Relative Charity

1. Relative Charity (a1(p) = 1  a1(p) = 0)? Fabien Schang schang.fabien@voila.fr fschang@hse.ru National Research University, HSE Seminar “Formal Philosophy” 24 February 2015

2. Content 1 The Background: Logic in Translation 2 An Alternative Referential Semantics 3 A Logic of Acceptance and Rejection 4 Coherence beyond Consistency 5 Truths in Meaning 6 Conclusion: The Answer is in the Question 7 Appendix: On Conditional

3. 1 The Background: Logic in Translation

4. by reference to assent and dissent we can state semantic criteria for truth-functions; i.e., criteria for determining whether a given native idiom is to be constructed as expressing the truth-function in question. The semantic criterion of negation is that it turns any short sentence to which one will assent into a sentence from which one will dissent, and vice versa. That of conjunction is that it produces compounds to which (so long as the component sentences are short) one is prepared to assent always only when one is prepared to assent to each component. That of alternation is similar with assent changed twice to dissent. Quine (1960): 57-8

5. For every proposition p: Assent v(p) = T Dissent v(p) = F

7. Negation v(p) = T iff v(p) = F Conjunction v(p  q) = T iff v(p) = v(q) = T Disjunction v(p  q) = F iff v(p) = v(q) = F … Conditional?

9. Negation v(p) = T iff v(p) = F Conjunction v(pq) = T iff v(p) = v(q) = T Disjunction v(p  q) = F iff v(p) = v(q) = F … Conditional?

11. Negation v(p) = T iff v(p) = F Conjunction v(pq) = T iff v(p) = v(q) = T Disjunction v(pq) = F iff v(p) = v(q) = F

12. Negation v(p) = T iff v(p) = F Conjunction v(pq) = T iff v(p) = v(q) = T Disjunction v(pq) = F iff v(p) = v(q) = F … Conditional?

13. The methodological advice to interpret in a way that optimizes agreement should not be conceived as resting on a charitable assumption about human intelligence that might turn out to be false. If we cannot find a way to interpret the utterances and other behavior of a creature as revealing a set of beliefs largely consistent and true by our own standards, we have no reason to count that creature as rational, as having beliefs, or as saying anything. Davidson (1973): 324

14. 2 An Alternative Referential Semantics

15. (1) Logical Absolutism A common rationality standard includes only one correct logic; there is only one rational standard; therefore, there is only one correct logic. (If A then B; A; therefore B) (2) Logical Relativism A common rationality standard includes only one correct logic; there are several correct logics; therefore, there are several rational standards. (If A then B; not-B; therefore A) (3) Relative charity A common rationality standard includes several correct logics not-(if A then B) = A and not-B

16. (1) Logical Absolutism A common rationality standard includes only one correct logic; there is only one rational standard; therefore, there is only one correct logic. (If A then B; A; therefore B) (2) Logical Relativism A common rationality standard includes only one correct logic; there are several correct logics; therefore, there are several rational standards. (If A then B; not-B; therefore not-A) (3) Relative charity A common rationality standard includes several correct logics not-(if A then B) = A and not-B

17. (1) Logical Absolutism A common rationality standard includes only one correct logic; there is only one rational standard; therefore, there is only one correct logic. (If A then B; A; therefore B) (2) Logical Relativism A common rationality standard includes only one correct logic; there are several correct logics; therefore, there are several rational standards. (If A then B; not-B; therefore not-A) (3) Relative charity A common rationality standard includes several correct logics not-(if A then B) = A and not-B

18. Cultural anthropologists usually maintain that there is no evidence that there exist cultures which adhere to different logics than we do. But I find this a strange claim. For one thing, even in my own country there is a subculture of people who try to adhere to intuitionistic logic rather than classical logic. Lokhorst (1998): 57

19. The proper “Conceptual Scheme”: A Logic of Statements - each lexicographer has to do with statements, rather than sentences - statements: speech-acts performed with sentential content Reference: Searle (1965) p: sentence with truth-conditions (truth, false) F(p): statement with satisfaction-conditions F: assertion, order, declaration, … (speech-act theory) In the following, 2 basic “transcendental” speech-acts: - assertion (the speaker says “yes”, commits himself, accepts p) - rejection (the speaker says “no”, does not commit himself, rejects p)

22. 3 A Logic of Acceptance and Rejection

23. Meaning corresponds to a logical value: a 3-dimensional object  answers = statements A() = a11(), … ,aij() questions = sentences parameters = quantifiers/modalities

24. AR4: Logic of Acceptance and Rejection (theory of normative answers) 2 main components:  Question Q: statement-forming operator upon a given sentence p Q(p) = q1(p), …, qn(p) The number of required questions about p is not predetermined  Answer A: set of ordered answers to a given question Q A(p) = a1(p), …, am(p) In AR4: n = 2 relevant questions, m = 2 possible sorts of answer Card(V) = mn = 22 = 4 Note: n and m are independent from each other (rejectivism!)

26. AR4: Logic of Acceptance and Rejection (theory of normative answers) 2 main components:  Question Q: statement-forming operator upon a given sentence p Q(p) = q1(p), …, qn(p) The number of required questions about p is not predetermined  Answer A: set of ordered answers to a given question Q A(p) = a1(p), …, am(p) In AR4: n = 2 relevant questions, m = 2 possible sorts of answers Card(V) = mn = 22 = 4 Note: n and m are independent from each other (rejectivism!)

28. Affirmation vs Negation (Question), Assertion vs Rejection (Answer) Affirmation: expresses the sentence that p Question: “Is it the case that p?” q1(p) Negation: expresses the sentence opposed to p Question: “Is it not the case that p?” = “Is it the case that not-p?” q2(p) = q1(p) Assertion: If I assert p, then I take p to be true (truth-claim) Answer: “Yes, it is the case that p (is true)” a1(p) = 1 Rejection: If I reject p, then I do not take p to be true (non-truth-claim) Answer: “No, it is not the case that p (is true)” a1(p) = 0

33. Unilateralism Equivalence Thesis (Frege, 1919): there is no difference between “No, it is not the case that p is true” and “Yes, it is the case that p is false” a1(p) = 0  a2(p) = 1 Logical theory: a theory of truth-preservation, only Denial is a by-product of assertion: a negative assertion

36. Correctness Under what conditions can a sentence be correctly said to be “true”? Correctness: assertibility-conditions 3 grades of “modal” involvement (3 modes of “being” true) (1) strong correctness: true as impossibly false (truth “by default”) (2) mild correctness: true as not false (truth “by default”) (3) weak correctness: being true as being possibly not false Frege assumed a mild view of truth: Bivalentism (Equivalence Thesis) AR4: no involvement about what truth means (abstract logic)!

37. Correctness Under what conditions can a sentence be correctly said to be “true”? Correctness: assertibility-conditions 3 grades of “modal” involvement (3 modes of “being” true) (1) strong correctness: true as impossibly false (truth “by default”) (2) mild correctness: true as not false (truth “by default”) (3) weak correctness: being true as being possibly not false Frege assumed a mild view of truth: Bivalentism (Equivalence Thesis) AR4: no involvement about what truth means (abstract logic)!

38. Correctness Under what conditions can a sentence be correctly said to be “true”? Correctness: assertibility-conditions 3 grades of “modal” involvement (3 modes of “being” true) (1) strong correctness: true as impossibly false (2) mild correctness: true as not false (truth “by default”) (3) weak correctness: being true as being possibly not false Frege assumed a mild view of truth: Bivalentism (Equivalence Thesis) AR4: no involvement about what truth means (abstract logic)!

42. Logical constants: for every sentence p such that A(p) = a1(p),a2(p) Negation A(p) = a2(p),a1(p) Conjunction A(pq) = a1(p)a2(q),a2(p)a2(q) Disjunction A(pq) = a1(p)a2(q),a2(p)a2(q)

43. p p 11 11 10 01 01 10 00 00

44. Logical constants Negation A(p) = a2(p),a1(p) Conjunction A(pq) = a1(p)a1(q),a2(p)a2(q) Disjunction A(pq) = a1(p)a2(q),a2(p)a2(q)

45. p q pq pq 11 11 11 11 11 10 11 10 11 01 01 11 11 00 01 10 10 11 11 10 10 10 10 10 10 01 01 10 10 00 00 10 01 11 01 11 01 10 01 10 01 01 01 01 01 00 01 00 00 11 01 10 00 10 00 10 00 01 01 00 00 00 00 00

46. Logical constants Negation A(p) = a2(p),a1(p) Conjunction A(pq) = a1(p)a1(q),a2(p)a2(q) Disjunction A(pq) = a1(p)a1(q),a2(p)a2(q)

47. p q pq pq 11 11 11 11 11 10 11 10 11 01 01 11 11 00 01 10 10 11 11 10 10 10 10 10 10 01 01 10 10 00 00 10 01 11 01 11 01 10 01 10 01 01 01 01 01 00 01 00 00 11 01 10 00 10 00 10 00 01 01 00 00 00 00 00

48. A Boolean translation of (non-)classical truth-values in AR4 v(p) = T p is true only A(p) = 1,0 v(p) = F p is false only A(p) = 0,1 v(p) = B p is both true and false A(p) = 1,1 v(p) = N p is neither true nor false A(p) = 0,0

53. 4 Coherence beyond Consistency

54.  3 grades of belief inconsistency (a) Bxp  Bxp (b) Bxp  Bxp (c) Bx(p  Bxp)  2 norms of rationality (CON) a1(p) = 1  a2(p) = 0 (COH) ai(p) = 1  ai(p)  0  What can “Yes and No” mean? A bivalentist answerhood from a non-bivalentist perspective “Yes” to p from one standpoint w1, “Yes” to p from another standpoint w2 (inconsistency) “Yes” to p from w1, “No” to p to w1 (incoherence)

57.  3 grades of belief inconsistency (a) Bxp  Bxp (b) Bxp  Bxp (c) Bx(p  p)  2 norms of rationality (CON) a1(p) = 1  a2(p) = 0 (COH) ai(p) = 1  ai(p)  0  What can “Yes and No” mean? A bivalentist answerhood from a non-bivalentist perspective “Yes” to p from one standpoint w1, “Yes” to p from another standpoint w2 (inconsistency) “Yes” to p from w1, “No” to p to w1 (incoherence)

61.  3 grades of belief inconsistency (a) Bxp  Bxp (b) Bxp  Bxp (c) Bx(p  p)  2 norms of rationality (CON) a1(p) = 1  a2(p) = 0 (COH) ai(p) = 1  ai(p)  0  What can “Yes and No” mean? Non-bivalentist answers to a (single) bivalentist question “Yes” to p from one standpoint w1, “Yes” to p from another standpoint w2 (inconsistency) “Yes” to p from w1, “No” to p from w1 (incoherence)

62. (At least) 3 sorts of ensuing propositional attitudes Bivalentism, Paracompletism Quine, Davidson, Lokhorst: (CON)  (COH) Paraconsistentism Da Costa: (CON)  (COH), but not (COH)  (CON) Dialethism Priest: (COH) does not hold (dialetheism) Does paracoherence make sense? ai(p) = {1,0}

65. (At least) 3 sorts of ensuing propositional attitudes Bivalentism, Paracompletism Quine, Davidson, Lokhorst: (CON)  (COH) Paraconsistentism Da Costa: (CON)  (COH), but not (COH)  (CON) Dialetheism Priest: (COH) does not hold (dialetheism) Does paracoherence make sense? ai(p) = {1,0}

66. 5 Truths in Meaning

67. 3 sublogics in AR4: 3 subvaluations in V Bivalentism (“classical” logic) V{11,00} = {10,01} Paracompletism (intuitionistic logic) V{10,11,01} = {10,00,01} Paraconsistentism V{10,11,01} = {10,11,01}

68. 3 sublogics in AR4: 3 subvaluations in V Bivalentism (“classical” logic) V{11,00} = {10,01} Paracompletism (intuitionistic logic) V{10,11,01} = {10,00,01} Paraconsistentism V{10,11,01} = {10,11,01}

69. 3 sublogics in AR4: 3 subvaluations in V Bivalentism (“classical” logic) V{11,00} = {10,01} Paracompletism (gappy logics) V{10,11,01} = {10,00,01} Paraconsistentism V{10,11,01} = {10,11,01}

70. 3 sublogics in AR4: 3 subvaluations in V Bivalentism (“classical” logic) V{11,00} = {10,01} Paracompletism (gappy logics) V{10,11,01} = {10,00,01} Paraconsistentism (glutty logics) V{10,00,01} = {10,11,01}

71. 6 Conclusion: The Answer is in the Question

72. Absolutists: (+) are right to claim that a general theory of meaning requires a universal standard for understanding (–) miss the point in focusing the problem upon truth and only truth Relativists: (+) are right to claim that alternative reasonings may prevail in different contexts of reasoning (-) lose track of common rationality by equating plurality with relativity

73. Absolutists: (+) are right to claim that a general theory of meaning requires a universal standard for understanding (–) miss the point in focusing the problem upon truth and only truth Relativists: (+) are right to claim that alternative reasonings may prevail in different contexts of reasoning (–) lose track of common rationality by equating plurality with relativity

74. QAS: (+) reconciles opposite standpoints within a common framework where the classical truth-functions v are replaced by verdict-functions A (+) the real bearer of meaning is not a sentence, but its statement (+) an answer to the problem of logical charity is in the question, viz. the statement-forming operator Q that is attached to any sentence

75. References  da Costa, Newton C.A. & French, S. (1989): “On the logic of belief”, Philosophy and Phenomenological Research, Vol. 49, pp. 431-46  Davidson, D. (1973): “On radical interpretation”, Dialectica, Vol. 27, pp. 313-28  Frege, G. (1919): “Die Verneinung”, published in M. Black and P. T. Geach (eds.), Translations from the Philosophical Writings of Gottlob Frege, Blackwell, Oxford (1960)  Lokhorst, G.J. (1998): “The Logic of Logical Relativism”, Logique et Analyse, 161-162- 163, pp. 57-65  Priest, G. (1979): “The Logic of Paradox”, Journal of Philosophical Logic, Vol. 8, pp. 219-41  Quine, W.V.O. (1960): Word and Object, MIT Press  Quine, W.V.O. (1973): The Roots of Reference, Open Court Publishing, La Salle (Illinois)  Quine, W.V.O. (2004): Philosophy of Logic, Harvard University Press (2nd edition)  Searle, J. (1969): Speech Acts, Cambridge University Press  Searle, J. Vanderveken, D. (1985): Foundations of Illocutionary Logic, N.-Y., Cambridge University Press  Suszko, R. (1977): “The Fregean axiom and Polish mathematical logic in the 1920’s”, Studia Logica, Vol. 36, pp. 377-80  Williams, J.N. (1981): “Inconsistency and contradiction”, Mind, Vol. 90, pp. 600-2

76. 7 Appendix: On Conditional

77.  A difference between conditional and the other logical constants: , ,  are committal upon their components: to give an answer to p, pq, pq is to give answers about p and q  is not committal upon its components: An answer can be given to pq without giving any one about p and q  Frege-Geach’s (Embedding) Problem, aka Frege’s Point: Force-indicators operate only on complete sentences, and never occur significantly within the scope of a logical or sentential connective. Frege’s Point: ‘├(p  q)’ is a correct operation, whereas ‘├p  ├q’ is not Logical constants do not connect statements but, rather, sentences Statements: sentences marked with force-indicators

78.  A difference between conditional and the other logical constants: , ,  are committal upon their components: to give an answer to p, pq, pq is to give answers about p and q  is not committal upon its components: An answer can be given to pq without giving any one about p and q  Frege-Geach’s (Embedding) Problem, aka Frege’s Point: Force-indicators operate only on complete sentences, and never occur significantly within the scope of a logical or sentential connective. Frege’s Point: ‘├(p  q)’ is a correct operation, whereas ‘├p  ├q’ is not Logical constants do not connect statements but, rather, sentences Statements: sentences marked with force-indicators

79.  A difference between conditional and the other logical constants: , ,  are committal upon their components: to give an answer to p, pq, pq is to give answers about p and q  is not committal upon its components: An answer can be given to pq without giving any one about p and q  Frege-Geach’s (Embedding) Problem, aka Frege’s Point: Force-indicators operate only on complete sentences, and never occur significantly within the scope of a logical or sentential connective. Frege’s Point: ‘├(p  q)’ is a correct operation, whereas ‘├p  ├q’ is not Logical constants do not connect statements but, rather, sentences Statements: sentences marked with force-indicat

80.  A difference between conditional and the other logical constants: , ,  are committal upon their components: to give an answer to p, pq, pq is to give answers about p and q  is not committal upon its components: An answer can be given to pq without giving any one about p and q  Frege-Geach’s (Embedding) Problem, aka Frege’s Point: Force-indicators operate only on complete sentences, and never occur significantly within the scope of a logical or sentential connective. Frege’s Point: ‘├(p  q)’ is a correct operation, whereas ‘├p  ├q’ is not Logical constants do not connect statements but, rather, sentences Statements: sentences marked with force-indicators ors

81.  A difference between conditional and the other logical constants: , ,  are committal upon their components: to give an answer to p, pq, pq is to give answers about p and q  is not committal upon its components: An answer can be given to pq without giving any one about p and q  Frege-Geach’s (Embedding) Problem, aka Frege’s Point: Force-indicators operate only on complete sentences, and never occur significantly within the scope of a logical or sentential connective Frege’s Point: ‘├(p  q)’ is a correct operation, whereas ‘├p  ├q’ is not Logical constants do not connect statements but, rather, sentences Statements: sentences marked with force-indicators

82.  A difference between conditional and the other logical constants: , ,  are committal upon their components: to give an answer to p, pq, pq is to give answers about p and q  is not committal upon its components: An answer can be given to pq without giving any one about p and q  Frege-Geach’s (Embedding) Problem, aka Frege’s Point: Force-indicators operate only on complete sentences, and never occur significantly within the scope of a logical or sentential connective Frege’s Point: ‘├(p  q)’ is a wff, whereas ‘├p  ├q’ is not Logical constants do not connect statements but, rather, sentences

83. Frege (1919): (a) “If the accused was in Rome at the time of the deed, he did not commit the murder. He was in Rome at this time. Therefore he did not commit the crime.” p: the accused was in Rome at the time of the deed q: he (the accused) did committ the murder An inference the form: ├(p  q), ├p, ├q (Modus Ponens) (b) “If the accused was in Rome at the time of the deed, he did not commit the murder. He did commit the murder. Therefore he was not in Rome at this time.” An inference the form: ├ (p  q), ├q, ├p (Modus Tollens)

84. Frege (1919): (a) “If the accused was in Rome at the time of the deed, he did not commit the murder. He was in Rome at this time. Therefore he did not commit the crime.” p: the accused was in Rome at the time of the deed q: he did commit the murder An inference the form: ├(p  q), ├p, ├q (Modus Ponens) (b) “If the accused was in Rome at the time of the deed, he did not commit the murder. He did commit the murder. Therefore he was not in Rome at this time.” An inference the form: ├ (p  q), ├q, ├p (Modus Tollens)

85. Frege (1919): (a) “If the accused was in Rome at the time of the deed, he did not commit the murder. He was in Rome at this time. Therefore he did not commit the crime.” p: the accused was in Rome at the time of the deed q: he did commit the murder An inference the form: ├(p  q), ├p, ├q (Modus Ponens) (b) “If the accused was in Rome at the time of the deed, he did not commit the murder. He did commit the murder. Therefore he was not in Rome at this time.” An inference the form: ├ (p  q), ├q, ├p (Modus Tollens)

86. Frege (1919): (a) “If the accused was in Rome at the time of the deed, he did not commit the murder. He was in Rome at this time. Therefore he did not commit the crime.” p: the accused was in Rome at the time of the deed q: he did commit the murder An inference the form: ├(p  q), ├p, ├q (Modus Ponens) (b) “If the accused was in Rome at the time of the deed, he did not commit the murder. He did commit the murder. Therefore he was not in Rome at this time.” An inference the form: ├ (p  q), ├q, ├p (Modus Tollens)

87. What of (c) “If the accused was in Rome at the time of the deed, he did not commit the murder. The accused was not in Rome at the time of the deed. (…)” An inference the form: ├ (p  q), ├p, … therefore ├ q ? (d) “If the accused was in Rome at the time of the deed, he did not commit the murder. I do not say that he was in Rome at the time of the deed. (…)” An inference the form: ├ (p  q), ┤q, … therefore ├ p ? 2 paradoxical side-effects of the “classical” (mainstream) conditional: (c) entails everything if p is asserted, in classical (bivalent) logics (d) is reducible to (c), according to Frege’s thesis of equivalence

88. What of (c) “If the accused was in Rome at the time of the deed, he did not commit the murder. The accused was not in Rome at the time of the deed. (…)” An inference the form: ├ (p  q), ├p, … therefore ├ q ? (d) “If the accused was in Rome at the time of the deed, he did not commit the murder. I do not say that he was in Rome at the time of the deed. (…)” An inference the form: ├ (p  q), ┤q, … therefore ├ p ? 2 paradoxical side-effects of the “classical” (mainstream) conditional: (c) entails everything if p is asserted, in classical (bivalent) logics (d) is reducible to (c), according to Frege’s thesis of equivalence

89. What of (c) “If the accused was in Rome at the time of the deed, he did not commit the murder. The accused was not in Rome at the time of the deed. (…)” An inference the form: ├ (p  q), ├p, … therefore ├ q ? (d) “If the accused was in Rome at the time of the deed, he did not commit the murder. I do not say that he did commit the murder. (…)” An inference the form: ├ (p  q), ┤q, … therefore ├ p ? 2 paradoxical side-effects of the “classical” (mainstream) conditional: (c) entails everything if p is asserted, in classical (bivalent) logics (d) is reducible to (c), according to Frege’s thesis of equivalence

93. What of (c) “If the accused was in Rome at the time of the deed, he did not commit the murder. The accused was not in Rome at the time of the deed. (…)” An inference the form: ├ (p  q), ├p, … therefore ├ q ? (d) “If the accused was in Rome at the time of the deed, he did not commit the murder. I do not say that he did commit the murder. (…)” An inference the form: ├ (p  q), ┤q, … therefore ├ p ? 2 paradoxical side-effects of the “classical” (mainstream) conditional: (c) entails everything if p is asserted, in classical (bivalent) logics (d) is reducible to (c), according to Frege’s Equivalence Thesis

94. An alternative definition of conditional: commitment as a bet QAS is a question-answer game upon components: - this game relies upon behavioral rules, i.e. answerhood-conditions - whoever does not reply correctly does not even play the game A strengthened set of rules for the conditional p q: - the answerer must assert the antecedent to use conditional meaningfully - whoever does not assert p is compelled to reject any commitment about the whole pq If the speaker does not assert p in pq, then: - (s)he is not committed at all about q - (s)he does not assert pq and, therefore, rejects it(s being true) - to reject (p q) is not tantamount to assert its negation!

97. An alternative definition of conditional: commitment as a bet QAS is a question-answer game upon components: - this game relies upon behavioral rules, i.e. answerhood-conditions - whoever does not reply correctly does not even play the game A strengthened set of rules for the conditional p q: - the answerer must assert the antecedent to use conditional meaningfully - whoever does not assert p is compelled to reject any commitment about the whole pq If the speaker does not assert p in pq, then: - (s)he is not committed at all about q - (s)he does not assert pq and, therefore, rejects it(s being true) - to reject (p q) is not tantamount to asserting its negation!

98. Yes: if p, then q p  q -------------- If (yes: p), then (yes: q) p q Yes: not-(If p, then q) (p  q) --------------- If (yes: p), then (yes: p) p q

99. a1(p  q) = 1 p  q -------------- a1(p) = 1  a1(q) = 1 p q a2(p  q) = 1 (p  q) --------------- a1(p) = 1  a2(q) = 1 p q

100. a1(p  q) = 1 p  q -------------- a1(p) = 1  a1(q) = 1 p q Yes: not-(if p, then q) (p  q) --------------- If (yes: p), then (yes: p) p q

101. a1(p  q) = 1 p  q -------------- a1(p) = 1  a1(q) = 1 p q a2(p  q) = 1 (p  q) --------------- a1(p) = 1  a2(q) = 1 p q

102. Logical constants Negation A(p) = a2(p),a1(p) Conjunction A(pq) = a1(p)a1(q),a2(p)a2(q) Disjunction A(pq) = a1(p)a1(q),a2(p)a2(q) … Conditional A(pq) = a1(p)a1(q),a1(p)a2(q)

103. p q pq pq 11 11 11 11 11 10 10 10 11 01 01 11 11 00 00 10 10 11 11 10 10 10 10 10 10 01 01 10 10 00 00 10 01 11 00 11 01 10 00 10 01 01 00 01 01 00 00 00 00 11 00 10 00 10 00 10 00 01 00 00 00 00 00 00

104. AR4 p q pq pq pq qp pq V = {11,10,01,00} D = {10,11} 11 11 11 11 11 11 10 10 10 10 11 01 01 11 01 11 00 00 10 00 10 11 11 10 11 10 10 10 10 10 10 10 10 01 01 10 01 00 00 10 00 00 10 00 00 00 01 11 01 11 00 01 10 01 10 00 01 00 01 01 01 01 00 00 00 01 00 01 00 00 00 00 00 11 01 10 00 00 10 00 10 00 00 00 00 01 01 00 00 00 00 00 00 00 00 00 00 00

105. p q pq pq 11 11 11 11 11 10 11 10 11 01 01 11 11 00 01 10 ? 10 11 11 10 10 10 10 10 10 01 01 10 10 00 00 10 01 11 01 11 01 10 01 10 01 01 01 01 01 00 01 00 00 11 01 10 ? 00 10 00 10 00 01 01 00 00 00 00 00

106. Let A(p) = 11 and A(q) = 00. Then: A(pq) = 01 a1(p) = 1 and a1(q) = 0, hence a1(pq) = 0 a2(p) = 1 and a2(q) = 0, hence a2(pq) = 1 Therefore A(1100) = 01 A(pq) = 01 a1(p) = 1 and a1(q) = 0, hence a1(pq) = 1 a2(p) = 1 and a2(q) = 0, hence a2(pq) = 0 Therefore A(1100) = 10

107. Ł3 p q pq pq pq qp pq V = {10,00,01} D = {10,11} 11 11 11 11 11 11 10 10 10 10 11 01 01 11 01 11 00 00 10 00 10 11 11 10 11 10 10 10 10 10 10 10 10 01 01 10 01 10 01 10 00 00 10 00 10 00 01 11 01 11 00 01 10 01 10 10 01 01 01 01 01 01 10 10 10 01 00 01 00 10 00 00 00 11 01 10 00 00 10 00 10 10 00 00 00 01 01 00 00 10 00 00 00 00 00 10 10 10

109. Ks3 p q pq pq pq qp pq V = {10,00,01} D = {10} 11 11 11 11 11 11 10 10 10 10 11 01 01 11 01 11 00 00 10 00 10 11 11 10 11 10 10 10 10 10 10 10 10 01 01 10 01 10 01 10 00 00 10 00 10 00 01 11 01 11 00 01 10 01 10 00 01 00 01 01 01 01 10 10 10 01 00 01 00 00 00 00 00 11 01 10 00 00 10 00 10 10 00 00 00 01 01 00 10 10 10 00 00 00 00 10 00 00

111. B3 p q pq pq pq qp pq V = {10,00,01} D = {10} 11 11 11 11 11 11 10 10 10 10 11 01 01 11 01 11 00 00 10 00 10 11 11 10 11 10 10 10 10 10 10 10 10 01 01 10 01 10 01 10 00 00 00 00 00 00 01 11 01 11 00 01 10 01 10 10 01 01 01 01 01 01 10 01 01 01 00 00 00 00 00 00 00 11 01 10 00 00 10 00 10 00 10 00 00 01 01 00 00 00 00 00 00 00 00 00 00 00

113. P3 p q pq pq pq qp pq V = {10,11,01} D = {10,11} 11 11 11 11 11 11 11 11 10 11 10 10 11 11 11 01 01 11 01 10 01 11 00 00 10 00 10 11 11 10 11 10 11 10 10 10 10 10 10 10 10 01 01 10 01 10 01 10 00 00 10 00 01 11 01 11 10 11 11 01 10 01 10 10 01 01 01 01 01 01 10 10 10 01 00 01 00 00 00 11 01 10 00 00 10 00 10 00 00 0 00 00 00 00 00

114. AR4 p q pq pq pq qp pq V = {11,10,01,00} D = {11,10} 11 11 11 11 11 11 11 11 10 11 10 10 11 11 11 01 01 11 01 00 01 11 00 00 10 00 10 11 11 10 11 10 11 10 10 10 10 10 10 10 10 01 01 10 01 00 01 10 00 00 10 00 01 11 01 11 00 01 00 01 10 01 10 00 01 00 01 01 01 01 00 00 00 01 00 01 00 00 00 11 01 10 00 00 10 00 10 00 00 0 00 00 00

115. K3 Ł3 P3 AR4 q ╞ p  q     p ╞ p  q     (p  q)  r ╞ (p  r)  (q  r)     (p  q)  (r  s) ╞ (p  s)  (r  q)     (p  q) ╞ p     p  r ╞ (p  q)  r     p  q, q  r ╞ p  r     p  q ╞ q  p     ╞ p  (q  q)     ╞ (p  p)  q    

116. Advantages of the strengthened : 1. It avoids the Paradoxes of Material Implication v( F  q) = T (“Ex falso sequitur quodlibet”) a1(p  q)  1 when a1(p) = 0 and a1(q) = 1 v(p  T) = T (“Verum sequitur ex quodlibet”) a1(p  q)  1 when a1(p) = 0 2. It requires rejectivism, by assigning an essential occurrence to denial Assertibility- and deniability-conditions are not the same for  and : A(p  q)  A(p  q) only in the light of QAS the difference holds only if denial and negative assertion differ

117. Advantages of the strengthened : 1. It avoids the Paradoxes of Material Implication v(F  q) = T (“Ex falso sequitur quodlibet”) a1(p  q)  1 when a1(p) = 0 and a1(q) = 1 v(p  T) = T (“Verum sequitur ex quodlibet”) a1(p  q)  1 when a1(p) = 0 2. It requires rejectivism, by assigning an essential occurrence to denial Assertibility- and deniability-conditions are not the same for  and : A(p  q)  A(p  q) only in the light of QAS the difference holds only if denial and negative assertion differ

118. Advantages of the strengthened : 1. It avoids the Paradoxes of Material Implication v(F  q) = T (“Ex falso sequitur quodlibet”) a1(p  q)  1 when a1(p) = 0 and a1(q) = 1 v(p  T) = T (“Verum sequitur ex quodlibet”) a1(p  q)  1 when a1(p) = 0 2. It requires rejectivism, by assigning an essential occurrence to denial Assertibility- and deniability-conditions are not the same for  and : A(p  q)  A(p  q) only in the light of QAS the difference holds only if denial and negative assertion differ

121. Inconvenients of the strengthened : 1. The difference is irrelevant, according to Suszko’s Thesis There is no substantial difference between 01 and 00 Both logical values belong to the same class of undesignated values play the same role in logic as a theory of consequence 2. Accordingly, it conflates conditional and biconditional a1(p  q) = 1 iff a1(q  p) = 1, therefore a1(p  q) = 1 iff a1(p  q) = 1

122. Inconvenients of the strengthened : 1. The difference is irrelevant, according to Suszko’s Thesis There is no substantial difference between 01 and 00 Both logical values belong to the same class of undesignated values play the same role in any theory of consequence 2. Accordingly, it conflates conditional and biconditional a1(p  q) = 1 iff a1(q  p) = 1, therefore a1(p  q) = 1 iff a1(p  q) = 1

123. Inconvenients of the strengthened : 1. The difference is irrelevant, according to Suszko’s Thesis There is no substantial difference between 01 and 00 Both logical values belong to the same class of undesignated values play the same role in any theory of consequence 2. Accordingly, it conflates conditional and biconditional a1(p  q) = 1 iff a1(q  p) = 1, therefore a1(p  q) = 1 iff a1(p  q) = 1

125. A way out: rejectivism against Suszko’s Thesis There is an essential difference between denial and assertion 01 and 00 are not only designated values, but also: A(pq) = 01 is the value of a negative assertion: an anti-designated value (pq)  D– A(pq) = 00 is not the value of an assertion at all: a non-designated value (pq)  D If A(pq) = 01, then ╞AR4 p  q and p ╞AR4 q I cannot assert p without rejecting q If A(pq) = 00, then ╡AR4 p  q but not p ╡AR4 q I can reject p without rejecting q

129. Prospects  A revision (not a mere extension) of classical logic, within QAS  2 basic relations in logic: - consequence (truth- and falsity-preservation) - rejection (non-truth and non-falsity preservation)  1 universal relation, upstream of consequence and rejection: Partition, as a structuration of universe: - formal ontology (predicative dimension of a logical value) - formal logic (answerhood dimension of a logical value) A universal theory of negation: - opposite-forming operators (predicative dimension) - iterated answerhood (beyond Pavlov’s Logic of Truth and Falsehood)

130. Prospects  A revision (not a mere extension) of classical logic, within QAS: - conditional (or implication) and sharing principle (in formal ontology) - co-implication (dual of implication)  2 basic relations in logic: - consequence (truth- and falsity-preservation) - rejection (non-truth and non-falsity preservation)  1 universal relation, upstream of consequence and rejection Partition, as a structuration of meaning: - formal ontology (predicative dimension of a logical value) - formal logic (answerhood dimension of a logical value) A universal theory of negation: - opposite-forming operators (predicative dimension) - iterated answerhood (beyond Pavlov’s Logic of Truth and Falsehood)

133. 5th World Congress and School on Universal Logic (20-30 June 2015) Workshop: “Non-Classical Abstract Logics” (Fabien Schang, James Trafford) http://www.uni-log.org/start5.html

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