Argument Evaluation Based on Proportionality Marcin Selinger - PowerPoint PPT Presentation

Argument Evaluation Based on Proportionality Marcin Selinger Department of Logic and Methodology of Sciences University of Wrocław Argument Strength Ruhr University, Bochum 2016

The strength of structured arguments I. Syntax II. Evaluation III. Attack relation 2

I. Syntax 3

Classical diagrams of arguments P 1 P 2 P 2 P 1 P 3 P 1 P 2 C C P 2 Simple argument P 4 P 5 P 6 P 7 Linked argument C P 8 P 1 P 2 Serial argument P C C C 1 C 2 Multilevel complex argument Convergent argument Divergent argument 4

Conductive arguments (i.e. pro - contra , cf . Walton & Gordon 2015) P 1 P 1 P 1 C P 3 P 2 P 2 Con-argument P 2 P 1 C C C problematic cases Conductive argument 5

Multilevel convergent argument (an example) P 1 P 3 P 2 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 P 13 P 14 P 15 P 16 P 17 P 18 P 19 P 20 • premise • conclusion • first premise C • intermediate conclusion • final conclusion, final argument • atomic argument (= either simple or linked) 6

Formal structure of arguments Two kinds of inference: • pro -premises support conclusions; • con -premises deny ( contradict ) conclusions. Formal representation (Selinger 2014, 2015) : • Let L be a language , i.e. a set of sentences.  Sequents are any tuples of the form < P , c , d >, where: • P is a finite, non-empty set of sentences of L ( premises ) ; • c is a single sentence of L ( conclusion ); • d is a Boolean value (1 in pro -sequents and 0 in con -sequents).  Argumentation structures ( arguments ) are any finite and non-empty sets of sequents. 7

Atypical argumentation structures: P 1 P 1 P P 1 P 2 C 1 P 2 P 2 C 2 C 1 C 2 divergence circularity incoherence • arguments can have less or more than one final conclusion • in what follows arguments will be assumed to be coherent, non- divergent and non-circular 8

II. Numerical evaluation 9

Evaluation. Formal preliminaries • We assume that L contains the negation and the conjunction connectives; v : L ’→[0, 1], where L ’  L , is evaluation function — v ( α ) is ( the degree of ) • acceptability of α ; w : L  L →[0, 1] is conditional acceptability — w ( α / β ) is the acceptability of α • under the condition that v ( β ) = 1; Question: should w be a partial function? v (  α ) = 1 – v ( α ) — postulate of rationality ; • • If some premises deny α then the evaluation of α is based on the evaluation of  α in the corresponding sequent, in which these premises support  α . 10

P 1 P 3 P 2 Evaluation as transforming of acceptability values P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 P 13 P 14 P 15 P 16 P 17 P 18 P 19 P 20 C • the evaluation function is defined for the first premises • the acceptability of the first premises is transformed step by step to the acceptability of the final conclusion • formally, in each step the domain of the evaluation function is extended to the set containing new conclusion 11

The principle of proportionality The strength of argument should vary proportionally to the values assigned to its components. 12

Evaluation of premises x, y are the acceptability values of some two premises x  y (𝑦  𝑧) = 𝑧 1 𝑦 x x  y = xy 0 1 ½ y 13

Evaluation of atomic arguments x is the acceptability of (the set) of premises; y is the conditional acceptability (conclusion/premises) x, y > ½ x  ’ y x  y y y 1 0 1 0 ½ ½ x x (𝑦  ′ 𝑧) − ½ (𝑦  𝑧) = 𝑧 − ½ = 𝑦 1 𝑦 − ½ ½ 𝑧 x  y = xy x  ’ y = 2 xy – x – y + 1 14

Evaluation of convergent pro -arguments x , y – acceptability values of two converging arguments x , y > ½ x  ’ y x  y Yanal’s algorithm (1991) x x 0 1 0 1 ½ ½ y y (𝑦  𝑧) − 𝑦 = 𝑧 − ½ (𝑦  ′ 𝑧) − 𝑦 = 𝑧 1 1 − 𝑦 ½ 1 − 𝑦 x  y = 2 x + 2 y – 2 xy – 1 x  ’ y = x + y – xy 15

Evaluation of convergent con -arguments x, y < ½ x  c y x 0 ½ 1 y (𝑦  c 𝑧) = 𝑧 ½ 𝑦 x  c y = 2 xy x  c y = 1 – [(1 – x )  (1 – y )] = 2 xy 16

Evaluation of conductive arguments x < ½ is the acceptability of all convergent con -arguments y > ½ is the acceptability of all convergent pro -arguments y  ’ x as the arithmetic mean of x and y y  x ½ 0 1 x y ½ 0 1 x y y  ’ x y  x = ( y – ½) – ( ½ – x ) + ½ y  x = y + x – ½ y – ( y  ’ x ) = ( y  ’ x ) – x y  ’ x = ½ ( y + x ) 17

Evaluation of atomic arguments Let ∧ A be the conjunction of all the propositions belonging to a finite set A . Let A = {< P , c , d >}, where P  dom ( v ), c  dom ( v ), and d is a Boolean value. The function v A is the following extension of v to the set dom ( v )  { c }: If d = 1 then v A ( c ) = v ( ∧ P ) ⋅ w ( c / ∧ P ); • If d = 0 then v A ( c ) = 1 – v ( ∧ P ) ⋅ w (  c / ∧ P ). • The value v A ( c ) can be called the (logical) strength or force of the argument A . Note: the strength of {< P , c , 0>} = 1 – the strength of {< A ,  c , 1>}. Acceptability of arguments: • If A is a pro -argument then it is acceptable iff v A ( c ) > ½; • If A is a con -argument then it is acceptable iff v A ( c ) < ½. 18

Evaluation of convergent arguments Let A be an argument, α its final conclusion, and let A = A 1  A 2 , where both A 1 and A 2 have the same final conclusion c , they are coherent and acceptable, and all their sequents whose conclusion is c are only either pro - or con -sequents. If both A 1 and A 2 are pro , and v A1 ( c ), v A2 ( c ) > ½ , then v A ( c ) = v A1 ( c )  v A2 ( c ) • If both A 1 and A 2 are con , and v A1 ( c ), v A2 ( c ) < ½ , then v A ( c ) = v A1 ( c )  c v A2 ( c ) = • = 1 – (1 – v A1 ( c ))  (1 – v A2 ( c )) where x  y = 2 x + 2 y – 2 xy – 1 x  c y = 2 xy . Note: The operations  and  c are both commutative and associative, therefore the strengths of any number of convergent arguments can be added in any order. 19

Evaluation of conductive arguments Let A be an argument, α its final conclusion, and let A = A pro  A con , where all the sequents of A pro whose conclusion is c are only pro -sequents and all the sequents of A con whose conclusion is c are only con -sequents. We assume that both A pro and A con are coherent and acceptable, i.e. v Apro ( c ) > ½ and v Acon ( c ) < ½ . • If v Apro ( c ) < 1, and v Acon > 0, then v A ( c ) = v Apro ( c )  v Acon ( c ) where y  x = y + x – ½ ; If v Apro ( c ) = 1, and v Acon ( c )  0, then v A ( c ) = 1; • If v Apro ( c )  1, and v Acon ( c ) = 0, then v A ( c ) = 0; • • If v Apro ( c ) = 1, and v Acon ( c ) = 0, then v A ( c ) is not computable. 20

III. Attack relation (elementary cases) 21

Attack relation between arguments. Rebuttals, underminers, undercutters (Prakken 2010) • Attack on argument conclusion : {< P 1 , c , d>} can attack ( rebut ) {< P 2 , c , 1 – d >} {< P 1 , c , d>} can attack ( rebut ) {< P 2 , c ’ , d >} , where c ’ =  c or  c ’ = c • Attack on argument premises : {< P 1 , c 1 , 0>} can attack ( undermine ) {< P 2 , c 2 , d >} if c 1  P 2 {< P 1 , c 1 , 1>} can attack ( undermine ) {< P 2 , c 2 , d >} if c 1 ’  P 2 , where c 1 ’ =  c 1 or  c 1 ’ = c 1 • Attack on the relationship between argument premises and argument conclusion : undercutting defeaters 22

Successful attack on conclusion. Rebuttals An argument A rebuts (the conclusion of) an argument B iff • A = {< P 1 , c , d >}, • B = {< P 2 , c , 1 – d >} either d = 0 and 1 – v A ( c )  v B ( c ), or d = 1 and 1 – v A ( c )  v B ( c ) • or • A = {< P 1 , c , d >}, B = {< P 2 , c ’, d >}, where ( c ’ =  c or c =  c ’) • either d = 0 and v A ( c )  v B ( c ’), or d = 1 and v A ( c )  v B ( c ’) . • Note: in the borderline cases, i.e. if the above values are equal, the conclusion is not rebutted, but it is merely questioned. 23

Successful attack on premises. Underminers An argument A undermines (a premise of) an argument B iff • A = {< P 1 , c 1 , 0>}, B = {< P 2 , c 2 , d >}, where c 1  P 2 is the attacked premise, • either d = 1 and v’ B ( c 2 ) = [ v ( c 1 )  v’ A ( c 1 )] · v ( ∧ P 2 – c 1 })  w ( c 2 / ∧ P 2 )  ½, or • d = 0 and v’ B ( c 2 ) = [ v ( c 1 )  v’ A ( c 1 )] · v ( ∧ P 2 – { c 1 })  w (  c 2 / ∧ P 2 )  ½, where v’ is the function obtained from v by deleting c 1 from its domain; or • A = {< P 1 , c 1 , 1>}, B = {< P 2 , c 2 , d >}, where c 1 ’  P 2 is attacked ( c 1 ’ =  c 1 or c 1 =  c 1 ’), • either d = 1 and v’ B ( c 2 ) = [ v ( c 1 )  (1 – v’ A ( c 1 ))]· v ( ∧ P 2 – { c 1 })  w ( c 2 / ∧ P 2 )  ½, • or d = 0 and v’ B ( c 2 ) = [ v ( c 1 )  (1 – v’ A ( c 1 ))] · v ( ∧ P 2 – { c 1 })  w (  c 2 / ∧ P 2 )  ½, where v’ is the function obtained from v by deleting c 1 and c 1 ’ from its domain. 24