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.

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

0	a	ag	ab	abg	aeg	abeg	afg	abfg
a	a	a	a	a	a	a	a	a
ag	a	ag	a	ag	ag	ag	ag	ag
ab	a	a	ab	ab	a	ab	a	ab
abg	a	ag	ab	abg	ag	abg	ag	abg
aeg	a	ag	a	ag	aeg	aeg	aeg	aeg
abeg	a	ag	ab	abg	aeg	abeg	aeg	abeg
afg	a	ag	a	ag	aeg	aeg	afg	afg
abfg	a	ag	ab	abg	aeg	abeg	afg	abfg

Table 5

$\wedge$	a	ag	ab	abg	aeg	abeg	afg	abfg
a	a	ag	ab	abg	aeg	abeg	afg	abfg
ag	ag	ag	abg	abg	aeg	abeg	afg	abfg
ab	ab	abg	ab	abg	abeg	abeg	abfg	abfg
abg	abg	abg	abg	abg	abeg	abeg	abfg	abfg
aeg	aeg	aeg	abeg	abeg	aeg	abeg	afg	abfg
abeg	abeg	abeg	abeg	abeg	abeg	abeg	abfg	abfg
afg	afg	afg	abfg	abfg	afg	abfg	afg	abfg
abfg	abfg	abfg	abfg	abfg	abfg	abfg	abfg	abfg

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 $\psi$.

$$\begin{gathered} f \to e \\ e \to g \\ \begin{array}{*{20}{c}} {}&{\neg f}&| & e&{} \\ {\neg \left. e \right|}&g&| & {\neg \left. e \right|}&g \end{array} \\ \begin{array}{*{20}{c}} {{{\Theta }_{1}}}&{{{\Theta }_{2}}}&{\Theta _{3}^{*}}&{{{\Theta }_{4}}} \end{array} \\ \end{gathered} $$

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 $\bar {R}_{{}}^{{''}}$.

$$\left. \begin{gathered} {{J}_{{\left\langle {1,0} \right\rangle }}}\left( {AB{{C}_{1}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle {1,0} \right\rangle }}}\left( {AB{{C}_{2}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle {1,0} \right\rangle }}}\left( {DG{{C}_{3}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle {1,0} \right\rangle }}}\left( {DG{{C}_{4}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle {1,0} \right\rangle }}}\left( {{{N}_{1}}{{L}_{1}}{{S}_{1}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle {1,0} \right\rangle }}}\left( {{{N}_{2}}{{L}_{2}}{{S}_{2}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ \end{gathered} \right\}D_{1}^{ + }$$

$$\left. \begin{gathered} {{J}_{{\left\langle { - 1,0} \right\rangle }}}\left( {{{P}_{1}}{{Q}_{1}}{{C}_{1}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle { - 1,0} \right\rangle }}}\left( {{{P}_{2}}{{Q}_{2}}{{C}_{2}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle { - 1,0} \right\rangle }}}\left( {{{K}_{1}}{{L}_{1}}{{C}_{3}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle { - 1,0} \right\rangle }}}\left( {{{K}_{2}}{{L}_{2}}{{C}_{4}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle { - 1,0} \right\rangle }}}\left( {RT{{S}_{1}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle { - 1,0} \right\rangle }}}\left( {RT{{S}_{2}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ \end{gathered} \right\}D_{1}^{ - }$$

$$\left. \begin{gathered} {{J}_{{\left\langle {\tau ,0} \right\rangle }}}(AB{{ \Rightarrow }_{2}}a) \hfill \\ {{J}_{{\left\langle {\tau ,0} \right\rangle }}}(DG{{ \Rightarrow }_{2}}a) \hfill \\ {{J}_{{\left\langle {\tau ,0} \right\rangle }}}(RT{{ \Rightarrow }_{2}}a) \hfill \\ \end{gathered} \right\}D_{2}^{\tau }$$

$$\begin{gathered} \left. \begin{gathered} {{J}_{{\left\langle {\tau ,0} \right\rangle }}}\left( {AB{{E}_{1}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle {\tau ,0} \right\rangle }}}\left( {DG{{E}_{2}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle {\tau ,0} \right\rangle }}}\left( {RTF{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle {\tau ,0} \right\rangle }}}\left( {MKN{{ \Rightarrow }_{1}}a} \right) \hfill \\ \end{gathered} \right\}D_{1}^{\tau } \hfill \\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \hfill \\ \end{gathered} $$

$$\left. \begin{gathered} {{J}_{{\left\langle {1,1} \right\rangle }}}\left( {AB{{ \Rightarrow }_{2}}a} \right) \hfill \\ {{J}_{{\left\langle {1,1} \right\rangle }}}\left( {DG{{ \Rightarrow }_{2}}a} \right) \hfill \\ \end{gathered} \right\}\tilde {D}_{2}^{ + }$$

$$\begin{gathered} \left. {{{J}_{{\left\langle { - 1,1} \right\rangle }}}\left( {RT{{ \Rightarrow }_{2}}a} \right)} \right\}\tilde {D}_{2}^{ - } \hfill \\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \hfill \\ \end{gathered} $$

$$\left. \begin{gathered} {{J}_{{\left\langle {1,2} \right\rangle }}}\left( {AB{{E}_{1}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ {{J}_{{\left\langle {1,2} \right\rangle }}}\left( {DG{{E}_{2}}{{ \Rightarrow }_{1}}a} \right) \hfill \\ \end{gathered} \right\}\tilde {D}_{1}^{ + }$$

$$\left. {{{J}_{{\left\langle { - 1,2} \right\rangle }}}\left( {RTF{{ \Rightarrow }_{1}}a} \right)} \right\}\tilde {D}_{1}^{ - }$$

$$\left. {{{J}_{{\left( {\tau ,2} \right)}}}\left( {MKN{{ \Rightarrow }_{1}}a} \right)} \right\}\tilde {D}_{1}^{\tau }$$

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.