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On the Heuristics of JSM Research (Additions to Articles)

Abstract

The logical means of detecting empirical regularities using the JSM method of automated research support are considered. Generators of hypotheses about the causes and hypotheses about predictions that are stored in sequences of expandable fact bases are determined. Many “histories of possible worlds” are considered, where “world” refers to an expandable fact base. This set is used to determine empirical regularities, that is, empirical laws, tendencies, and weak tendencies. Empirical regularities are used to determine empirical modalities of necessity (for empirical laws), possibilities (for empirical tendencies), and weak possibilities (for weak empirical tendencies). The Propositional calculi of the class ERA are proposed, that is, modal logics with two empirical modalities of necessity and possibility such that they imitate abductive inference through the axioms of abduction (◻(pq) & Tq) → ◻p), (◇(pq) & Tq) → ◇p), where ◻, ◇, T are operators of necessity, possibility, and truth (“it is true that…”). A series of definitions related to the characterization of data mining using heuristics of the JSM method of automated research support is given.

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Fig. 5.

Notes

  1. 1.

    D.V. Vinogradov in [9] established that, for finite models, JSM rules are expressible in the predicate logic of the first order.

  2. 2.

    ⇌ is equality by definition.

  3. 3.

    We note that there are many attempts to formalize the ideas of C.S. Peirce on abduction by means of logic and programming using deduction [2022].

  4. 4.

    \({{\bar {\rho }}^{\sigma }} \leqslant 1\), in recognition problems often get \({{\bar {\rho }}^{\sigma }}\) = 0.8.

  5. 5.

    We can assume that CCA(σ) is the principle of induction (J.S. Mill in [14] considered the law of uniformity of nature to be such).

  6. 6.

    (τ, 1) and (τ, 2) are sets of truth values.

  7. 7.

    According to the terminology of I. Kant in “Critique of Pure Reason” [32], ICF are the conditions of “possible experience”.

  8. 8.

    In [3], Int and Ext were considered for the initial predicates of the JSM method and the plausible inference rules.

  9. 9.

    For simplicity, we will use the number i instead of \(HP{{W}_{i}}\).

  10. 10.

    In [35], a description is given of an intelligent system that implements the ASSR JSM method for gastroenterology data. This computer system has 16 JSM strategies.

  11. 11.

    Conditions a and ad0 formalize inductive canons of similarity and difference [14]. The canons of similarity-differences are formalized in [13, 36].

  12. 12.

    An interpretation of [15] is available in [18], where the condition of the best explanation is added.

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Correspondence to V. K. Finn.

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Translated by S. Avodkova

[1, 2].

APPENDIX

APPENDIX

1. Abduction of the second kind can be formulated in the following equivalent way using parcels (1), (2) and (3):

(1) \(A_{\chi }^{\sigma }\)(C ʹ, Q), where χ ∈ E;

(2) Mχ ∀Z ∀p ∀hn (J〈1,n〉H2(C ʹ, Q, p, h) → J〈1,n+1〉H1(Z, Q, p, h));

(3) ∀Z((C ' ⊂ Z) → Ver[\({{J}_{{\langle 1,\bar {n} + 1\rangle }}}\)H1(Z, Q, \(\bar {s},\bar {h}\))] = t);

(4)Mχ ∀phnJ〈1,nH2(C ', Q, p, h),

where Mχ is ◻χ, ◇χ and ∇χ, and χ ∈ E = {a, b, …, m, n}.

We note that parcels (1) and (2) have truth values according to the coherent theory of truth, and parcel (3) uses the correspondent theory of truth. Therefore, corollary (4) is obtained according to the interaction of two theories of truth.

We also note that (2) is a consequence of (1), and the ERA1 derived rule is Mχ(pq), TqMχp is a propositional imitation of abduction of the second kind.

2. In [1, 2], the principle of the modal trace M1M2Mk was formulated, generated by the continuation of the sequence of nested FB(p) and the formation of the corresponding sequence of histories of possible worlds \({{\overline {HPW} }_{1}}\), \({{\overline {HPW} }_{2}},...,{{\overline {HPW} }_{k}},\) which correspond to modalities M1, M2,…, Mk.

Since the modal operators Mχ corresponding to the Tree T and the set of integral causal forcings \(\overline {ICF} \) are partially ordered, then the sequence M1, M2, …, Mk will be called regular if M1M2 ⊑ … ⊑ Mk – 1Mk.

The sequences of Mχ-operators corresponding to Strx,y will be denoted by \({\mathbf{\tilde {M}}}\)(x, y). Obviously, the set of all \({\mathbf{\tilde {M}}}\)(x, y), corresponding to the set \(\overline {Str} \) of all strategies of JSM reasoning Strx,y [13], can be ordered as follows: \({{{\mathbf{\tilde {M}}}}_{1}}\)(x1, y1) ⊒ \({{{\mathbf{\tilde {M}}}}_{2}}\)(x2, y2), if and only if \(M_{i}^{{(1)}}\)\(M_{i}^{{(2)}}\) for i = 1, …, k and 〈x1, y1〉 ≥ 〈x2, y2〉 [13], where \(M_{i}^{{(1)}}\) and \(M_{i}^{{(2)}}\) are modal sequence operators \({{{\mathbf{\tilde {M}}}}_{1}}\)(x1, y1) and \({{{\mathbf{\tilde {M}}}}_{2}}\)(x2, y2), respectively.

Let M be the set of all sequences of Mχ-operators; then in M there exist the largest and the smallest elements.

We now state the principle of a successful modal trace:

the modal trace is successful for k-histories of possible worlds HPW that are sequentially expandable and generate \(\overline {HPW} \), if there is a strategy of JSM reasoning Strx,y such that the corresponding sequence \({\mathbf{\tilde {M}}}\)(x, y) obtained by an acceptable JSM research according to the definition Df.20-4.

A propositional imitation of a successful JSM research is the nonfinite S4 and S5 similar ERA1 amplifications by adding the axioms ◻p → ◻◻…◻kp and ◇p → ◻◻…◻kp for all k that correspond to regular Cd codes of empirical regularities.

3. We now state the conditions for an ideal JSM research.

(1) There exists Strx,y such that the condition holds: if Ω(p) ⊆ Ω(q), then \({{\bar {O}}_{{x,y}}}\)(Ω(p)) ⊆ \({{\bar {O}}_{{x,y}}}\)(Ω(q)). Then, the JSM operator \({{\bar {O}}_{{x,y}}}\)(Ω(p)) is a closure.

(2) For Strx,y, satisfying Condition (1), the following statement holds: for any 〈V, Y〉 and all p, h if J〈1,n〉H2(V, Y, p, h) ∨ J〈−1,nH2(V, Y, p, h) holds, then 〈VY〉 ∈ Gx,y = (\(\bigcup\nolimits_{\chi \in E} \{ \)V, Y〉|\(A_{\chi }^{ + }\)(V, Y)}) ∪ (\(\bigcup\nolimits_{\chi \in E} \{ \)V, Y〉|\(A_{\chi }^{ - }\)(V, Y)}), where\(A_{\chi }^{\sigma }\)(C ', Q) are realization ICF for 〈C ', Q〉, χ ∈ E, and ¬(Gx,y = Λ).

(3) For Strx,y, satisfying Condition (1), the causal completeness axioms CCA(σ) are true, where σ ∈+, – [6].

(4) \({{\bar {O}}_{{x,y}}}\)(Ω(s))| ≥ |Ωτ(0)| and m0= l0, where s is the number of the last expansion of FB(p), m0 = |Ωτ(0)|, and l0 is the number of correct predictions of the studied effect Q.

(5) For Strx,y satisfying Condition (1), there exists a successful sequence \({\mathbf{\tilde {M}}}\)(x, y) such that k ≥ 3 (k successful \({\mathbf{\tilde {M}}}\)(x, y)).

(6) Complete JSM research for all Strx,y from a given set \(\overline {Str} \) is characterized by the following scheme.

10. Ω(0, 1), Ω(1, 1), …, Ω(s, 1); Ω(0, 1) ⊂ Ω(1, 1) ⊂ … ⊂ Ω(s, 1),

20. Ωτ(0, 1), Ωτ(0, 1) = Ωτ(p, h) for all p and h, where 0 ≤ ps, h\(\overline {HPW} \);

30. \(\overline {HPW} \), |\(\overline {HPW} \)| = (s + 1)!;

40. \(\overline {Str} \),

50. \(\overline {ICF} \),

60. \(\overline {{{6}^{0}}.\,\Im ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \),

where \(\Im \) = {[\(\Im \)x,y]E|(xI+) & (yI)}, [\(\Im \)x,y]E = 〈Σ∪ ΣE, \(\tilde {\Omega }\)x,y(\(\bar {s}\), (\(\bar {s}\) + 1)!) ∪ \(\tilde {\Delta }\)(\(\bar {s}\),(\(\bar {s}\) + 1)!, R)〉, ΣE, many of all \(A_{\chi }^{\sigma }\)(C ', Q), corresponding to Gx,y, where χ ∈ E.

We suppose that there exists a QAT such that for Strx,y Conditions (1)–(6) are satisfied.

The following condition holds: \(\Im \) belong to \({{\left[ {{{\Im }_{{\left\langle {{{{(a{{d}_{0}}b)}}^{ + }},\neg {{a}^{ - }}} \right\rangle }}}} \right]}_{E}}\) and \({{\left[ {{{\Im }_{{\left\langle {\neg {{a}^{ + }},{{{(a{{d}_{0}}b)}}^{ - }}} \right\rangle }}}} \right]}_{E}}\), where 〈(ad0b)+, ¬a〉 and 〈¬a+, (ad0b)〉 are the largest elements of direct products of lattices Int(L+x¬L) and IntL+xL) for inductive inference rules (I+) and (I), respectively [13]. Wherein \(A_{a}^{ + }\left( {C_{1}^{'},{{Q}_{1}}} \right)\) and \(A_{a}^{ - }\left( {C_{2}^{'},{{Q}_{2}}} \right)\) correspond to \({{\Im }_{{\left\langle {{{{(a{{d}_{0}}b)}}^{ + }},\neg {{a}^{ - }}} \right\rangle }}}\) and \({{\Im }_{{\left\langle {\neg {{a}^{ + }},{{{(a{{d}_{0}}b)}}^{ - }}} \right\rangle }}}\), where a is the index \(A_{a}^{ + }\) and \(A_{a}^{ - }\) is the largest element of the partially ordered sets E+ and E, respectively, where E = E+Е.

Conditions (1)–(6) have various attenuations that characterize real JSM research, which correspond to ExtER and a specific forest generated by this complete JSM research for \(\overline {Str} \).

It is important to note that ΣE contains empirical nomological statements (ENS) of three types of modalities ◻χ, ◇χ and ∇χ, which in a sense expresses the degree of nomology while maintaining universality using quantifiers ∀Zp. ENS express the knowledge discovery, which is the goal of data mining as a means of research support and the formation of open theories (by virtue of this, open data is more important than big data).

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Finn, V.K. On the Heuristics of JSM Research (Additions to Articles). Autom. Doc. Math. Linguist. 53, 250–282 (2019) doi:10.3103/S0005105519050078

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Keywords:

  • JSM reasoning
  • JSM research
  • JSM method of automated research support
  • induction
  • analogy
  • abduction
  • empirical regularity
  • modalities of necessity
  • possibility and weak possibility
  • nomological statements