### 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 (◻(*p* → *q*) & *Tq*) → ◻*p*), (◇(*p* → *q*) & *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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## Notes

- 1.
D.V. Vinogradov in [9] established that, for finite models, JSM rules are expressible in the predicate logic of the first order.

- 2.
⇌ is equality by definition.

- 3.
- 4.
\({{\bar {\rho }}^{\sigma }} \leqslant 1\), in recognition problems often get \({{\bar {\rho }}^{\sigma }}\) = 0.8.

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

- 7.
According to the terminology of I. Kant in “Critique of Pure Reason” [32], ICF are the conditions of “possible experience”.

- 8.
In [3],

*Int*and*Ext*were considered for the initial predicates of the*JSM*method and the plausible inference rules. - 9.
For simplicity, we will use the number

*i*instead of \(HP{{W}_{i}}\). - 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.
- 12.

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## Additional information

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) *M*χ ∀*Z* ∀*p* ∀*h*∃*n* (*J*_{〈1,n〉}H_{2}(*C* ʹ, *Q*, *p*, *h*)
→ *J*_{〈1,n+1〉}*H*_{1}(*Z*, *Q*, *p*, *h*));

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

(4)*M*χ ∀*p*∀*h*∃*n**J*_{〈1,n〉}*H*_{2}(*C* ', *Q*, *p*, *h*),

where *M*_{χ} 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 *ERA*_{1} derived rule is *M*_{χ}(*p* → *q*), *Tq* ⊢ *M*_{χ}*p* is a propositional imitation of abduction of the second kind.

**2.** In [1, 2], the principle of the modal trace *M*_{1}*M*_{2}…*M*_{k} 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 *M*_{1}, *M*_{2},…, *M*_{k}.

Since the modal operators *M*_{χ} corresponding to the Tree *T* and the set of integral causal forcings \(\overline {ICF} \) are partially ordered, then the sequence *M*_{1}, *M*_{2}, …, *M*_{k} will be called regular if *M*_{1} ⊑ *M*_{2} ⊑ … ⊑ *M*_{k – 1} ⊑ *M*_{k}.

The sequences of *M*_{χ}-operators corresponding to *Str*_{x,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 *Str*_{x,y} [13], can be ordered as follows: \({{{\mathbf{\tilde {M}}}}_{1}}\)(*x*_{1}, *y*_{1}) ⊒ \({{{\mathbf{\tilde {M}}}}_{2}}\)(*x*_{2}, *y*_{2}), if and only if \(M_{i}^{{(1)}}\) ⊒ \(M_{i}^{{(2)}}\) for *i* = 1, …, *k* and 〈*x*_{1}, *y*_{1}〉 ≥ 〈*x*_{2}, *y*_{2}〉 [13], where \(M_{i}^{{(1)}}\) and \(M_{i}^{{(2)}}\) are modal sequence operators \({{{\mathbf{\tilde {M}}}}_{1}}\)(*x*_{1}, *y*_{1}) and \({{{\mathbf{\tilde {M}}}}_{2}}\)(*x*_{2}, *y*_{2}), respectively.

Let *M* be the set of all sequences of *M*_{χ}-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 *Str*_{x,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 *ERA*_{1} amplifications by adding the axioms ◻*p* → ◻◻…◻_{k}*p* and ◇*p* → ◻◻…◻_{k}◇*p* 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 *Str*_{x,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 *Str*_{x,y}, satisfying Condition (1), the following statement holds: for any 〈*V*, *Y*〉 and all *p*, *h* if *J*_{〈1,n〉}*H*_{2}(*V*, *Y*, *p*, *h*) ∨ *J*_{〈−1,n〉}*H*_{2}(*V*, *Y*, *p*, *h*) holds, then 〈*V*, *Y*〉 ∈ G_{x,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 ¬(G_{x,y} = Λ).

(3) For *Str*_{x,y}, satisfying Condition (1), the causal completeness axioms CCA^{(σ)} are true, where σ ∈+, – [6].

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

(5) For *Str*_{x,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 *Str*_{x,y} from a given set \(\overline {Str} \) is characterized by the following scheme.

1^{0}. Ω(0, 1), Ω(1, 1), …, Ω(*s*, 1); Ω(0, 1) ⊂ Ω(1, 1) ⊂ … ⊂ Ω(*s*, 1),

2^{0}. Ω^{τ}(0, 1), Ω^{τ}(0, 1) = Ω^{τ}(*p*, *h*) for all *p* and *h*, where 0 ≤ *p* ≤ *s*, *h* ∈ \(\overline {HPW} \);

3^{0}. \(\overline {HPW} \), |\(\overline {HPW} \)| = (*s* + 1)!;

4^{0}. \(\overline {Str} \),

5^{0}. \(\overline {ICF} \),

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

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

We suppose that there exists a QAT such that for *Str*_{x,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 〈(*ad*_{0}*b*)^{+}, ¬*a*^{–}〉 and 〈¬*a*^{+}, (*ad*_{0}*b*)^{–}〉 are the largest elements of direct products of lattices *Int*(*L*^{+}*x*¬*L*^{–}) and *Int*(¬*L*^{+}*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 Σ_{E} 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 ∀*Z*∀*p*. *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

**).**

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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