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 (◻(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.

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 ∀h∃n (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 ∀p∀h∃nJ_〈1,n〉H₂(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 ERA₁ derived rule is M_χ(p → q), Tq ⊢ M_χp is a propositional imitation of abduction of the second kind.

2. In [1, 2], the principle of the modal trace M₁M₂…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 M₁, M₂,…, 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 M₁, M₂, …, M_k will be called regular if M₁ ⊑ M₂ ⊑ … ⊑ 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: ${{{\mathbf{\tilde {M}}}}_{1}}$(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 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 ERA₁ amplifications by adding the axioms ◻p → ◻◻…◻_kp 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〉H₂(V, Y, p, h) ∨ J_〈−1,n〉H₂(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 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 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.

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

2⁰. Ω^τ(0, 1), Ω^τ(0, 1) = Ω^τ(p, h) for all p and h, where 0 ≤ p ≤ s, 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|(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 〈(ad₀b)⁺, ¬a^–〉 and 〈¬a⁺, (ad₀b)^–〉 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 big data).