<p>There are two possible approaches to treating problems in the history of philosophy: (a) understanding and interpreting texts and the problems contained therein (hermeneutic approach), (b) transforming philosophic ideas into concepts by means of modern scientific language and of the emerging research problems. In this work we use the second approach. However, we should elaborate on the term “idea” in the title of the lecture: the term “idea” is not used in Kant’s sense as a tool of the mind; it refers to a proto-concept, with unformed knowledge but definite intent. Exact epistemology (ExEp) is scientific reflection that offers logical tools for studying cultural and civilizational problems of information society while refining and developing the ideas of epistemology in connection with solving the problems of artificial intelligence (AI). It is important to note that ExEp refines the ideas of the classical epistemological triad: object (O) – cognitive process (CP₁) – cognizing subject (S).</p>

However, the refining and developing makes essential use of Immanuel Kant’s epistemological theory that contains the original triad of “sensible intuition (<em>sinnlicher</em><em> </em><em>Anschauung</em>) – understanding (<em>Verstand</em>) – mind (<em>Gemüt</em>)”, the priority of cognitive faculties while addressing philosophical problems (transcendency), achieving necessary and universal knowledge by means of <em>a priori</em> concepts that are used in synthetic <em>a priori</em> and <em>a posteriori</em> judgements, establishing indirect connections between a phenomenon and knowledge about it that is obtained using transcendental schemata, and the distinction between general logic (logic of understanding) and transcendental logic (logic of mind).

ExEp is a conceptual framework for creating a transcendental basis for the heuristics of artificial intelligence (AI) by offering a methodology for the intellectual process of obtaining new knowledge in computer systems.

Let us enumerate the principles of ExEp: Pr1-Pr5.

Pr1. ExEp considers the priority of <strong>theoretic intelligence</strong>, its aspects being understanding and mind; sequences of evolvable fact bases of intelligent systems (IS) correspond to “sensible intuition” (in Kant’s sense).

Pr2. The tool of ExEp is synthesis of cognitive procedures: induction, analogy, abduction with consequent deduction, which involves solving the problem of induction (“D. Hume’s problem”) for intelligent systems by generating sequences of empirical modalities defined by means of empirical nomological statements (as a result of knowledge discovery) and by means of control over monotonic nondecrease of these ordered sequences. The nondecrease is evidence of discovery of empirical regularities (synthetic <em>a posteriori</em> judgements).

Pr3. ExEp uses two theories of truth: the <strong>correspondence</strong> theory of truth (Aristotelian) and the <strong>coherence</strong> theory of truth, which establishes correlation and consistency of new knowledge with respect to the knowledge that already exists. The coherence theory of truth is needed when evaluating hypotheses and explaining the nature of applications of <em>a priori</em> knowledge as rationally well-founded (this is, basically, I. Kant’s answer to D. Hume’s sceptical challenge). The coherence theory of truth is required as a tool of meta-language for expressing reflection and as a semantic tool for formalizing C.S. Peirce’s abduction[1].

Pr4. ExEp provides logical foundation for <strong>transcendental logic</strong>, which is the logic of <strong>synthesis of cognitive procedures</strong> (induction, analogy and abduction), as it combines the logic of reasoning (an analogue of I. Kant’s general logic) and the logic of research (analogue of I. Kant’s transcendental logic that generates new knowledge on the object of research). <strong>Logic of reasoning</strong> contains the four-valued logic of argumentation A4, and transcendental logic uses the hierarchy of formal languages: J-logic of knowledge representation for formalizing induction and analogy, ML language for formalizing empirical regularities discovery and abduction, MML-language for representation of an ordered set of empirical regularities, the intention for which consists of synthetic <strong>a priori</strong> judgements [5].

Pr5. ExEp considers knowledge growth according to K.R. Popper’s [6] ideas on evolutionary epistemology P1-TT-EE-EMR-P2, where P1 is a problem, TT is an open (quasi-axiomatic) theory with rules of plausible inference including induction, analogy and abduction, EE is error elimination for TT, EMR is a set of empirical regularities, and P2 is a new problem. Let us note that unlike K.R. Popper’s theory ExEp is epistemology of the cognizing subject, which can be an intelligent system such that it is a <strong>man-machine partnership</strong> system implemented in two modes: <strong>automated</strong> (imitation of understanding) and <strong>interactive</strong> (support of mind) [5].

The principles Pr1-Pr5 of ExEp are detailed in its postulates P1-P6.

P1. We assume K.R. Popper’s three worlds distinction: 1. W1 (physical world), W2 (the world of mental states), W3 (the world of objective knowledge, autonomous world); ExEp details the structure of world W1; W1 consists of three types of worlds (ontologies): these are W1-1 (the world of random events), W1-2 (the world of directed influences, both positive and negative, <em>i.e.</em> the world of such events that their effects are due to causes in the broad sense); W1-3 (the world with conditioned events and random factors), a combination of W1-1 and W1-2.

<p>ExEp characterizes the nature of W1-2 and W1-3, but primarily W1-2, since ExEp solves the induction problem for its implementation in intelligent systems that are based on knowledge <strong>of conditions of a possible experience</strong> (in the sense of I. Kant’s <em>Critique of Pure Reason</em> [2]). P2. Cognizing subject S performs the intellectual process (cognitive faculties in the sense of I. Kant) by means of faculties (1)-(13) (IP):&nbsp;</p> <p>(1) discovery of the essential in the data (K. Jaspers’ idea when defining intelligence in his&nbsp;<em>General Psychopathology</em>&nbsp;[3]), (2)^*&nbsp;generating the “goal-plan-action” sequence (Miller G.A., Galanter E., Pribram K.H. [4]), (3) selecting premises relevant to the goal of reasoning, (4) reasoning: inferring consequences from the premises while applying the rules of ampliative inference generation synthetic&nbsp;<em>a posteriori</em>&nbsp;judgements by means&nbsp;<em>a priori</em>&nbsp;stated rules, (5) synthesis and interaction of cognitive procedures while applying the rules of ampliative inference to implement induction, analogy and abduction, followed by application of deduction, (6)^*&nbsp;reflection: evaluation of knowledge and actions, (7) explanation: answering the question “why?” (implementation of explanation is possible by applying abduction that uses two theories of truth, the correspondence theory and the coherence theory), (8) argumentation for decision making, formalized by means of four-valued logic (four combinations of&nbsp;<em>pro</em>&nbsp;and&nbsp;<em>contra</em>&nbsp;arguments), (9) cognitive curiosity and image recognition, (10) learning and the use of memory, (11)^*&nbsp;knowledge integration for theory creation, (12)^*&nbsp;clarifying vague ideas: transforming them into concepts, (13)^*&nbsp;changes in the system of knowledge when new knowledge is obtained or cognitive situation is changed.</p>

The following terms are essential for the conceptual framework of ExEp. 1. <strong>Reasoning</strong> is a sequence of statements such that each element of the sequence is a premise or a consequence received by applying some rules of inference to the previous elements of the sequence, with the rules including rules of ampliative inference; besides, there is a procedure of control of the consequences and a condition for acceptance of the final result of reasoning.

<strong>Thought process (TP):</strong> intention (questions, imperatives, goals and attitudes), choice of premises relevant to the goal of the reasoning, reasoning, reflection.

<strong>Cognitive process (CP):</strong> data (fact) analysis, prediction (generation of hypotheses), generation of empirical regularities (EMR), explanation and acceptance of the results.

<strong>Intellectual process (IP):</strong> interaction between TP (world W2) and CP (world W3) for expanding W3. IP implements intellectual faculties (1)-(13), where TP={(1), (3), (4), (5), (7)-(10)}, CP={(2)^*, (6)^*, (11)^*, (12)^*, (13)^*}.

TP implements <strong>procedural</strong> and <strong>declarative</strong> knowledge that represent <strong>undestanding</strong>; CP implements <strong>conceptual</strong> knowledge that represents <strong>mind</strong>.

Conceptual knowledge characterizes the idea of mind as organizing production of well-founded and necessary knowledge, which means [5]:

1. generating the principles of knowledge production, 2. creating the means of knowledge evaluation for reflection (faculty (6)^*), 3. generating ideas and transforming them into concepts (faculty (12)^*), 4. management of understanding (i.e., faculties (3) and (4): finding premises and reasoning, respectively), 5. integrating and correcting knowledge, i.e. EE according to [6] (faculties (11)^* and (13)^*, respectively), 6. applying the means of leaving the hermeneutic circle [7], which implies the choice of undefined concepts and the corresponding protocol statements [8], 7. conceptual knowledge is contained in the autonomous world W3 of objective knowledge [6], 8. conceptual knowledge is a meta-logical tool of <strong>meta-theoretic</strong> scrutiny of procedural and declarative knowledge, which represent the subject field and the reasoning of mind thereon.

Let us now define the main concept of ExEp: <strong>theoretic (natural) intelligence</strong> (TI): Df.1. TI is a system of knowledge, the set of intellectual faculties (1)-(13) constituting the intellectual process (IP), and the higher mental functions (HMF), where the HMF are intention, intuition, initiative, imagination and reflection (HMF are included in a person’s subjective world (PSW)).

It follows from Df.1 that <strong>understanding</strong> and <strong>mind</strong> are aspects of <strong>intelligence</strong>, thus TI is the <strong>primary</strong> concept of epistemology. The functioning of TI is implemented by means of HMF and the cognizing subject’s system of knowledge, while the constructive characterization of the subject results in determining its automated representation: an intelligent system.

Df.2. <strong>Intelligent system (IS)</strong> is a computer system such that it has the architecture defined below and in automated and interactive modes implements the <strong>intellectual process</strong> represented by intellectual faculties (1)-(13): IS=(FB, KB) – problem solver – comfortable interface, where BF means fact base, KB means knowledge base, and the Problem Solver has three modules: a Reasoner, a Calculator and a Synthesizer [6].

Important consequences of Df.1 and Df.2: faculties (1), (3)-(5), (7)-(10) that correspond to <strong>understanding</strong> can be implemented in <strong>automated mode</strong>, while faculties (2)*, (6)*, (11)*-(13)* correspond to <strong>mind</strong> and can be implemented in <strong>interactive mode</strong>; and IS is a <strong>man-machine partnership</strong> system representing an automated cognitive subject of exact epistemology, which is a <strong>technological</strong> implementation of the transcendental principle in Kant’s sense. Furthermore, Df.1 implies that it is impossible to create a <strong>supermind</strong> which generates the principle of singularity.

The next postulate of ExEp is P3.

P3. Reasoning by means of which new knowledge emerges is a <strong>synthesis of cognitive procedures</strong> that uses ampliative inferences by means of <em>a priori</em> synthetic inference rules.

Implementation of such a synthesis involves interaction of induction, analogy and abduction.

An ideal of rationality achievable by means of TI is <strong>research</strong> which consists in solving problems by means of method expressed in a language possessing the <strong>descriptive</strong> and the <strong>argumentative</strong> functions (K.R. Popper [6]): applying the method generates <strong>empirical regularities</strong>.

P4. Imitation and enhancement of research is implemented by means of:

1. a sequence of expanding bodies of facts (fact bases) to which reasoning are applied, 2. discovering empirical regularities, 3. supplementation of open theories (QAT, quasi-axiomatic theories [5]) by means of empirical nomological statements (ENS).

P5. The cognitive process (CP) of ENS generation is implemented by means of interaction between the <strong>correspondence</strong> theory of truth [6] and the <strong>coherence</strong> theory of truth [9].

P6. Research results are accepted by means of <strong>non-singular</strong> evaluation of the quality of reasoning and the quality of generated hypotheses while using two corresponding scales [5].

P7. The modal trace postulate, which is a means of control for <strong>prolonged </strong>research: research should be made by applying reasoning to sequences of s+1 fact bases FB_j(0),…, FB_j(s), where j=1,…,s+1, s≥3, in such a way that it can be continued r times, where r≥2; for each of r stages of research there is a corresponding <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>M</mi></mrow><mrow><mi>χ</mi></mrow></msub></math> - operator of empirical modalities (necessity, possibility and weak possibility) [5] such that it is generated by empirical nomological statements (ENP).

ENP are synthetic <em>a posteriori</em> judgements that represent the extension of the “empirical regularity” intention and create the concept “empirical regularity” by means of procedural implementation of intention [5].

P7 is the solution for the problem of induction (“D. Hume’s problem”) for IS by use of the tools of ExEp.

P1-P7 comprise the basis for a definition of artificial intelligence (AI).

Df.3. AI is a research and practice field of imitation and enhancement of the <strong>intellectual process</strong> <strong>(IP)</strong> (according to Df.1) and of <strong>rational</strong> human behaviour by means of AI systems, intelligent systems (IS), and AI-robots; these imitations and enhancements in AI implement the ExEp postulates P1-P7, based on which <strong>intelligent data analysis</strong> is performed: creating a model of the subject field, discovery of empirical regularities and making well-founded decisions using the heuristics of obtaining new knowledge [10].

AI systems are computer implementations of various knowledge obtaining procedures, AI-robots are IS, the sensorics module and the mechatronics module, and IS are the main product of AI that implements the <strong>conditions for possible experience</strong> for obtaining knowledge from fact bodies that represent an analogue of Kant’s “sensible intuition” (in AI it is a sequence of fact bases).

<strong>JSM-method for automated support of research (JSM-method for ASR)</strong> is a methodology and a constructive tool for creation of intelligent systems, the main product of AI.

JSM-method for ASR consists of applicability conditions, JSM-reasoning that formalize J.S. Mill’s rules of inductive inference [11] and are a synthesis of induction, analogy and abduction [1]; of representation of knowledge in the form of quasi-axiomatic theories, metatheoretic tools for analysis of reasoning and subject field and for the ordering of cognitive procedures , tools for discovering empirical regularities (EMR), and finally intelligent systems (Df.2) [5, 10].

JSM-method for ASR includes two constructive stages: the stage of JSM-reasoning application and the stage of JSM-investigation for obtaining new knowledge in the form of empirical regularities that are added to the open quasi-axiomatic theories QAT, where , where is the set of axioms being added to (EMR included), is the open set of facts and hypotheses on causes and hypotheses on predictions, and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="fraktur">R</mi></math> is the set of rules of inference containing ampliative inference rules for induction, analogy and abduction.

The goal of JSM-method for ASR is knowledge discovery for open quasi-axiomatic theories (QAT) with empirical regularities (EMR) being added to, where EMR is conservation of cause-effect relations and of predictions for Y effects for objects X, which is representable by predicates <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>V</mi><msub><mrow><mo>⇒</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>Y</mi></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>X</mi><msub><mrow><mo>⇒</mo></mrow><mrow><mn>1</mn></mrow></msub><mi>Y</mi><mo>,</mo></math> respectively: V is the cause of Y and X possesses Y.

Causal relations are created by a set of formalized and modified J.S. Mill’s inductive rules [11], that are a tool of JSM-reasoning created by <strong>synthesis of cognitive procedures</strong> “induction → analogy → abduction” such that the rules of inference (induction and analogy) contain argumentation (<em>pro</em> and <em>contra</em>) and falsification.

The aforementioned synthesis is a <strong>constructive</strong> tool for obtaining new knowledge, which means that the rules of inference are ampliative, generating synthetic <em>a posteriori</em> statements.

JSM-method for ASR formalizes modified J.S. Mill’s rules of inductive inference [11]: similarities, similarities with counterexamples forbidden, similarities-and-differences, differences with counterexamples forbidden, programmed in intelligent systems. These rules are formalized using M-predicates of similarity for positive (+) and negative (-) facts which are predicates and such that they establish similarities on (+) and (-) facts, respectively. Subscripts <em>x</em> and <em>y</em> are names of predicates for representation of the corresponding inference rules formed by four boolean combinations <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>M</mi></mrow><mrow><mi>x</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math> & <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>¬</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>y</mi></mrow><mrow><mo>-</mo></mrow></msubsup><mo>,</mo><mo>¬</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>x</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math> & <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>M</mi></mrow><mrow><mi>y</mi></mrow><mrow><mo>-</mo></mrow></msubsup></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>M</mi></mrow><mrow><mi>x</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math> & <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>M</mi></mrow><mrow><mi>y</mi></mrow><mrow><mo>-</mo></mrow></msubsup><mo>,</mo><mo>¬</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>x</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math> & <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>¬</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>y</mi></mrow><mrow><mo>-</mo></mrow></msubsup></math> such that they correspond to four types of factual truth values “1”, “-1”, “0”, “τ” (respectively: “true”, “false”, “inconsistent”, “uncertain”). These factual truth values belong to the semantics of four-valued argumentation logic A₄ [10]. M-predicates are partially ordered and are formed by addition to the least predicates of M⁺-similarity and M^--similarity of some additional conditions for the inductive inference rules that formalize modified J.S. Mill’s rules [11]. Application of inductive rules of inference of the JSM-method for ASR results in hypotheses on causes of effects in the world W1-2: cause V (positive or negative) of effect Y obtained by applying the inductive inference rules with the name <em>x</em>, <em>y</em> (x is the name of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>M</mi></mrow><mrow><mi>x</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math> , y is the name of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>M</mi></mrow><mrow><mi>y</mi></mrow><mrow><mo>-</mo></mrow></msubsup></math> ). These hypotheses can be represented by the predicate <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>V</mi><msub><mrow><mo>⇒</mo></mrow><mrow><msub><mrow><mn>2</mn></mrow><mrow><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></mrow></msub></mrow></msub><mi>Y</mi><mtext>.</mtext></math>

Furthermore, hypotheses on <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfenced separators="|"><mrow><mo>±</mo></mrow></mfenced></math> -causes are used for inference by analogy that can be represented using predicates <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>П</mi></mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></mfenced></math> , where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>σ</mi><mtext>=+</mtext><mo>,</mo><mo>-</mo><mo>,</mo><mn>0</mn><mo>,</mo><mi>τ</mi><mo>,</mo></math> the application of which generates predictions that can be represented using predicate (object <em>X</em> has effect <em>Y</em>). The premise for the rules of inference by analogy is the following proposition: evaluation for <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>X</mi><msub><mrow><mo>⇒</mo></mrow><mrow><mn>1</mn></mrow></msub><mi>Y</mi></math> is uncertainty (τ).

In intelligent systems (IS) the body of initial data – fact bases (for (+)-facts, (-)-facts, (τ)-facts) — is the analogue of “sensible intuition” in the sense of [2]. The procedure used for <strong>acceptance</strong> of predictions is abduction-1 (abduction of the 1^{<span style="text-decoration: underline;">st</span>} kind): whether each (σ)-fact, where σ = +,-, has a (σ)-cause. There are three types of outcome:

1. each (σ)-fact has a (σ)-cause, <em>i</em><em>. e</em>. an explanation; 2. (σ)-facts of the initial data have a degree of explanation <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>ρ</mi></mrow><mrow><mi>σ</mi></mrow></msup></math> such that it is greater than or equal to an assigned threshold <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mover accent="true"><mrow><mi>ρ</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>σ</mi></mrow></msup><mfenced separators="|"><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>≥</mo><msup><mrow><mover accent="true"><mrow><mi>ρ</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>σ</mi></mrow></msup><mo>,</mo><mn>0</mn><mo>≤</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mi>σ</mi></mrow></msup><mo>≤</mo><mn>1</mn></mrow></mfenced></math> ; 3. <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>ρ</mi></mrow><mrow><mi>σ</mi></mrow></msup></math>

Consecutive application of inductive inference rules, rules of inference by analogy and the abduction procedure implemented using -predicates constitutes a <strong>strategy</strong> of JSM-reasoning Str_x_,_y, the name of which is the pair <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfenced open="⟨" close="⟩" separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> . The set <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover></math> of all possible strategies is partially ordered and offers the corresponding types of causality for the world W1-2 presentable as <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>V</mi><msub><mrow><mo>⇒</mo></mrow><mrow><mn>2</mn><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></mrow></msub><mi>Y</mi><mtext>.</mtext></math>

Let us note that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover></math> determines the <strong>conditions of possible experience</strong> in the sense of [2], implemented by means of JSM-reasoning as a synthesis of cognitive procedures imitating and enhancing the <strong>thought process</strong><strong> </strong>(<strong>TP</strong>). But this is just the first stage of the JSM-method of ASR: the stage of imitating understanding (intellectual process faculties (1), (3), (4), (5), (7), (8)-(10)) by means of a transformation of declarative knowledge of fact bases applying procedural knowledge of inference rules. However, at this stage inaccessibility of thresholds <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mover accent="true"><mrow><mi>ρ</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>σ</mi></mrow></msup></math> is possible, where σ = +,-, which necessitates further JSM-reasoning for expandable sequences of fact bases for achieving a state <em>s</em>, such <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>ρ</mi></mrow><mrow><mi>σ</mi></mrow></msup><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>≥</mo><msup><mrow><mover accent="true"><mrow><mi>ρ</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>σ</mi></mrow></msup></math> , where <em>p</em> = 0,1,…,<em>s</em>.

Dynamic consideration of a sequence of fact bases is relevant not only to achievability of the threshold for abductive acceptance of hypotheses <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>ρ</mi></mrow><mrow><mi>σ</mi></mrow></msup></math> , but also to providing <strong>foundation</strong> for JSM-reasoning contained in JSM-investigation: conservation of hypotheses on causes and hypotheses on predictions in sequences of expandable fact bases.

We consider fact bases of (+)-, (-)-, (τ)- facts such that each fact base FB(p,h) belongs to a history of their expansions, p = 0,1,…,s; 1≤h≤(s+1)!, where FB(p,h) is a possible world, HPW_h is a history of possible worlds <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtext>FB</mtext><mfenced separators="|"><mrow><mn>0</mn><mo>,</mo><mi>h</mi></mrow></mfenced><mo>⊂</mo><mtext>...</mtext><mo>⊂</mo><mtext>FB</mtext><mfenced separators="|"><mrow><mi>s</mi><mo>,</mo><mi>h</mi></mrow></mfenced><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtext>FB</mtext><mfenced separators="|"><mrow><mn>0,1</mn></mrow></mfenced></math> is the initial possible world such that these sequences expand it by adding nonintersecting sets of facts. In order to <strong>minimize the randomness</strong><strong> </strong>of expansion we consider (s+1)! permutations of expansions of FB(0,1). <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>HPW</mtext></mrow><mo>¯</mo></mover></math> is the set of all histories of all possible worlds ([5], Appendix 1).

Thus, we have two sets <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover></math> и <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>HPW</mtext></mrow><mo>¯</mo></mover></math> : the set of all possible strategies of JSM-reasoning and the set of all histories of all possible worlds, respectively, to which strategies of JSM-reasoning Str_x_,_y are applied that characterize the type of causality relation.

In order to formalize the second stage of JSM-method, <strong>JSM-investigation</strong>, we define the predicates of conservation for hypotheses on causes and hypotheses on predictions <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>V</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>h</mi></mrow></mfenced></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>Z</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>h</mi></mrow></mfenced></math> , respectively: <em>V</em> is the cause of effect <em>Y</em> in fact base FB(p,h) with the number <em>р</em> in the history of possible worlds HPW_h and with final bases FB(s,h), where FB(s,1)=…= FB(s,(s+1)!.

For some <em>p</em> and <em>s</em>, and all <em>p</em><em>, </em><em>V</em><em>, </em><em>Y</em><em>, </em><em>Z</em> the predicate of conservation of cause <em>V</em> for effect <em>Y</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>V</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>h</mi></mrow></mfenced></math> implies the predicate of conservation of prediction of effect <em>Y</em><em> </em>for object<em> </em><em>Z</em>. Depending on the behaviour of these predicates on the set <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>HPW</mtext></mrow><mo>¯</mo></mover></math> we get pre-regularities <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></math> , where σ = +,-, and 1≤j≤6 ([5], Appendix 1).

On the set of all histories of possible worlds <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>HPW</mtext></mrow><mo>¯</mo></mover></math> we define logical combinations of pre-regularities <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></math> , where σ = +,-, and 1≤j≤6, such that they constitute a set of <strong>empirical regularities (EMR)</strong> E with the elements <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mo>,</mo><mi>χ</mi><mo>∈</mo><mi>E</mi><mo>,</mo><mi> </mi><mi> </mi></math> <em>E</em> = {a,b,…,m,n}, where a,b,…,m,n are names of EMR. <em>E</em> is a partially ordered set with a greatest element and a smallest element (<em>a</em> и <em>n</em>, respectively). Let us now represent the concept of empirical regularity by a generalization of G. Frege’s triangle [12], where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></math> - is intention of EMR, <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><mi>Q</mi></mrow></mfenced></math> extension of EMR, <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup></math> is the cause, <em>Q</em> is the effect, σ = +,-:

<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>χ</mi></math> is the name of EMR, and Str_x_,_y is a <strong>procedural</strong><strong> </strong>implementation of intention <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></math> such that it generates extension of EMR <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><mi>Q</mi></mrow></mfenced></math> . <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></math> is a synthetic <em>a</em><em> </em><em>priori</em> judgement, and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><mi>Q</mi></mrow></mfenced></math> is a synthetic <em>a</em><em> </em><em>posteriori</em> judgement [2]. Pair <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfenced open="⟨" close="⟩" separators="|"><mrow><msub><mrow><mtext>Str</mtext></mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></mrow></mfenced></math> is a <strong>transcendental schema</strong><strong> </strong>according to [2], and the set of all such possible transcendental schemata can be represented by a Cartesian product <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover><mo>×</mo><mi>E</mi></math> which sets the <strong>conditions of possible experience</strong>, I. Kant’s fundamental idea [2]. <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover><mo>×</mo><mi>E</mi></math> characterizes the possible domain of application results for intelligent systems (cognizing subjects) that implement JSM-method for ASR.

We can regard <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>HPW</mtext></mrow><mo>¯</mo></mover></math> as the set of phenomena, <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfenced open="⟨" close="⟩" separators="|"><mrow><msub><mrow><mtext>Str</mtext></mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></mrow></mfenced></math> is a transcendental schema [2], and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><mi>Q</mi></mrow></mfenced></math> is conceptual knowledge representing EMR. We then get the following triad: <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>HPW</mtext></mrow><mo>¯</mo></mover></math> - <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfenced open="⟨" close="⟩" separators="|"><mrow><msub><mrow><mtext>Str</mtext></mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></mrow></mfenced></math> - <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><mi>Q</mi></mrow></mfenced></math> , where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfenced open="⟨" close="⟩" separators="|"><mrow><msub><mrow><mtext>Str</mtext></mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></mrow></mfenced><mo>∈</mo><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover><mo>×</mo><mi>E</mi></math> .

Let us note that we can represent <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>HPW</mtext></mrow><mo>¯</mo></mover></math> as <strong>declarative</strong><strong> </strong>knowledge, <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover></math> as <strong>procedural</strong> knowledge, and <em>E</em> as <strong>conceptual</strong><strong> </strong>knowledge. Therefore, we have interaction between imitations of “understanding”, applied to “sensible intuition”, and support of “mind” (by means of intention <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></math> which belongs to W3), which manages “understanding”, both implementing Str_x_,_y.

K.R. Popper in [6] expressed some doubts concerning the existence of synthetic <em>a priori</em> judgements. Intentions EMR <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></math> remove those doubts.

We use extensions of EMR <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><mi>Q</mi></mrow></mfenced></math> , where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><mi>Q</mi></math> are constants representing cause and effect, respectively, to define modalities □ <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub></math> (necessity), (possibility), and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mo>∇</mo></mrow><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msub></math> (weak possibility), where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi><mi> </mi></mrow></mfenced></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi><mi> </mi></mrow></mfenced></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>χ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></mfenced></math> . These modalities are generated by JSM-investigation of rank <em>r</em>, where <em>r</em><em>=1</em>, and they are <strong>empirical</strong><strong> </strong>modalities representing in a one-to-one way EMR with intentions <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></math> , where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>χ</mi><mo>∈</mo><mi>E</mi></math> . Due to that the set of empirical modalities <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi>M</mi></mrow><mo>¯</mo></mover></math> is partially ordered with the greatest element □_<em>a</em> and the smallest element <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mo>∇</mo></mrow><mrow><mi>n</mi></mrow></msub></math> .

Let us note that intentions <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></math> are <strong>ideal types</strong><strong> </strong>of EMR in Max Weber’s sense.

Let us suppose that we continue JSM-investigations <em>r</em> times with step <em>s</em> such that <em>r</em><em>>1</em> and they use sequences of nested FB(p,h) and the corresponding histories of possible worlds HPW_h, 1≤h≤(s+1)!; they then generate sequences of empirical modalities <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mtext>...</mtext><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></math> such that they represent prolonged EMR of rank <em>r</em>, where <em>r</em><em>>1</em>, implementing <strong>control </strong> over the consequences of JSM-reasoning. Evidently the set <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mover accent="true"><mrow><mi>M</mi></mrow><mo>¯</mo></mover></mrow><mo>¯</mo></mover></math> of rank <em>r</em> sequences also is partially ordered and represents possible prolonged EMR, some of which are monotonically non-decreasing. The set <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mover accent="true"><mrow><mi>M</mi></mrow><mo>¯</mo></mover></mrow><mo>¯</mo></mover></math> and the partial ordering thereof provide a solution of the induction problem for IS and rank <em>r</em> JSM-investigation, which can be expressed by postulate P7 from the postulates list of ExEp.

Let us recall that JSM-reasoning that generate hypotheses on causes of effects and hypotheses on predictions and their conservation with respect to the set of histories of possible worlds <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>HPW</mtext></mrow><mo>¯</mo></mover></math> are constructively implemented by strategies Str_x_,_yfrom the set <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover></math> . Each Str_x_,_yis concluded by an abductive inference (abduction-2) such that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>M</mi></mrow><mrow><mi>χ</mi></mrow></msub><mfenced separators="|"><mrow><mi>p</mi><mo>→</mo><mi>q</mi></mrow></mfenced><mo>,</mo><mtext>Tq</mtext></math> ⊢ <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mtext>TM</mtext></mrow><mrow><mi>χ</mi></mrow></msub><mi>q</mi></math> , where <em>T</em> is an EMR verification operator, ⊢ is the symbol for the relation of deductibility, <em>р</em> is the premise representing a hypotheses on the cause, <em>q</em> is the consequence, and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>M</mi></mrow><mrow><mi>χ</mi></mrow></msub></math> is the modality (□, ).

Let us note that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>M</mi></mrow><mrow><mi>χ</mi></mrow></msub><mfenced separators="|"><mrow><mi>p</mi><mo>→</mo><mi>q</mi></mrow></mfenced></math> has a <strong>coherent</strong> truth value with respect to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>HPW</mtext></mrow><mo>¯</mo></mover></math> (<em>t</em>,<em> </em><em>f</em> , external truth values: true, false), <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtext>Tq</mtext></math> has a <strong>correspondent</strong> truth value (<em>t</em>_<em>c</em>, <em>f</em>_<em>c</em>, external truth values: true, false) [6]. Then <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mtext>TM</mtext></mrow><mrow><mi>χ</mi></mrow></msub><mi>q</mi></math> has truth value (, , <em /ememf/em, emf/emsubemc/em/sub>), where is <strong>distinguished </strong>truth value (as in multivalued logics). Thus generation of EMR is well-founded by means of solutions of induction and abduction problems using two theories of truth, the coherence theory and the correspondence theory, with the coherence theory of truth being required for the definition of <strong>synthetic </strong><em><strong>a</strong></em><em><strong> </strong></em><em><strong>priori</strong></em><strong> statement</strong><strong>s</strong><strong> </strong> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">χ</mi></mrow><mrow><mi mathvariant="bold-italic">σ</mi></mrow></msubsup></math> , where σ = +,-, and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>χ</mi><mo>∈</mo><mi>E</mi></math> .

Let us note that evaluation of JSM-investigation results by applying two theories of truth ensures representability of the conditions of possible experience [2] and inevitability of using <em><strong>a priori</strong></em> tools of knowledge representation.

Let us sum up our discussion of Immanuel Kant’s epistemological ideas set forth in [2] and [13] from the point of view of ExEp, the theoretical foundation of AI.

1. ExEp gives a formal and constructive definition of synthetic <em>a priori</em> judgements: those are <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup></math> , where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>χ</mi><mo>∈</mo><mi>E</mi></math> , σ = +,-; and also synthetic <em>a posteriori</em> judgements: those are <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><mi>Q</mi></mrow></mfenced></math> .

2. ExEp elaborates on and gives a formal characterization of the transcendental schema of indirect connectivity between event and conceptual knowledge of understanding by means of . 3. ExEp defines the conditions of possible experience for intelligent systems by means of the Cartesian product <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover><mo>×</mo><mi>E</mi></math> .

4. JSM-method of ASR (an ExEp tool) is <strong>transcendental logic</strong> in Kant’s sense, because by using <em>a priori</em> means (induction, analogy, abduction) the JSM-method discovers new well-founded knowledge for quasi-axiomatic theories, axioms thereof being EMR.

5. ExEp characterizes concepts as a means for representing and organizing knowledge ([5], Chapter 3), which conforms to Immanuel Kant’s ideas on subordination of marks which generate concepts [14].

6. Kant’s concept of a thing-in-itself and of appearance clarifies the <strong>impossibility of automated imitation of consciousness</strong>, since in accordance with Df. 1 of TI TI includes many higher mental functions (intuition, intention, initiative, imagination, reflexion; HMF) which cannot be expressed by means of formal languages. Therefore, consciousness, dependent on HMF, is a <strong>thing-in-itself</strong>. AI products, such as AI systems, IS, AI-robots ([5], [12]) can imitate in automated or interactive mode only the <strong>phenomenology of intellectual processes</strong>, that is, <strong>of TI appearances</strong>.

Interpretation of the ideas of Immanuel Kant’s epistemology ([2], [13], [14]) by means of ExEp is complemented by developing new conceptual and logical tools used for solving AI problems (Df. 3).

Let us enumerate new tools of ExEp and the consequences of their application <em>A</em> – <em>J</em>.

<em>A</em><em>.</em> JSM-method of ASR – a tool of ExEp – uses a hierarchy of formal languages JL, MJL and MMJL. JL is used to formalize JSM-reasoning, MJL is used to formalize JSM-investigations (MJL is a metalanguage of JL), MMJL is used for representation and ordering of the set of EMR that supplement QAT, the result of application of JSM-investigations.

<em>В</em><em>.</em> JSM-reasoning is a<strong> synthesis of cognitive procedures</strong>: induction, analogy and abduction. <em>C. </em>With the tools of JSM-method of ASR we can solve the problems of induction ([11], [6], [5]) and abduction [1].

<p>Abduction-1 is a formalization of a well-known idea of C.S. Peirce represented as follows:</p>

<em>D</em>: the set of facts <em>H</em>: the set of hypotheses <em>E</em>(<em>D,H</em>): <em>H</em> explain <em>D</em> _____________________ For any <em>h</em> such that h&isin;H, <em>h</em> is plausible. &nbsp;

JSM-method formalizes generating the set <em>H</em> (results of induction and analogy), it also characterizes <em>E</em><em>(</em><em>D</em><em>,H) </em>by means of abduction-1 that uses the axioms of causal completeness ACC⁽^σ⁾, where σ = +,-, and functions of degree of abductive explanation of fact bases <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>ρ</mi></mrow><mrow><mi>σ</mi></mrow></msup></math> . The scheme outlined above is abduction-1 applied to <strong>acceptance</strong> of hypotheses in JSM-reasoning, being a means of <strong>control</strong> thereof<strong> </strong>[5].

Let us note that due to the existence of a partially ordered set of strategies <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover></math> for JSM-reasoning the condition of obtaining the <strong>best explanation</strong><strong> </strong>(often used in works on abduction) is <strong>redundant</strong>.

<p>Finally, abduction-2 is expressible as rules of inference that use two theories of truth and empirical modalities.</p>

Solution for the induction problem is the ExEp postulate P7 discussed above. Let us note that the solution of the induction and abduction problems is made in relation to the main product of AI: IS.

<em>D</em>. <strong>The JSM-method of ASR </strong><strong>is used</strong><strong> to create the concept of empirical qualitative causality (CEQC)</strong>. This implies a <strong>constructive</strong> consideration of Kant’s theory of causality. The point is that by means of synthesis of cognitive procedures the partially ordered set of strategies <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover></math> of JSM-method ([5], [12]), and of JSM-investigation we specify <strong>types of causality</strong> and obtain <strong>validation</strong> of the hypothesis of causality, both used for generating EMR. This leads to the definition of a <strong>stronger</strong> (in comparison with [6]) demarcation criterion: knowledge is scientific if it is not only falsifiable, but also obtained by using EMR.

CEQC is a constructive elaboration on the world W1-2, which makes <strong>ExEp</strong> into a<strong> non-speculative</strong> theory and answers the question of <strong>how is AI possible</strong>.

<em>E</em>. The central concept of ExEp is TI (Df.1), as it is <strong>primary</strong>, <strong>understanding</strong> and <strong>mind</strong> being merely its aspects.

In Df.1 we use a characterization of the<strong> intellectual process </strong><strong>(</strong><strong>IP)</strong> as an analogue of coginitive faculties [2]. Df.1 includes intellectual faculties (1)-(13), pertaining to understanding and mind, and the higher mental functions, HMF: intuition, intention, initiative, imagination, reflection. It is essential that implementing Df.1 in IS (Df.2) uses three kinds of knowledge: declarative, procedural (understanding) and conceptual (mind).

<em>F</em>. Intelligent systems (IS), which are the final component of the JSM-method of ASR, represent <strong>experimental </strong>application of the <strong>conditions of possible experience </strong>[2], being <strong>man-machine partnership systems</strong> such that they imitate understanding in <strong>automated</strong> mode and support mind in<strong> interactive </strong>mode.

<em>G</em>. ExEp uses two theories of truth: the <strong>correspondence</strong> theory of truth [6] and the <strong>coherence</strong> theory of truth [9]. The latter is mainly related to the world W3 of objective knowledge [6] and is a natural consequence of the necessity of applying <em>a priori</em> knowledge, that is synthetic <em>a priori</em> judgements.

We can suppose that the antinomies considered by Immanuel Kant in [2] constitute a conflict of opposite <em>a priori</em> judgements. While each part of the antinomies is not correspondently true (false or uncertain), under some assumptions one part of the antinomy is coherently true and the other is coherently false.

<em>H</em>. Extensions of the concept of EMR <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><mi>Q</mi></mrow></mfenced></math> are synthetic <em>a posteriori</em> judgements defined by means of empirical modalities. At the same time <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>χ</mi></mrow><mrow><mi>σ</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>'</mi></mrow></msup><mo>,</mo><mi>Q</mi></mrow></mfenced></math> are empirical nomological statements generated by JSM-investigations [5], which distinguishes them from modalities defined in [15a], [15b].

<em>I</em>. ExEp uses K.R. Popper’s idea of growth of knowledge [5] in accordance with the P1-TT-EE-P2 scheme of his evolutionary epistemology. In JSM-method TT (tentative theories) are replaced by QAT (open quasi-axiomatic theories with JSM-reasoning), and after EE (error elimination) we add the set of empirical regularities <em>E</em>. We then get: P1-QAT-EE-E-P2.

In [13] Immanuel Kant (thinker and <strong>naturalist</strong>!) considered the impossibility of adequate representation of sensible intuition’s empirical data using general concepts of understanding which represent<strong> discursive </strong>knowledge. He postulated the existence of <strong>intuitive</strong><strong> </strong>understanding capable of integral cognition unlike discursive understanding. By adding an evolutionary scheme of the growth of knowledge ExEp makes it possible to solve the paradox of research, which happens when existing theoretical knowledge in QAT containing synthetic <em>a priori</em> judgements contradicts facts obtained by experiment. In this case we should change the strategies (heuristics) of research and the language of representation of knowledge. Such two-way connectivity of theory and experience generates a new problem P2 and growth of knowledge.

<em>J</em>. JSM-method is heuristics of discovery of new knowledge, which is evaluated on five levels of its acceptance [5] using two scales for evaluating the quality of reasoning and the quality of investigations which implies non-singular evaluation of results, unlike evaluating only by means of truth values.

Heuristics of the JSM-method of ASR consists of generating hypotheses on causes in CEQC (induction), forming hypotheses on predictions, using abduction-1 for hypotheses acceptance, conducting JSM-investigations for the set <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="false"><mrow><mtext>Str</mtext></mrow><mo>¯</mo></mover></math> of rank <em>r</em><em>=1</em>, and finally, conducting prolonged JSM-investigation of rank <em>r</em><em>>1</em> for the generation and validation of EMR that are added to QAT, the final product of IS. This heuristics is constructive transcendency of the JSM-method of ASR which is the <strong>transcendental logic of AI</strong>.

Thus ExEp is <strong>exact neo-Kantianism</strong> which transforms transcendency into a constructive intellectual process representable in IS, the main product of AI.

In conclusion let us note that in <em>Critique of Pure Reason</em> [2], <em>Critique of the Power of Judgement</em> [13] and <em>Opus</em><em> </em><em>Postumum</em> [16] Immanuel Kant attempted to bring <em>a priori</em> knowledge into accordance with empirical knowledge obtained using <em>a priori</em> knowledge. In [16] to this end he was going to justify physics by means of metaphysics, which was, of course, impossible. Thus constructivization of transcendency became theoretically required and practically in demand for the problems of AI. And it is natural to consider the emergence of exact epistemology, which is <strong>exact neo-Kantianism</strong>, as a <strong>reflection of the culture of information society on its civilizational (technological) aspects</strong>.