3. Structural correctness of arguments.

Strukturalna poprawność argumentu.

Marcin Selinger

Uniwersytet Wrocławski Katedra Logiki i Metodologii Nauk

marcisel@uni.wroc.pl

1. Definition of argument and further notions.

2. Operations on arguments.

3. Structural correctness of arguments.

Table of contents:

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.

• Argument = konkluzja + przesłanki.

• Przesłanki mogą wspierać konkluzję:

α₂ α

α₁ α₁ α₂

α α

• rozłącznie

(rozdzielnie, równolegle)

• łącznie

(zespołowo, szeregowo)

• w sposób mieszany α₁ α₂ α₃ α₄

• Czasami przesłanki wspierają inne przesłanki:

α₄ α₅ α₆ α₁ α₂ α₃

α₉

α₈

α

α₇

A A

A

₁

,

₂

,...,

n_A

} ,

,..., ,

, ,

{ <

^{1 1}

> <

^{2 2}

> < >

=

^m

α

^{m m}

α

^{m imm}

α

^imm

m

P P P

A

A

is an argument iff the following conditions hold:

Thus

α α

α

¹₁

⁼

₁²

^{= ... =}

₁ⁱ¹

k P

j ≤ i_m ∃ _m^j ∈ ^k_m−1

∀

α

(i)

for 2 ≤ m ≤ n_A. (i.e. for m = 1);

(ii)

for m ≤ n_A .

Def. 1.

Let S be a set of sentences of a given language.

Let

A

^{= < >} be a finite sequence of non-

empty,

finite relations defined on the set P

_fin

(S) × S.

Further definitions.

α α

α ¹₁ ⁼ ₁^{2 = ... =} ₁ⁱ¹

The final conclusion of

A

is the sentence:

Assume that

A

^{= <} > 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:

A A

A

₁

,

₂

,...,

_nA

A A

A

₁

,

₂

,...,

n_A

Def. 2.

Def. 3.

.

Examples

<{<{α₁}, α >, <{α₂}, α >}>

<{<{α₁, α₂}, α >}>

<{<{α₁, α₂, α₃}, α >, <{α₄}, α >}>

α₂

α α₁

α₁ α₂

α

α

α₁ α₂ α₃ α₄

convergent argument

linked argument

<

^{<{α₉}, α >},

{<{α₄, α₅, α₆}, α₉ >, <{α₈}, α₉ >},

{<{α₁}, α₅ >, <{α₂}, α₅ >, <{α₃}, α₅ >}, <{α₇}, α₈ >}

>

^.

α₄ α₅ α₆ α₁ α₂ α₃

α₉

α₈

α

α₇

The final argument of

A

is the one-element sequence <A₁>.

The m-th level of

A

is the relation A_m (for m ≤ n_A).

An argument <{<P, β >}> is an atomic argument of

A

^iff

there exists m ≤ n_A such that <P, β > ∈ A_m . Def. 6.

Def. 5.

Def. 4.

A

^{= <}

A

₁

, A

₂

,..., A

n_A>

An argument is direct iff it consists of one level only.

Def. 7.

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 . or

— it belongs to an element of the domain of (for m < n_A), but it does not belong to the counterdomain of .

A

n_A

A

m

A

m +1

<

^{<{α₉}, α >},

{<{α₄, α₅, α₆}, α₉ >, <{α₈}, α₉ >},

{<{α₁}, α₅ >, <{α₂}, α₅ >, <{α₃}, α₅ >}, <{α₇}, α₈ >}

>

^.

Examples

• final argument

• level of argument

• atomic argument

• direct argument

• intermediate conclusion

• first premise

α₄ α₅ α₆ α₁ α₂ α₃

α₉

α₈

α

3

2

1 α₇

=

α₁

α₂ α₃

α

α₁

α₂ α₃

α

α₁

<{<{α₂, α₃}, α >}, {<{α₁}, α₂ >, <{α₁}, α₃ >}>

α₁

α₄

α₅ α₇

α

α₃ α₂

α₆

α₁

α₄

α₅ α₇

α

α₃ α₂

α₆

α₁

α₄

α₅ α₇

α

α₃ α₂ α₆

α₁

α₄

α₅ α₇

α

α₃ α₂

α₆

α₁

=

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

^.

<

^{<{α₉}, α >},

{<{α₄, α₅, α₆}, α₉ >, <{α₈}, α₉ >},

{<{α₁}, α₅ >, <{α₂}, α₅ >, <{α₃}, α₅ >}, <{α₇}, α₈ >}

>

^.

Example

α₄ α₅ α ₆ α₁ α₂ α₃

α₉

α₈

α

3

2

1

α₇ Domain:

{α₁, α₂, α₃, α₄, α₅, α₆, α₇, α₈, α₉} Counterdomain:

{α, α₅, α₈, α₉} Range:

{α, α₁, α₂, α₃, α₄, α₅, α₆, α₇, α₈, α₉}

Def. 13

A sentence δ directly supports a sentence δ’ in

A

^iff

there exists an atomic argument of

A

, such that δ’ belongs to its domain, and δ belongs to its counterdomain.

Def. 14

A sentence δ_n indirectly supports a sentence δ₁ in

A

^iff

there exists a sequence of sentences <δ₁, δ₂, … δ_n>,

where n ≥ 3, such that each of its elements (except for δ₁) directly supports (in

A

) the preceding element.

Def. 15

A sentence δ supports a sentence δ’ in

A

^iff

δ directly or indirectly supports δ’ in

A

^.

Def. 16

An argument is circular iff its range contains a sentence, which supports itself (in this argument).

α₄ α₅ α ₆ α₁ α₂ α₃

α₉

α

α

α₄ α₅ α ₆ α₁ α₂ α₃

α₉

α₁

α

circular non-circular

Assume that

A

^{= <}

A

₁

, A

₂

,..., A

n_A^{> and}

B

^{= <}

B

₁

, B

₂

,..., B

n_B^>

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

( B

₁

⊆ A

_k^,

B

₂

⊆ A

_{k 1+} ^,...,

B

_nB

⊆ A

_{k+nB −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}

, ,...,

)

^.

A

B

₁

⊆

_k

B

₂

⊆ A

_{k 1+}

B

_nB

⊆ A

_{k+nB −1}

are arguments.

Remark 2:

"

B

⊂⊂⊂⊂

A

" doesn’t mean "

B

⊆⊆⊆⊆

A

^and

B

≠

A

^".

Remark 1: A

⊆⊆⊆⊆

A

^{, for all}

A

^.

Example

α₄ α₅ α ₆ α₁ α₂ α₃

α₉

α₈

α

α₇

α₄ α₅ α ₆ α₂

α₉

α

⊆

⊆

⊆

⊆

⊄

⊄

⊄

⊄

2. Operations on arguments.

• Addition.

• Maximal subarguments.

• Subtraction.

Addition of arguments

"conclusional" "premisal"

α₂

α α₁

α

α₁ α₂

α

+

^↓

=

α

α₁ α₂

α₁

α α₁ α₂

+

^↑

=

Assume that

A

^{= < >,}

B

^{= < >}

and

C

^{= <} > are arguments.

Assume that

1 ≤ m ≤ n

_A^.

A A

A

₁

,

₂

,...,

n_A

B

₁

, B

₂

,..., B

n_B

C C

C

₁

,

₂

,...,

n_C

Def. 19

A

⁺_↓m

B

⁼

C

^iff

• either the final conclusion of

B

is not contained in the counter- domain 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

α β γ

α α α

α β

γ

α α

α α α

α β

γ

α α

α

=

=

+

_↓1

+

_↓2

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

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

^).

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

^.

Def. 23

A

⁺

B

^{= (}

A

⁺_↑

B

^{) +}_↓1

B

Maximal subarguments

determined by a conclusion

determined by

an atomic argument

α₄ α₅ α₆ α₁ α₂ α₃

α₉

α₈

α

α₇

α₄ α₅ α ₆ α₁ α₂ α₃

α₉

α₈

α

α₇

conclusion:

α₉

atomic argument:

<{<{α₄, α₅, α₆}, α₉>}>

Def. 24

Assume that

A

^{= <}

A

₁

, A

₂

,..., 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.

C

^{= max(}

A

^,

B

^{, m) iff}

C

is the longest (e.i. containing the largest number of levels) of the arguments

C*

= < >, such that satisfy the following conditions:

(i) n_C* ≤ n_A – m + 1;

(ii) C*₁ = B₁;

(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}.

,..., * , *

*

₁

C

₂

C

_C*

C

_n

Def. 25

C

^{= max(}

A

^{, δ, m) iff}

C

is the longest of the arguments

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}.

,..., * , *

*

₁

C

₂

C

_C*

C

_n

Assume that

A

^{= <}

A

₁

, A

₂

,..., A

n_A> is an argument.

Assume that δ is an element of the counterdomain of

A

_m

for the level number m ≤ n_A.

Remark 2: If {B¹, B², …, B^k} is the set of all atomic arguments of the m-th level of A, which have the same conclusion δ, then:

max(A, δ, m) = max(A, B¹, m) +_↓1 max(A, B², m) +_↓1 …+_↓1 max(A, B^k, m).

Remark 1: If

B

is the final argument of

A

^{, then max(}

A

^,

B

^{, 1) =}

A

^.

If δ is the final conclusion in

A

^{, then max(}

A

^{, δ, 1) =}

A

^.

Subtraction of arguments

α₄ α₅ α₆ α₁ α₂ α₃

α₉

α₈

α

α₇

α₄ α₅ α₆ α₁ α₂ α₃

α₉ α₉

α₈

α α₇

– =

Assume that

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^).

A A

A

₁

,

₂

,...,

n_A

Assume that

C

= < > = max(

C

₁

, C

₂

,..., C

_nC

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.

3. Structural correctness of arguments.

For a structurally correct argument

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

^).

(3) For every sentence δ:

if there are i, j ≤ n_A, such that the counterdomains of A_i and A_j contain δ, then max(

A

^{, δ, i) = max(}

A

^{, δ, j).}

A A

A

₁

,

₂

,...,

n_A

An open problem: Are these conditions sufficient?

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.

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.

Example

α α₁₆

α₁₂ β α

β γ α₇ α₆

α₇ α₆

α₉ α₈

α₁₀ α₁₁ γ α₄ α₁

α₅ γ α₂ α₃

α₁₅ α₁₄

α₁₃

6

5

3 4

1 2

α α₁₆

α₁₂ γ α₄

α₁

α₅ γ α₂ α₃

α₁₅ α₁₄

α₁₃

6

5

3 4

1 2 β γ

α α₁₆

α₁₂ β γ

α₇ α₆

γ α₄

α₁

α₅ γ α₂ α₃

α₁₅ α₁₄

α₁₃

6

5

3 4

1 2

α α₁₆

α₁₂ β γ

α₇ α₆

γ α₄

α₁

α₅ γ α₂ α₃

α₁₅ α₁₄

α₁₃

6

5

3 4

1 2 α₉

α₈

α₁₀ α₁₁

α α₁₆

α₁₂ β γ

α₇ α₆

γ α₄

α₁

α₅

γ α₂ α₃

α₁₅ α₁₄

α₁₃

6

5

3 4

1 2 α₉

α₈

α₁₀ α₁₁ α₄

α₁

α₂ α₃

α₂ α₃ α₄

α₁

7

- Dziękuję za uwagę.