistic negation an involutive operator ?

1 Catuskoti, and 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, … by means of {,,} in L - semantic version: increasingly complex truth-values p, together with T, F, B, N, … in D  Duality: between answers to questions about truth-values

1 Catuskoti, and 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, and 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, and 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, and 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, and 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, and 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 “x is a” stand for? A statement of the form Xp = “p is X” X: semantic predicate Example: “p is true” Xp 𝐗p Xp

Tp Fp Bp Np

“told” value complementation “marked” value

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 ⨅ 𝐅p 𝐓p ⨅ Fp Tp ⨅ Fp 𝐓p ⨅ 𝐅p

2 Question-Answer Semantics

What is a “truth-value” ?  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 true”:

“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 “truth-value” ?  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 true”:

“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 “truth-value” ?  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 true”:

“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 “truth-value” ? A generalization of truth-values, beyond monovalence and bivalence See Zaitsev & Shramko (2013) Referential truth-values: “p is true/false” Inferential truth-values: “p is held true/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)  How to deal with “truth-values” in catuskoti and saptabhangi?

2 Question-Answer Semantics

 What is a “truth-value” ? A generalization of truth-values, beyond monovalence and bivalence See Zaitsev & Shramko (2013) Referential truth-values: “p is true/false” Inferential truth-values: “p is held true/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)  How to deal with “truth-values” in catuskoti and saptabhangi?

2 Question-Answer Semantics

 What is a “truth-value” ? A generalization of truth-values, beyond monovalence and bivalence See Zaitsev & Shramko (2013) Referential truth-values: “p is true/false” Inferential truth-values: “p is held true/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)  How to deal with “truth-values” in 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 “truth” and “falsity” mean in various domains of values V? Non-falsity = truth Non-falsity  truth

in V = {T,F} in V = {T,F,N}

𝐓p = Fp 𝐓p = 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 “truth” and “falsity” mean in various domains of values D? Non-falsity = truth Non-falsity  truth

in V = {T,F} in V = {T,F,N}

𝐓p = Fp 𝐓p = 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 𝐓p Tp 𝐓p Tp 𝐓p Tp

⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅

𝐅p Fp Fp 𝐅p 𝐅p Fp Fp

⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅

𝐀p 𝐀p 𝐀p 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 𝐓p Tp 𝐓p Tp 𝐓p Tp

⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅

𝐅p Fp Fp 𝐅p 𝐅p Fp Fp

⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅

𝐀p 𝐀p 𝐀p 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 𝐓p 𝐓p 𝐓p Tp Tp Tp 𝐓p 𝐓p 𝐓p Tp Tp Tp 𝐓p Tp

⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅

𝐅p Fp 𝐅p 𝐅p Fp 𝐅p 𝐅p Fp Fp 𝐅p Fp Fp 𝐅p Fp Fp

⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅

𝐁p 𝐁p Bp 𝐁p 𝐁p Bp 𝐁p Bp 𝐁p Bp Bp 𝐁p Bp Bp Bp

⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅ ⨅

𝐍p 𝐍p 𝐍p Np 𝐍p 𝐍p Np 𝐍p Np Np 𝐍p 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

n does not equate with P(n) = 2 m m = 2, in Shramko & Wansing’s framework

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 𝐅p, so that A(p) = a1(p),a2(p) = 10 Fp = 𝐓p ⊓ Fp, so that A(p) = a1(p),a2(p) = 01 Xp  Xp whenever n > 1 (more than 1 statement about p’s truth-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 𝐅p, so that A(p) = a1(p),a2(p) = 10 Fp = 𝐓p ⊓ Fp, so that A(p) = a1(p),a2(p) = 01 Xp  Xp whenever n > 1 (more than 1 statement about p’s truth-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 𝐅p, so that A(p) = a1(p),a2(p) = 10 Fp = 𝐓p ⊓ Fp, so that A(p) = a1(p),a2(p) = 01 Xp  Xp whenever n > 1 (more than 1 statement about p’s truth-value)

2 Question-Answer Semantics

An example of 4-valuedness: bilateralist logic AR22 = AR4 AR4 = L, A,V4,⨅,⨆ L

set of formulae {p, q, …} set of logical functions {,,,}

A(p)

valuation function, mapping from L to V4 A(p) = a1(p), a2(p)

V4

{11, 10, 00, 01}

1⨅0 = 0⨅1 = 0⨅0 = 0, 1⨅1 = 1 1⨆1 = 1⨆0 = 0⨆1 = 1, 0⨆0 = 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)⨅a2(q), a1(p)⨆a2(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)⨅a2(q), a1(p)⨆a2(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(pq) = a1(p  q), a2(p  q) = a1(p)⨆a2(q), a1(p)⨅a2(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(pq) = a1(p  q), a2(p  q) = a1(p)⨆a2(q), a1(p)⨅a2(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 (“strenghened”) A(p  q) = a1(p  q), a2(p  q) = a1(p)⨅a1(q), a1(p)⨅a2(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 (“strenghened”) A(p  q) = a1(p  q), a2(p  q) = a1(p)⨅a1(q), a1(p)⨅a2(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 (“strenghened”) A(p  q) = a1(p  q), a2(p  q) = a1(p)⨅a1(q), a1(p)⨅a2(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 ‘no’ a force-indicator? Yes, sooner or later!” (to be submitted)

2 Question-Answer Semantics

0-valuedness? AR0n = ARm0 = AR0 m ?

or a1(p) { }

n ?

or { }

A(p)  { } Priest’s “silence”? 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} 1 for “yes” Saptabhangi (Balcerowicz 2011)

2 Question-Answer Semantics

1-valuedness? AR1n = AR1 m ai(p){0} n ? A(p)  {0} 0 for “no” 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 “yes” 0 for “no”

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} 1 for “yes” ½ for “both yes and no”, if qi(p) = Bp 0 for “absolutely no” 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} 1 for “yes” ½ for “both yes and no”, if qi(p) = Bp 0 for “absolutely no” 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} 1 for “yes” ½ for “neither yes nor no”, if qi(p) = Np 0 for “no” 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} 1 for “yes” ½ for “neither yes nor no”, if qi(p) = Np 0 for “no” 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} 1 for “yes” 0 for “no” 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} 1 for “yes” 0 for “no” 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} 1 for “yes” 0 for “no” 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} 1 for “yes” 0 for “no” Glutty logics

2 Question-Answer Semantics

4-valuedness? AR41 = AR4 m ai(p)  {1,⅔,⅓,0} n q1(p) = Tp ? A(p)  {1,2/3,1/3,0} 1 for “yes” 2/ 3 for “yes and no” 1/ 3 for “neither yes nor no” 0 for “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} 1 for “yes” 0 for “no”

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} 1 for “yes” 0 for “no” 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} 1 for “yes” 0 for “no” 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} 1 for “yes” 0 for “no” 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} 1 for “yes” 0 for “no” 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} 1 for “yes” 2/ 3 for “yes and no” 1/ 3 for “neither yes nor no” 0 for “no”

(or “only yes”) (or “yes, but not only”) (or “no, but not only”) (or “only no”)

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 for “yes” 0 for “no”

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 be proper “logics”? - 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 (“paryudāsapratiṣedha”): 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 (“paryudāsapratiṣedha”): 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? (2) locutionary negation (“paryudāsapratiṣedha”): 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? (3) locutionary negation (“paryudāsapratiṣedha”): 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? (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 “no, p is not T” = “yes, p is 𝐓 = F” = “yes, p is T” “no, p is not F” = “yes, p is 𝐅 = T” = “yes, p is T” - rejection is complementation in AR2n “no, p is not X” = “yes, p is 𝐗”

3 Dialectical negation

What is negation in the catuskoti and saptabhangi? (2) 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 “no, p is not T” = “yes, p is 𝐓 = F” = “yes, p is T” “no, p is not F” = “yes, p is 𝐅 = T” = “yes, p is T” - rejection is complementation in AR2n “no, p is not X” = “yes, p is 𝐗”

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 “Hegelian” 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 “Hegelian” 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  “Hegelian” dialectics: thesis-anthesis-synthesis - dialectics leads to Aufhebung (overcome the negative): X, 𝑋, 𝑋𝑋 - 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

antithesis in L2

TTTT

synthesis in L2 = thesis in L3 = {p,q,r}

TTTT

antithesis in L3

TTTTTTTT

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

antithesis in L2

TTTT

synthesis in L2 = thesis in L3 = {p,q,r}

TTTT

 Priest’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

XD

AR21 = AR2

1

0

XD

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

XD

XD 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

XD

XD 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 XD

XD 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) = 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 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) = 000000000000 … 0 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” version of the Catuskoti:

(a)

v(p)  T

(b)

v(p)  T

(c)

v(p)  TT

(d)

v(p)  TT

3 Dialectical negation ⁞ ⁞

n times

 Law of n-th negation: cyclic negation modulo n, s.t. X = 𝑋 in ARmn  Designated values = accepted values For every value X in V, XpD iff p is accepted XpD iff p is not accepted, i.e. rejected  Meaning of truth-values (see Suszko (1977)): “truth”? “falsity”? - 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 = 𝐓  For every truth-value X, X is a designated value X is a non-designated value

iff iff

TX TX

3 Dialectical negation ⁞ ⁞

n times

 Law of n-th negation: cyclic negation modulo n, s.t. X = 𝑋 in ARmn  Designated values = accepted values For every value X in V, XpD iff p is accepted XpD iff p is not accepted, i.e. rejected  Meaning of truth-values (see Suszko (1977)): “truth”? “falsity”? - 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 = 𝐓  For every truth-value X, X is a designated value X is a non-designated value

iff iff

TX TX

4. Conclusion (and Prospects)

4 Conclusion

 Are LC and LS dual logics? if there is no designated value XD in LC, LC is both paraconsistent and paracomplete For every truth-value Xp: Xp╞/ Xq Xp╞/ Xq, Xq Tarskian trivial logic: every sentence follows from every other one if there is no non-designated value XD in LC, LS is neither paraconsistent nor paracomplete: Tarskian trivial logic For every truth-value X: Xp╞ Xq Xp╞ Xq, Xq Tarskian trivial logic: no sentence follows from no other one dualized by Boolean negation: p in LC = b(p) in LS b(1111…1) = (0000…0)

4 Conclusion

 Are LC and LS dual logics? if there is no designated value XD in LC, LC is both paraconsistent and paracomplete For every truth-value Xp: Xp╞/ Xq Xp╞/ Xq, Xq Tarskian trivial logic: every sentence follows from every other one if there is no non-designated value XD in LC, LS is neither paraconsistent nor paracomplete: Tarskian trivial logic For every truth-value X: Xp╞ Xq Xp╞ Xq, Xq Tarskian trivial logic: no sentence follows from no other one dualized by Boolean negation: p in LC = b(p) in LS b(1111…1) = (0000…0)

4 Conclusion

 Are LC and LS dual logics? if there is no designated value XD in LC, LC is both paraconsistent and paracomplete For every truth-value Xp: Xp╞/ Xq Xp╞/ Xq, Xq Tarskian trivial logic: every sentence follows from every other one if there is no non-designated value XD in LC, LS is neither paraconsistent nor paracomplete: Tarskian trivial logic For every truth-value X: Xp╞ Xq Xp╞ Xq, Xq Tarskian trivial logic: no sentence follows from no other one dualized by Boolean negation: p in LC = b(p) in LS b(1111…1) = (0000…0)

4 Conclusion

 Are LC and LS dual logics? if there is no designated value XD in LC, LC is both paraconsistent and paracomplete For every truth-value Xp: Xp╞/ Xq Xp╞/ Xq, Xq Tarskian trivial logic: every sentence follows from every other one if there is no non-designated value XD in LC, LS is neither paraconsistent nor paracomplete: Tarskian trivial logic For every truth-value X: Xp╞ Xq Xp╞ Xq, Xq Tarskian trivial logic: no sentence follows from no other one dualized by Boolean negation: p in LC = b(p) in LS b(1111…1) = (0000…0)

4 Conclusion

 Are LC and LS dual logics? if there is no designated value XD in LC, LC is both paraconsistent and paracomplete For every truth-value Xp: Xp╞/ Xq Xp╞/ Xq, Xq Tarskian trivial logic: every sentence follows from every other one if there is no non-designated value XD in LC, LS is neither paraconsistent nor paracomplete: Tarskian trivial logic For every truth-value X: Xp╞ Xq Xp╞ Xq, Xq Tarskian trivial logic: no sentence follows from no other one dualized by Boolean negation: p in LS = b(p) in LC b(1111…1) = (0000…0)

4 Conclusion

 Are LC and LS dual logics? if there is no designated value XD in LC, LC is both paraconsistent and paracomplete For every truth-value Xp: Xp╞/ Xq Xp╞/ Xq, Xq Tarskian trivial logic: every sentence follows from every other one if there is no non-designated value XD in LC, LS is neither paraconsistent nor paracomplete: Tarskian trivial logic For every truth-value X: Xp╞ Xq Xp╞ Xq, Xq Tarskian trivial logic: no sentence follows from no other one dualized by Boolean negation: p in LS = b(p) in LC b(1111…1) = (0000…0)

4 Prospects

 Limits of rationality in dialogue Are there impossible answers in every dialogical situation? A indefinite range of logics within extreme answers: all “yes” vs all “no”  Non-classical answers What is the logical meaning of “yes and no”, or “neither yes nor no”? ARmn (with m > 2)  Future works Inclusive algebras in ARmn: ARmn  ARmn+1, ARmn  ARm+1n Equivalent algebras in ARmn: ARmn = ARm’n’ Non-classical answers and many-valued modal logics Many-valuedness: set of questions n in ARmn Modalities: modes of answers m in ARmn

? ?

4 Prospects

 Limits of rationality in dialogue Are there impossible answers in every dialogical situation? A indefinite range of logics within extreme answers: all “yes” vs all “no”  Non-classical answers What is the logical meaning of “yes and no”, or “neither yes nor no”? ARmn (with m > 2)  Future works Inclusive algebras in ARmn: ARmn  ARmn+1, ARmn  ARm+1n Equivalent algebras in ARmn: ARmn = ARm’n’ Non-classical answers and many-valued modal logics Many-valuedness: set of questions n in ARmn Modalities: modes of answers m in ARmn

? ?

4 Prospects

 Limits of rationality in dialogue Are there impossible answers in every dialogical situation? A indefinite range of logics within extreme answers: all “yes” vs all “no”  Non-classical answers What is the logical meaning of “yes and no”, or “neither yes nor no”? ARmn (with m > 2)  Future works Inclusive algebras in ARmn: ARmn  ARmn+1, ARmn  ARm+1n Equivalent algebras in ARmn: ARmn = ARm’n’ Non-classical answers and many-valued modal logics Many-valuedness: set of questions n in ARmn Modalities: modes of answers m in ARmn

? ?

4 Prospects

 Limits of rationality in dialogue Are there impossible answers in every dialogical situation? A indefinite range of logics within extreme answers: all “yes” vs all “no”  Non-classical answers What is the logical meaning of “yes and no”, or “neither yes nor no”? ARmn (with m > 2)  Future works Inclusive algebras in ARmn: ARmn  ARmn+1, ARmn  ARm+1n Equivalent algebras in ARmn: ARmn = ARm’n’ Non-classical answers and many-valued modal logics Many-valuedness: set of questions n in ARmn Modalities: modes of answers m in ARmn

? ?

References Bahm, A.J. (1958). “Does Seven-Fold Predications equal Four-Cornered Negation Reversed?”, Philosophy East and West, Vol. 7: 127-130 Balcerowicz, P (2011). “Do attempts to formalize the syād-vāda make sense”, paper presented at the 11th Jaina Studies Workshop: Jaina Scriptures and Philosophy, SOAS, 151. London. Ganeri, J. (2002). “Jaina Logic and the Philosophy Basis of Pluralism”. History and Philosophy of Logic, Vol. 23: 267-281 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 Jaina contribution to logic”. In Ganeri, J. & Tiwari, H. (eds)., The Character of Logic in India. State University of New Press, 1998: 127-139.

Priest, G. (2010). “The Logic of the Catuskoti”. Comparative Philosophy, Vol. 1: 24-54 Schang, F. (2013). “A Non-One Sided Logic for Non-One-Sidedness”. International Journal of Jaina Studies (Online) Vol. 9: 1-25 Suszko, R. (1977). “The Fregean Axiom and Polish mathematical logic in the 1920s”, Studia Logica, Vol. 36: 87–90 Sylvan, R. (1987). “A Generous Jainist Interpretation of Core Relevant Logics”, Bulletin of the Section of Logic, Vol. 16: 58-66 Zaitsev, D. & Shramko, Y. (2013). “Bi-facial truth: a case for generalized truth-values”, Studia Logica, Vol. 101: 1299-1318

Extra: Two cases of deaf dialogues

 Deaf dialogue #1: Aristotle (A), Heraclites (H) A: “Is it the case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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 case that p, i.e. p is true?” H: “Yes, p is true.” A: “Hence p is false, right?” H: “No, it is not.” A: “Do you mean that both p and p are true?” H: “Yes, I do.” A: “Well, let us assume that p and 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 you accept 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

endorse now what you just refuted a couple of minutes ago. I am right, and you are wrong.” H: “Why on earth?!” A: “Because 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 agree with the first part of your conclusion. Not the second, however.” A: “You cannot proceed in such a way!” H: “Yes, I do and prove it as follows. I told you that p and 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 you are wrong.” H: “Why on earth?!” A: “Because 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 agree with the first part of your conclusion. Not the second, however.” A: “You cannot proceed in such a way!” H: “Yes, I do and prove it as follows. I told you that p and 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 you are wrong.” H: “Why on earth?!” A: “Because 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 agree with the first part of your conclusion. Not the second, however.” A: “You cannot proceed in such a way!” H: “Yes, I do and prove it as follows. I told you that p and 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 you are wrong.” H: “Why on earth?!” A: “Because 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 agree with the first part of your conclusion. Not the second, however.” A: “You cannot proceed in such a way!” H: “Yes, I do and prove it as follows. I told you that p and 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 you are wrong.” H: “Why on earth?!” A: “Because 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 agree with the first part of your conclusion. Not the second, however.” A: “You cannot proceed in such a way!” H: “Yes, I do and prove it as follows. I told you that p and 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 you are wrong.” H: “Why on earth?!” A: “Because 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 agree with the first part of your conclusion. Not the second, however.” A: “You cannot proceed in such a way!” H: “Yes, I do and prove it as follows. I told you that p and 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 you are wrong.” H: “Why on earth?!” A: “Because 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 agree with the first part of your conclusion. Not the second, however.” A: “You cannot proceed in such a way!” H: “Yes, I do and prove it as follows. I told you that p and 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 you are wrong.” H: “Why on earth?!” A: “Because 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 agree with the first part of your conclusion. Not the second, however.” A: “You cannot proceed in such a way!” H: “Yes, I do and prove it as follows. I told you that p and 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 you are wrong.” H: “Why on earth?!” A: “Because 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 agree with the first part of your conclusion. Not the second, however.” A: “You cannot proceed in such a way!” H: “Yes, I do and prove it as follows. I told you that p and 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 you are wrong.” H: “Why on earth?!” A: “Because 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 agree with the first part of your conclusion. Not the second, however.” A: “You cannot proceed in such a way!” H: “Yes, I do and prove it as follows. I told you that p and 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-confidently.” A: “What you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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-confidently.” A: “What you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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-confidently.” A: “What you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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-confidently.” A: “What you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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-confidently.” A: “What you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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-confidently.” A: “What you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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 you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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 you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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 you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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 you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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 you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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 you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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 you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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 you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.”

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 you are saying does not make sense. You cannot accept any contradiction as you do here, because whoever proceeds in such a way does not say anything relevant. It is just noise, nothing meaningful here.” H: “I do not accept ‘contradictions’, once again. These are your contradictions, not mine.” A: “You are playing with words, and there is no point to go further with you by doing so.” H: “As you please. I do not want to contradict you, anyway.

A: “… .”

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” p: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” p: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” P: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” p: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” p: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” p: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” p: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” p: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” p: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” p: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again.

 Deaf dialogue #2: Aristotle (A), Pyrrho (P) A: “Do you think that p is true?” P: “No, I don’t.” A: “Alright. So you think that p is false?” P: “I do not, either.” A: “Again, after Heraclites this morning? He claimed he did not 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 true, albeit true together?” p: “Not any more.” A: “I’ll eventually know what you think, anyway! And this is the following, namely: that neither p nor p are true or false, because they are neither true nor false. Right?” P: “No.” A: “I am fed up. I give up. Again …