JSM Reasoning and Knowledge Discovery: Ampliative Reasoning, Causality Recognition, and Three Kinds of Completeness^#

Abstract

Inductive inferences and inferences, by analogy with JSM reasoning, are characterized as ampliative inferences generating new knowledge. New predicates for inductive inference rules and their ordering are considered. The case of a single-element effect for the predicate “X has effect Y” is also investigated.

1. The fifth level of acceptance of the generated hypotheses is an M-sequence of modal operators of rank r, where r > 1, representing empirical regularities that are obtained by applying JSM reasoning r-fold to sequences of extensible fact bases [1, p. 22].

2. Thus, the empirical patterns are a formalization of the idea of the knowledge-discovery process formulated in [2].

3. To understand the definition of knowledge in a computer system, one should become familiarized with its definition in [5, p. 31].

4. FB(0), FB(1), …, FB(s) is the sequence of expandable fact bases: FB(0) ⊂ FB(1) ⊂ … ⊂ FB(s).

5. Note that the degree of plausibility of a fact is 0 and that of a hypothesis is greater than 0, since it represents the number of steps of a plausible inference.

6. Since the truth values of JL formulas have the form \(\bar {v} = \left\langle {v,n} \right\rangle \), where \(v \in \left\{ {1, - 1,0,\tau } \right\}\), and \(n \in N\), the logic is infinite-valued.

7. In the case |Y| =1, \({{J}_{{\left\langle {1,1} \right\rangle }}}\left( {{{V}_{i}} \Rightarrow _{2}^{{(P)}}{{Y}_{i}}} \right)\) and \({{J}_{{\left\langle {1,0} \right\rangle }}}\left( {X \Rightarrow _{1}^{{(P)}}Y} \right)\) are used.

8. That is, similarities for (+)-examples and (–)-examples.

9. For convenience of notation, we will represent finite sets {A, B, ..., P} as words AB...P; for example, we will represent {A, B, C} as АВС.

10. Therefore, the relation \( \Rightarrow _{2}^{*}\) is not functional.

11. The premises of p.i.r.-1 are the corresponding elements of the diagram for \({{\Re }_{1}}\), and the premises of p.i.r.-2 are the corresponding elements of the diagram for \({{\bar {\Re }}_{1}}\).

12. In IS–JSM intelligent systems that implement the ARS JSM method [4, 19], D_1,0(p) form a fact base.

13. If CCA^(σ) are true, then the admissible JSM reasoning is strong [5].

14. This condition can be generalized to all Str_{x, y} from [7]. The same holds true for the sufficient condition (**).

15. Obviously, for strong admissible JSM reasoning ρ^σ(s) = 1, where σ = +,–.

16. In Section 4 of this paper, possible strengthenings will be considered.

17. Since \(X{{ \Rightarrow }_{1}}Y\) and \(V{{ \Rightarrow }_{2}}Y\) are defined with respect to relational systems and their corresponding D_0,1(P), these primitive predicates depend on the parameter Р.

18. In [8], J.S. Mill used the term agreement–difference. In the ARS JSM, the term similarity–difference is preferred.

19. The elements of M have a superscript σ, where σ = +, –, which, for the sake of notation, will sometimes be omitted.

20. Definitions of operations and are contained in the Appendix.

21. φ is obtained in the Appendix.

22. Indices (P) and (x, y) will sometimes be omitted for convenience of notation.

23. In the Appendix, the relational system \({{\bar {\Re }}_{f}}\) is considered such that the predicates \(M_{{{{a}_{{12}}}fg,0}}^{\sigma }\left( {V,Y} \right)\) are satisfiable in it.

24. Part II of this article will deal with the cases when |Y| > 1 and there is ¬α.

25. The content of this section assumes familiarity with papers [1, 5].

26. They are empirical nomological statements [1, 26].

27. This means strengthened realization of the scientific research demarcation criterion [27].

28. This definition is formulated for computer systems that implement the ARS JSM method.

29. We intend to consider the Boolean function F₂() for the case |Y | > 1 in the second part of this article.

30. The sets \(\overline {Str} \times A_{E}^{\sigma }\) are partially ordered with the largest and smallest element for σ = +, –, respectively.

31. Recall that \(A_{\chi }^{\sigma }\) is an element of the intension of the concept of empirical regularity and \(A_{\chi }^{\sigma }(C{\kern 1pt} ',\,\,Q)\) is an extension element of this concept.

32. Situational JSM reasoning is obviously in demand for the analysis of sociological data. Note also that there are cases when V = Ø or S = Ø.

33. We intend to consider this problem in the second part of this article.

Authors and Affiliations

1. Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, 119333, Moscow, Russia

   V. K. Finn

2. Russian State University for the Humanities, 125993, Moscow, Russia

   V. K. Finn

Corresponding author

Correspondence to V. K. Finn.

The author declares that he has no conflicts of interest.

Translated by S. Avodkova

^#In the presented part of the article, we study the case of a singleton effect for the predicate “X has effect Y”. We intend to consider the Boolean function F₂(\(\alpha ,\beta ,\gamma \)) for the case |Y| > 1 in the second part of this article.

APPENDIX

1. Using the method of analytic tables [21], we prove the proposition from Section 3: \({{(\exists )}^{{(\sigma )}}}\) ⊢ \((*)\), where σ = +, –, “⊢” is the deducibility ratio, \((*)\) is ∃V∃Y (\((M_{{{{a}_{{12,0}}}}}^{ + }\left( {V,Y} \right) \vee M_{{{{a}_{{12,0}}}}}^{ - }\left( {V,Y} \right))\), and \({{(\exists )}^{{( + )}}}\) is ∃V∃X∃Y \(({{J}_{{\left\langle {1,1} \right\rangle }}}(V{{ \Rightarrow }_{2}}Y)\) & \({{J}_{{\left\langle {1,0} \right\rangle }}}(X{{ \Rightarrow }_{1}}Y)\) & \((V \subset X))\).

∃V∃Y(\(M_{{{{a}_{{12}}},0}}^{ + }\left( {V,Y} \right)\) \( \cap , \cup ,\) \(M_{{{{a}_{{12}}},0}}^{ - }\left( {V,Y} \right)\)) ↔ (∃V∃Y\(M_{{{{a}_{{12}}},0}}^{ + }\left( {V,Y} \right)\) \(\left\langle {C,a} \right\rangle \in R_{1}^{ + };\) ∃V∃Y\(M_{{{{a}_{{12}}},0}}^{ - }\left( {V,Y} \right)\)).

Let us denote the closed formula ∃V∃Y\(M_{{{{a}_{{12}}},0}}^{ - }\left( {V,Y} \right)\) through \(\varphi \); then \((*)\) we represent as ∃V∃Y.

For the sake of simplicity of notation, we introduce the following notation:

Р(V,Y) instead of,

S(V, Y) instead of,

Q(V, Y) instead of,

L(V, X) instead of (V X),

a K(X, Y) instead of.

Then we obtain ∃V∃X∃Y(S(V, Y) & P(V, Y) & Q(V, Y) & K(X, Y) & L(X, Y) ⊢ ∃V∃YP(V, Y).

Let us construct an analytical table F.

∃V∃X∃Y(S(V, Y) & P(V, Y) & Q(V, Y) & K(X, Y) & L(X, Y)

¬(∃V∃YP(V, Y) \( \vee \,\,\varphi \))

¬∃V∃YP(V, Y)

¬\(\varphi \)

S(a, b) & P(a, b) & Q(a, b) & K(c, b) & L(c, b)

S(a, b)

P(a, b)

Q(a, b)

K(c, b)

L(c, b)

¬P(a, b)

\(*\)

F is a closed analytical table. a, b, c are the parameters.

In a similar way, we will show that (∃)^–⊢. The above proposition can be extended to any JSM reasoning strategy Str_{x, y}.

2. \((**)\) is \(\exists X\exists Y\left( {\Pi _{{{{a}_{{12}}}}}^{ + }\left( {X,Y} \right) \vee } \right.\left. {\Pi _{{{{a}_{{12}}}}}^{ - }\left( {X,Y} \right)} \right),\) \({{\left( \exists \right)}^{{\left( \sigma \right)}}}\) is \(\exists V\exists X\exists Y\left( {{{J}_{{\left\langle {v,1} \right\rangle }}}\left( {V{{ \Rightarrow }_{2}}Y} \right)} \right.\) & \({{J}_{{\left\langle {v,o} \right\rangle }}}\left( {X{{ \Rightarrow }_{1}}Y} \right)\) & \((V \subset X))\), where \(v = \left\{ \begin{gathered} 1,\,\,\,\,{\text{if}}\,\,\,\sigma = + \hfill \\ - 1,\,\,\,\,{\text{if}}\,\,\,\sigma = - \hfill \\ \end{gathered} \right.\) \(\exists X\exists Y\left( {\Pi _{{{{a}_{{12}}}}}^{ + }\left( {X,Y} \right)} \right.\) \( \vee \) \(\left. {\Pi _{{{{a}_{{12}}}}}^{ - }\left( {X,Y} \right)} \right)\) \( \leftrightarrow \) \(\left( {\exists X\exists Y\Pi_{{{{a}_{{12}}}}}^{ + }\left( {X,Y} \right) \vee } \right.\) \(\exists X\exists Y\left. {\Pi _{{{{a}_{{12}}}}}^{ - }\left( {X,Y} \right)} \right)\).

Let us show that the truth of \((**)\) implies the truth of \({{(\exists )}^{{(\sigma )}}}\), where σ = +, –.

Let us assume that \((**)\) is true. Without loss of generality, let us assume that \(\exists X\exists Y\Pi _{{{{a}_{{12}}}}}^{ + }(X,Y)\). Then \(\Pi _{{{{a}_{{12}}}}}^{ + }(C,a)\) is true, where С are the individual constants. Let us consider \({{J}_{{\left\langle {1,1} \right\rangle }}}(C{\kern 1pt} '{{ \Rightarrow }_{2}}a)\) & \((C{\kern 1pt} ' \subset C)\) & \(\neg \exists {{V}_{0}}\left( {{{J}_{{\left\langle { - 1,1} \right\rangle }}}({{V}_{0}}{{ \Rightarrow }_{2}}a)} \right.\) \( \vee {{J}_{{\left\langle {0,1} \right\rangle }}}({{V}_{0}}{{ \Rightarrow }_{2}}a))\) & \(({{V}_{0}} \subset C))\), since \(\Pi _{{{{a}_{{12}}}}}^{ + }(C,a)\) ⇌ \(\exists V({{J}_{{\left\langle {1,1} \right\rangle }}}(V{{ \Rightarrow }_{2}}a)\) & \((V \subset C)\& \neg \exists {{V}_{0}}\) \((({{J}_{{\left\langle { - 1,1} \right\rangle }}}({{V}_{0}}{{ \Rightarrow }_{2}}a) \vee {{J}_{{\left\langle {0,1} \right\rangle }}}({{V}_{0}}{{ \Rightarrow }_{2}}a))\) & \(({{V}_{0}} \subset C))\), let the value of the variable V be \(C{\kern 1pt} '\). Then \({{J}_{{\left\langle {1,1} \right\rangle }}}(C{\kern 1pt} '{{ \Rightarrow }_{2}}a)\) is true. Since \(({{J}_{{\left\langle {\tau ,0} \right\rangle }}}(C{\kern 1pt} '{{ \Rightarrow }_{2}}a)\& \) \(M_{{{{a}_{{12}}},0}}^{ + }\left( {C{\kern 1pt} ',a} \right)\) & ¬\(M_{{{{a}_{{12}}},0}}^{ + }\left( {C{\kern 1pt} ',a} \right) \leftrightarrow {{J}_{{\left\langle {1,1} \right\rangle }}}(C{\kern 1pt} '{{ \Rightarrow }_{2}}a)\) (this follows from the definition of p.i.r.-1 and the theorem of the reversibility of the inference rules p.i.r.-1 and p.i.r.-2 in the ARS JSM method [22, Ch. 5]).

Therefore, \(M_{{{{a}_{{12}}},0}}^{ + }\left( {C{\kern 1pt} ',a} \right)\) & ¬\(M_{{{{a}_{{12}}},0}}^{ - }\left( {C{\kern 1pt} ',a} \right)\) is true and \(M_{{{{a}_{{12}}},0}}^{ + }\left( {C{\kern 1pt} ',a} \right)\) is true. Then \(\exists {{X}_{1}} \ldots \) \(\exists {{X}_{k}}\left( {\left( {\& _{{i = 1}}^{k}} \right.} \right.{{J}_{{\left\langle {1,0} \right\rangle }}}\left( {{{X}_{i}}{{ \Rightarrow }_{1}}a} \right)\) & \(( \cap _{{i = 1}}^{k}{{X}_{i}} = C{\kern 1pt} ')\& \) \(\left( {k \geqslant 2} \right)\& \forall X\left( {\left( {{{J}_{{\left\langle {1,0} \right\rangle }}}\left( {X{{ \Rightarrow }_{1}}a} \right)\& } \right.} \right.\) \(\left. {\left( {C{\kern 1pt} ' \subset X} \right)} \right) \to \) \(\left. {\left( { \vee _{{i = 1}}^{k}\left( {{{X}_{i}} = X} \right)} \right)} \right)\) is true. Therefore, \(\exists {{X}_{i}}\left( {{{J}_{{\left\langle {1,0} \right\rangle }}}({{X}_{1}}{{ \Rightarrow }_{1}}a)} \right.\) & \((C{\kern 1pt} ' \subset X)\) & \({{J}_{{\left\langle {1,1} \right\rangle }}}\left( {\left. {\left. {C{\kern 1pt} '{{ \Rightarrow }_{2}}a} \right)} \right)} \right. \leftrightarrow \) \(\exists V\exists X\exists Y\left( {{{J}_{{\left\langle {1,1} \right\rangle }}}\left( {V{{ \Rightarrow }_{2}}Y} \right)} \right.\) & \((V \subset X)\) & \({{J}_{{\left\langle {1,0} \right\rangle }}}(X{{ \Rightarrow }_{1}}Y))\).

Therefore, the truth of \((**)\) implies the truth of \({{(\exists )}^{{( + )}}}\). It is obvious that a similar reasoning holds for the assumption of the truth of \(\exists X\exists Y\exists Y\Pi _{{{{a}_{{12}}}}}^{ - }(X,Y)\).

Obviously, since the truth of \((**)\) implies the truth of \({{(\exists )}^{{(\sigma )}}}\) and the truth of \({{(\exists )}^{{(\sigma )}}}\) implies the truth of \((*)\), the truth of \((**)\) implies the truth of \((*)\).

3. Operations of the lattice of intensions of M-predicates.

Table 4

Table 5

4. Let us construct an analytical table for (f → e) & (e → g) and obtain its disjunctive normal form \(\varphi \). We reduce \(\varphi \) to the perfect disjunctive normal form \(\psi \).

The branch ϴ₃ is closed \((*)\).

\(\varphi = \neg e\neg f \vee \neg fg \vee eg,\) \(\varphi \leftrightarrow \psi \), where \(\psi = {{F}_{3}}(b,f,e,g)\) = \(bfeg \vee b\neg feg \vee b\neg f\neg eg \vee \) \(b\neg f\neg e\neg g \vee bfeg \vee b\neg feg\) \( \vee \,\,b\neg f\neg feg \vee \neg b\neg f\neg eg \vee \) \(b\neg f\neg e\neg g.\)

5. Relational system \(\bar {R}_{{}}^{{''}}\).

Thus, we obtain \({{\bar {D}}_{1}} = D_{1}^{ + } \cup D_{1}^{ - } \cup \) \(D_{1}^{\tau } \cup \tilde {D}_{1}^{ + } \cup \) \(\tilde {D}_{1}^{ - } \cup \tilde {D}_{1}^{\tau },\) \({{\bar {D}}_{2}} = \tilde {D}_{2}^{\tau } \cup \tilde {D}_{2}^{ + } \cup \tilde {D}_{2}^{ - }\), where \(V{{ \Rightarrow }_{{2(x,y)}}}Y\) is generated for x = (a₁₂fgb)⁺ and y = (a₁₂ fgb)^–. It can be shown that \(\bar {\Re }{\kern 1pt} {''}\left| = \right.M_{{{{a}_{{12}}}fgb,0}}^{ + }\left( {AB,a} \right)\), \(\bar {\Re }{\kern 1pt} {''}\) ⊨ \(M_{{{{a}_{{12}}}fgb,0}}^{ + }\left( {DG,a} \right)\), \(\bar {\Re }{\kern 1pt} {''}\) ⊨ \(M_{{{{a}_{{12}}}fgb,0}}^{ - }\left( {RT,a} \right)\), \(\bar {\Re }{\kern 1pt} {''}\) ⊨ \(\left( {\left( {{{J}_{{\left( {\tau ,0} \right)}}}\left( {AB{{ \Rightarrow }_{2}}a} \right)} \right.} \right.\) & \(M_{{{{a}_{{12}}}fgb,0}}^{ + }\left( {AB,a} \right)\) & ¬\(\left. {M_{{{{a}_{{12}}}fgb,0}}^{ - }\left( {AB,a} \right)} \right)\) \( \to \) \(\left. {{{J}_{{\left\langle {1,1} \right\rangle }}}\left( {AB{{ \Rightarrow }_{2}}a} \right)} \right)\), where the predicate \(V{{ \Rightarrow }_{{2(x,y)}}}Y\) is generated.

The same holds for the pairs 〈DG, a〉 and 〈RT, a〉, which correspond to \(D_{2}^{\tau }\). We also obtain \(\bar {\Re }{\kern 1pt} {''}\)⊨ \(\left( {\left( {{{J}_{{\left( {\tau ,0} \right)}}}\left( {AB{{E}_{1}}{{ \Rightarrow }_{1}}a} \right)} \right.} \right.\) & \(\left. {\Pi _{1}^{ + }\left( {AB{{E}_{1}},a} \right)} \right) \to \) \(\left. {{{J}_{{\left\langle {1,2} \right\rangle }}}\left( {AB{{E}_{1}}{{ \Rightarrow }_{1}}a} \right)} \right)\).

The same holds for the pairs 〈DGE₂, a〉, 〈RTF, a〉, and 〈MKN, a〉, which correspond to \(D_{1}^{\tau }\).

5. In [1], for each JSM reasoning strategy Str_{x ,y}, the intension of the concept of “empirical regularity” A_E was defined, the elements of which are \(A_{\chi }^{\sigma }\), where \(\sigma \in \left\{ { + , - } \right\}\), and \(\chi \in E\). \(A_{\chi }^{\sigma }\) were strengthened by an added existential condition, which meant replacing \(A_{\chi }^{\sigma }\) with \(\tilde {A}_{\chi }^{\sigma }\). This addition and replacement of \(A_{\chi }^{\sigma }\) with \(\tilde {A}_{\chi }^{\sigma }\) is redundant to determine the intension. However, such an addition of an existential condition is necessary to determine the extension of the concept of “empirical regularity”, the elements of which should be \(\tilde {A}_{\chi }^{\sigma }\left( {C{\kern 1pt} ',Q} \right)\), containing an existential condition. An example of an existential condition is

For each Str_{x, y} we specify the corresponding relational system. Then, we consider \(\bar {\Re }_{{x,y}}^{{''}}\), where x = (a₁₂ fgb)⁺ and y = (a₁₂fgb)^–.

It can be shown that \(\bar {\Re }_{{x,y}}^{{''}}\) ⊨ \(A_{\chi }^{\sigma }\), where \(\sigma \in \left\{ { + , - } \right\}\), \(\chi \in E\), and \(\bar {\Re }_{{x,y}}^{{''}}\) ⊨ \(\tilde {A}_{\chi }^{\sigma }\left( {C{\kern 1pt} ',Q} \right)\), if the existential condition is true.