Strukturalna poprawność argumentu.

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1 Strukturalna poprawność argumentu. Marcin Selinger Uniwersytet Wrocławski Katedra Logiki i Metodologii Nauk marcisel@uni.wroc.pl

2 Table of contents: 1. Definition of argument and further notions. 2. Operations on arguments. 3. Structural correctness of arguments.

3 1. Definition of argument and further notions. References: [1973] S. N. Thomas (second edition, 1986), Practical reasoning in natural language. [2001] K. Szymanek, Sztuka argumentacji. Słownik terminologiczny. [2003] K. Szymanek, K. A. Wieczorek, A. Wójcik, Sztuka argumentacji. Ćwiczenia w badaniu argumentów. [2006] M. Tokarz, Argumentacja, perswazja, manipulacja.

4 Argument = konkluzja + przesłanki. Przesłanki mogą wspierać konkluzję: łącznie (zespołowo, szeregowo) rozłącznie (rozdzielnie, równolegle) w sposób mieszany Czasami przesłanki wspierają inne przesłanki:

5 Let S be a set of sentences of a given language. A1, A2,..., An A > be a finite sequence of nonempty, finite relations defined on the set P fin (S) S. Let A = < Thus A m = { < 1, >, <, >,..., < m, m P P P > } m m m m i m i m for m n A. Def. 1. A is an argument iff the following conditions hold: (i) 1 2 i 1 = 1 =... = 1 (i.e. for m = 1); (ii) j i k j k m m P m 1 for 2 m n A.

6 Assume that A = < Def. 2. Def. 3. Further definitions. The final conclusion of A is the sentence: 1 A1, A2,..., A 2 i 1 1 = 1 =... = 1 > is an argument. A sentence is a premise of A iff it is an element of a set belonging to the domain of some of relations: A1, A2,..., A n A n A.

7 Examples 1 2 <{<{ 1 }, >, <{ 2 }, >}> 1 2 convergent argument <{<{ 1, 2 }, >}> linked argument <{<{ 1, 2, 3 }, >, <{ 4 }, >}>

8 <{<{ 9 }, >}, {<{ 4, 5, 6 }, 9 >, <{ 8 }, 9 >}, {<{ 1 }, 5 >, <{ 2 }, 5 >, <{ 3 }, 5 >}, <{ 7 }, 8 >}>.

9 A = < A, A 2,..., A 1 > n A Def. 4. The final argument of A is the one-element sequence <A 1 >. Def. 5. The m-th level of A is the relation A m (for m n A ). Def. 6. An argument <{<P, β >}> is an atomic argument of A iff there exists m n A such that <P, β > A m.

10 Def. 7. An argument is direct iff it consists of one level only. Def. 8. A sentence is an intermediate conclusion of A iff it belongs to the counterdomain of some of its levels, which are higher then 1. Def. 9. A sentence is a first premise of A iff it belongs to an element of the domain of A n A. or it belongs to an element of the domain of A m (for m < n A ), but it does not belong to the counterdomain of A m + 1.

11 Examples final argument level of argument atomic argument direct argument intermediate conclusion first premise 9 1 <{<{ 9 }, >}, {<{ 4, 5, 6 }, 9 >, <{ 8 }, 9 >}, {<{ 1 }, 5 >, <{ 2 }, 5 >, <{ 3 }, 5 >}, <{ 7 }, 8 >}>.

12 1 1 1 = <{<{ 2, 3 }, >}, {<{ 1 }, 2 >, <{ 1 }, 3 >}>

13 =

14 Def. 10 The domain of A is the set of all the premises of A. Def. 11 The counterdomain of A is the set of all the conclusions of A. i.e. the set of intermediate conclusions {final colclusion} Def. 12 The range of A is the sum of the domain and counterdomain of A.

15 Example Domain: { 1, 2, 3, 4, 5, 6, 7, 8, 9 } Counterdomain: {, 5, 8, 9 } 9 1 Range: {, 1, 2, 3, 4, 5, 6, 7, 8, 9 } <{<{ 9 }, >}, {<{ 4, 5, 6 }, 9 >, <{ 8 }, 9 >}, {<{ 1 }, 5 >, <{ 2 }, 5 >, <{ 3 }, 5 >}, <{ 7 }, 8 >}>.

16 Def. 13 A sentence δ directly supports a sentence δ in A iff there exists an atomic argument of A, such thatδ belongs to its domain, and δ belongs to its counterdomain. Def. 14 A sentenceδ n indirectly supports a sentenceδ 1 in A iff there exists a sequence of sentences <δ 1, δ 2, δ n >, where n 3, such that each of its elements (except for δ 1 ) directly supports (in A) the preceding element. Def. 15 A sentenceδsupports a sentenceδ in A iff δ directly or indirectly supportsδ in A.

17 Def. 16 An argument is circular iff its range contains a sentence, which supports itself (in this argument) circular non-circular

18 Assume that A = < A1, A2,..., A > and B = < B1, B2,..., B > are arguments. Def. 17 (B A) B is a subargument of A iff the following conditions hold: (i) n B n A ; (ii) k n A n B +1 ( B1 Ak, B2 Ak + 1,..., Bn Ak + n 1 ). Def. 18 (B A) B is an internal subargument of A iff the following conditions hold: (i) n B < n A ; (ii) k n A n B +1 (k > 1 and B A n A B B n B, B2 k+ 1,..., Bn Ak + n 1 ). 1 k A B B

19 Example Remark 1: A A, for all A. Remark 2: "B A" doesn t mean "B A and B A".

20 2. Operations on arguments. Addition. Maximal subarguments. Subtraction.

21 Addition of arguments "conclusional" "premisal" = = Assume that A = < A1, A2,..., An A >, B = < B1, B2,..., Bn B > and C = < C1, C 2,..., C > are arguments. n C Assume that 1 m n A.

22 Def. 19 A + m B = C iff either the final conclusion of B is not contained in the counterdomain of A m and A = C. or the final conclusion of B is contained in the counterdomain of A m and the following condisions hold: (i) n C = max {n A, m + n B 1}; (ii) C i = A i, if 1 i < m (for m 2) or i > m + n B ; (iii) C i = A i B i-m+1, if m i n A ; (iv) C i = B i-m+1, if n A < i n C. Def. 20 A + B = ( ((A + na B) + na-1 B) + na-2 ) + 1 B

23 γ β + 1 = γ β + 2 = γ β

24 Def. 21 A + m B = C iff either the final conclusion of B is not contained in any element of the domain of A m and A = C or the final conclusion of B is contained in some element of the domain of A m and the following condisions hold: (i) n C = max {n A, m + n B }; (ii) C i = A i, if 1 i m (for m 2) or i > m + n B ; (iii) C i = A i B i-m, if m < i n A ; (iv) C i = B i-m, if n A < i n C. Def. 22 A + B = ( ((A + na B) + na-1 B) + na-2 ) + 1 B

25 Remark 1: Let m > 1. Then A + m B = A + m-1 B iff the final conclusion of B is contained in the counterdomain of A m or the final conclusion of B is not contained in any element of the domain of A m-1. (i.e. the above equation holds iff the final conclusion of B is not any of the first premises on the level m-1 of A) Remark 2: The operations of addition are neither commutative nor associative, but if A, B (and C) have identical final conclusions, then the following equations hold: A + 1 B = B + 1 A; (A + 1 B) + 1 C = A + 1 (B + 1 C).

26 Def. 23 A + B = (A + B) + 1 B Remark 1: If the final conclusion of B is not in the range of A, then: A + B = A. Remark 2: If A is not circular, then A + A = A.

27 Maximal subarguments determined by a conclusion determined by an atomic argument conclusion: atomic argument: <{<{ 4, 5, 6 }, 9 >}>

28 Assume that A = < A1, A2,..., A n A > and B = <B 1 > are arguments. Assume that B is an atomic argument in A, where B 1 A m for the level number m n A. Def. 24 C = max(a, B, m) iff C is the longest (e.i. containing the largest number of levels) of the arguments C* = < C* 1, C* 2,..., C* n >, such that satisfy C* the following conditions: (i) n C* n A m + 1; (ii) C* 1 = B 1 ; (iii) if n C* 2, then for every 2 i n C* : C* i = {<P, δ*> A i+m-1 : δ* is contained in some element of the domain of C* i-1 }.

29 Assume that A = < A1, A2,..., A n A > is an argument. Assume thatδ is an element of the counterdomain of A m for the level number m n A. Def. 25 C = max(a, δ, m) iff C is the longest of the arguments C* = < C, C*,..., * >, * nc* 1 2 C such that satisfy the following conditions: (i) n C* n A m + 1; (ii) C* 1 = {<P, δ*> A m : δ* =δ}; (iii) if n C* 2, then for every 2 i n C* : C* i = {<P, δ*> A i+m-1 : δ* is contained in some element of the domain of C* i-1 }.

30 Remark 1: If B is the final argument of A, then max(a, B, 1) = A. Ifδis the final conclusion in A, then max(a, δ, 1) = A. Remark 2: If {B 1, B 2,, B k } is the set of all atomic arguments of the m-th level of A, which have the same conclusionδ, then: max(a,δ, m) = max(a, B 1, m) + 1 max(a, B 2, m) max(a, B k, m).

31 Subtraction of arguments =

32 Assume that A = < A1, A2,..., An A > is an argument, and that B = <B 1 > is an atomic (non-final) argument in A (B 1 A m for m n A ). Assume that C = < C1, C2,..., C > = max(a, B, m). Def. 26 A m B = D iff (i) m 1 n D n A ; (ii) if m 2, then D i = A i, for every i < m; (iii) if n A = n C + m 1, then: n D = max{j < n A : A j C j-m+1 }; D i = A i C i-m+1, for every m i n D ; (iv) if n A > n C + m 1, then: n D = n A ; D i = A i C i-m+1, for every m i n C + m 1; D i = A i, for every n C + m 1 < i n D. n C

33 3. Structural correctness of arguments. For a structurally correct argument A = < n A > it is necessary that the following conditions hold: (1) For every argument B: if B A, then (the counterdomain of B) (the domain of B) = = {the final conclusion of B}. (2) (The domain of A) (the counterdomain of A) = = (the set of all the first premises of A). A1, A2,..., A (3) For every sentence δ: if there are i, j n A, such that the counterdomains of A i and A j containδ, then max(a, δ, i) = max(a, δ, j). An open problem: Are these conditions sufficient?

34 Remark 1: The condition (1) doesn t hold iff A is circular. Remark 2: The condition (2) doesn t hold iff there is a sentence in the domain of A, which is one of the first premises and a conclusion (final or intermediate) at the same time. Remark 3: The condition (3) doesn t hold iff there is a sentence in the domain of A, which is supported by different subarguments, when it appears on different levels. Remark 4: If the condition (1) doesn t hold, then at least one of the conditions: (2) or (3) doesn t hold either. The converse implication is not true.

35 Example γ γ 5 5 β γ β

36 1 4 γ 2 3 γ β γ

37 γ 6 7 γ 5 5 β γ

38 γ 6 7 γ 5 5 β γ

39 γ γ 5 5 β γ

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