THE FUZZY LOGIC OF TEXT UNDERSTANDING G ... - Science Direct

Fuzzy Sets and Systems 23 (1987) 347-360 North-Holland

347

THE F U Z Z Y LOGIC OF TEXT U N D E R S T A N D I N G

G. GRECO* and A.F. ROCHA Department of Physiology and Biophysics, UNICAMP, 13081, Campinas, Brasil Received January 1986 Fuzzy graphs and fuzzy logic are used to describe the reasoning of verbal understanding of a text played to students of Unicamp. The results demonstrate that each phrase in the text is associated to two different truth values. One of them, here called the confidence degree, measures the truth of the information as related to the individual sets of beliefs; the other, called correlation degree, measures the truth of each phrase as regards the structure of the text itself. Also, empirical relations were established for each of the logical operators used by the volunteers.

Keywords: Empirical research, Graph theory, Logic, Approximate reasoning, Psychology.

1. Introduction

Human reasoning is inductive, because it handles heuristic information, having its truth guaranteed by the repetition of facts; and it is deductive, because it constructs reality as chains of logical relations, having their origins in intuitively (or inductively) accepted, but not logically proved axioms (Rocha [12, 13]). Conditioning is the most successful biological paradigm for studying the rules and the electrophysiology (brain activity) of inductive (heuristic) learning. However, today there exists no methodology that permits physiologists to investigate the brain mechanisms involved with the logical process. Rocha [12] has proposed that fuzzy set theory now provides the necessary background for such an enterprise, and its application to the study of the brain mechanisms underlying language perception may provide a first and very important paradigm to make logic the subject of experimentation in physiology laboratories. The brain activity related with a given process may be recorded in an electroencephalogram (EEG). The statistical analysis of the activity, evoked in the EEG by a particular task, may provide the key elements for the study of the cerebral rules for meeting that task. However, any attempt in this direction requires the possibility to correlate the EEG activity with behavioral components of the task under investigation (Ritter et al. [9, 10], Ruckin [15]). In the case of language perception, this means that it is necessary to correlate the EEG activity during the decoding of the verbal information with its comprehension (Rocha [13]). A good picture of the verbal comprehension may be obtained from its recall. Hence, it is possible to correlate the statistical analysis of the EEG * Supported by FAPESP. 0165-0114/87/$3.50 t~) 1987, Elsevier Science Publishers B.V. (North-Holland)

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G. Greco, A.F. Rocha

recorded during the verbal decoding with a statistical analysis of the recall of the verbal information. Fuzzy graph theory provides the necessary tools to describe each recall related to a heard or read text (Theoto et al. [18]). The graph of the recalled text joins the recalled phrases as they are assumed to be linked in order to provide the conclusion for a chosen theme T (the main subject of the text) and rheme R (what is said about T) ([12, 18], Rocha and Rocha [14]). The mean graph associated to the 'mean text decodification by a given population' is obtained depending on the frequency of nodes and phrases in the recalled graphs and texts of this population [18]. The logical structure underlying the text decodification may be obtained by asking people to assign, to each node of the recalled graph, the logical operator they assumed to be suited best to permit the logical deduction of the chosen T and R. There is a clear distinction between language as a set of symbols and meanings, shared by a population, and speech as the use of this language to express personal 'senses' in society (Luria [4], Saussure [16]). As a matter of fact, each text may be viewed as a partially closed structure [14] depending, on one hand, on meaning, guaranteeing minimal coherence among the symbols used, and, on the other hand, on sense, permitting people to speak about individuality of ideas and models. Each piece of information in the text is correlated with the structure or meaning used to convey the desired information or sense, and triggers a degree of confidence with the receiver, depending on both environment and personal contexts ([4, 14], Oleron [7], Olson [8]). Logical operators at each node therefore have to handle two different sets of truths: one relating the information represented at the node to the structure of the text itself, and the other associating this information with the set of beliefs of the subject trying to understand the text. This hypothesis is supported by the findings of Theoto et al. [18], showing that confidence in the decoded information is different from the importance it may have to define the theme or rheme of the text. The first of these truth values here will be called confidence degree, and the second will be referred to as correlation degree. The investigation of the logical structure fully describing text understanding here concerns both correlation of and confidence in the verbal information conveyed by a pre-recorded text, played twice to volunteers before asking them to recall (RT) the text, to construct the graph of this RT and to state the confidence and correlation degrees for each node of the graph.

2. Methods

The experiments were carried out with 120 students of Unicamp. The text (Table 1) was played during two experimental sessions, with no resting interval between them. The electroencephalogram (EEG) was recorded while the volunteers were listening to the text.

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349

Table 1. The text Brazil spends about three billion dollars each year on technology importation (1), that is, the equivalent of eight times the total value of investments in applied and basic research in the country (2). There is no selectivity in foreign technology purchase (3). Even Lubrax-4 from Petrobras has a foreign formula (4). External dependency increases exponentially (5). Scientific and technology research do not receive the necessary backing (6) and do not obtain the priority demanded by exchange pressures (7). Oil importation burdens the commercial balance in a heavier fashion (8). Despite some good examples of technology creation such as CTA, IPT, Escola Politecnica de Silo Paulo, Unicamp and others (9), Brazil faces today the problems of a narrow-minded technocracy at CNPq (10), the authority responsible for direct and indirect application of twenty five billion cruzeiros in research and technological development (11). Numbers in brackets are used to represent the phrases of the text in the graphs of the recalled texts and for statistical purposes.

The general procedures to construct the RT are the same as those used by Theoto et al. [18] in this issue. Also, the mean graphs, describing the 'mean' understanding of the text of the different experimental groups, were obtained with the same technology used by those authors. Finally, the chosen theme and rheme were compared to the original phrases, and were represented by the numbers of the phrases describing them best. Once graph G was constructed, the individuals were asked to associate one of the logical operators NOT, AND, OR, IF and IF AND ONLY IF to each non-terminal node of G. Also, subjects were given a set of five words: NULL, ALMOSTNULL, MEDIUM, STRONG, and EXTREMELYSTRONGto quantify the degree of confidence or correlation of the information represented at each node of G. This was done using an approach similar to that proposed by Giles [2]. At the end of the experiment, the volunteers were demanded to rank these adjectives in the closed interval [0, 1]. On one hand, this approach allowed the individuals to easily quantify the truth values using their daily (verbal) fuzzy measures, and, on the other hand, provided the researchers with the numbers they need for statistics (Freska [1]). Because it was necessary to reduce the time required for each interview in order to maintain the cooperation of the volunteers, they were divided into two experimental groups. Group I was composed of students handling only the correlation between the information represented at each node and the chosen T and R, whereas Group II enrolled people quantifying only the confidence in the received information. The frequency of each logical operator in the recalled graphs as well as its distribution in G where calculated to furnish the logical structure associated to each possible 'mean' understanding of the text. Confidence and correlation were averaged for each logical operator, and for each node of the possible 'mean graphs'. Regression analysis was employed to assess any correlation between truth values and node position, and any dependence of the truth values at each node on the values associated to the arguments linked to it.

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3. Results 3.1. The 'mean' decodings of the text

The analysis of node frequency and node connectivity in G, and of phrase distribution (Figure 1) in RT revealed similar text decodings by both experimental groups. The number of phrases in RT averaged six, and only three terminal nodes in either theme or rheme subgraphs (see Figures 4, 5 and 6) crossed the frequency threshold level of 70%, necessary for a node to be included in the mean graph [18]). Besides this, the first and second terminal nodes were linked to the first node of level 2 for 95% of the subjects, in the case of both theme and rheme subgraphs. The study of the correlation between the chosen theme or rheme and the original phrases of the text (Figure 2) showed that three different themes and rhemes were the most frequent choices, and that they were correlated with the ideas that: "We spend a lot of money on importation of technology" (Theme 1 and Rheme 1), embodied in phrase 1; "There is no backing of national science" (Theme 2 and Rheme 2), correlated with phrase 6; "We do have important research institutes" (Theme 3), as pointed out by phrase 9, and "We have a low budget for national science" (Rheme 3), associated with phrase 2. The phrase distribution on terminal nodes (Figure 3) further supports the above correlation, since phrases 1, 6 and 9 predominated at terminal node number 1 in the theme subgraph, and phrases 1, 2 and 6 were the most frequent at the same position in the rheme subgraph. As a matter of fact, the theme (Ft) and rheme (F0 frequencies were linearly correlated with the phrase frequencies at terminal nodes 1 (Ptl or P J and 2 (et2 or Pr2): Ft = 0.052 + 0.94Pt1 + 0.58Ptz,

Prob = 8%;

Fr = 1.25Pra - 0.05P~2, Prob = 1%.

O.9 O.II O.7 0.6

O

QNOUP !

,4

~I~OI,,,IP2

Fig. 1. Frequency of the phrases in the recalled texts of the two experimental groups.

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351

0.2:1 0.2

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0.17

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0.12

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Fig. 2. Frequency of the chosen themes and rhemes.

As a matter of fact, Ft was much more related to Ptl (correlation coefficient r = 0 . 6 1 ) than to Pt2 (r=0.27). On the other hand, F~ was almost equally dependent on Prl (r = 0.82) and Pr2 (r ----0.71). These results point to three 'mean' texts (see Table 2) derived from three theme and rheme subgraphs, having their terminal nodes occupied by the three most probable phrases at each position. It is interesting to note that distribution of the phrases in the theme subgraph was more crisp than in the rheme subgraphs. Because of this, it was easier to obtain the theme texts than to decide on the theme ones. The conclusion is that decoding the rheme was much more fuzzy than understanding the theme. The possible 'mean' decodings of the text by the volunteers were obtained from the most frequent associations between these theme and rheme texts. This analysis stressed the following associations: theme and rheme phrases

T1 1, 2 and 9

R2

T2

R]

T3

R3

6, 10 and 4

6, l0 and 4

1,2 and 9

9, 1 and 10

2, 4 and 8

Therefore, they were used to compose three 'mean' graphs G (Figures 4, 5 and 6) describing the decoding of the text by our population of students. 3.2. The logical operators and truth values in G

The Frequency and distribution of the logical operators were the same for the graphs of both experimental groups. Operators AND and IF drastically predominated over the operators OR and IF AND ONLY ~F (see Figure 7), whereas operator NOT was never used. AND and IF were most frequent at terminal nodes, although the operator IF also had a considerable occurence at the root.

G. Greco, A.F. Rocha

352 0.6

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Fig. 3. Distributions of the phrases on the terminal nodes of the theme and rheme subgraphs: (a) theme, (b) rheme.

When the distribution of the logical operators was analysed, regarding the associations of phrases suggested by the three mean theme and rheme graphs, it was observed that: (a) Phrases 6 and 10 were related by an IF THEN condition, pointing at the 'technocracy of CNPq' as being responsible for the problems of national science; (b) Phrases 1 and 2, 9 and 1, as well as 2 and 4 were associated through the logic operator AND; (C) Operator AND joined almost all information at the second order node, except the phrases 1, 2 to phrase 9 (see Figure 4) in the case of the theme T3. In this theme, half of the subjects used an IF THENcondition to associate the idea of a

The fuzzy logic of text understanding

353

Table 2. The mean themes and rhemes Theme 1 and Rheme I Brazil spends about three billion dollars each year in technology importation, that is, the equivalent to eight times the total value of investments in applied and basic research in the country. Despite this, we have some good examples of technology creation such as CTA, IPT, Escola Politecnica de Silo Paulo, Unicamp and others. Theme 2 and Rheme 2 Scientific and technology research do not receive the necessary backing, because Brazil faces today the problems of a narrow-minded technocracy at CNPq. Even Lubrax-4 from Petrobras has a foreign formula. Theme 3 Despite some good examples of technology creation such as CTA, IPT, Escola Politecnica de S~o Paulo, Unicamp and others, Brazil spends about three billion dollars each year in technology importation. Brazil faces today the problems of a narrowminded technocracy at CNPq. Rheme 3 This is the equivalent to eight times the total value of investments in applied and basic research in the country. Even Lubrax-4 from Petrobras has a foreign formula. Oil importation burdens the commercial balance in a heavier fashion.

poor financial support of the national research institutes with the necessity to import hugh amounts of technology. (d) Operator IF associated T 3 and R1 as well as T2 and R2, whereas TI and R 3 were linked by means of an AND. These findings determined the distribution of the operators on the 'mean' graphs of the text 'mean' understandings (Figures 4, 5 and 6), and preserved the general trends of the operator frequencies illustrated by Figure 7. Averaging the correlation degree for each node of G has shown that correlation statistically decreased (significance at the 1% level) from the root to the terminal nodes and from the first to the last node at the terminal level, in the case of both theme and rheme subgraphs (Figure 6). As a matter of fact, correlation (C) linearly decreased with node position on G, from top to bottom (N) or left to right (P): Ct = 0.79 + 0.08N,

Prob = 1%;

Ct = 0.84 - 0.45N,

Prob = 0.5%,

Cr = 0.67 + 0.70P,

Prob = 5%;

Cr = 0.82 - 0.60P,

Prob = 0.1%.

The observation that confidence decreases from left to right is in accordance with the fact that people were asked to order the terminal nodes according to their importance for defining the chosen T or R. On the other hand, there were no such trends in the case of the averaging of confidence (Figures 4, 5 and 6), since there was no statistical difference in confidence between the root and terminal nodes, nor between nodes at this level. These findings seem to confirm the suggestion of Theoto et al. [18] that two different truth values have to be associated to each node, since verbal decoding is

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G, Greco, A.F. Rocha

if

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Fig. 4. The graph of the mean text composed by T~ and R 2. At each terminal node (level N1): numbers at the left are the confidence degree of the phrase (number at the fight). At the other nodes (levels N2 to N4): numbers are the confidence degree of the information represented at the node. Also the most frequent logical operator is shown.

dependent on both the individual context as well as on the structure of language itself. Additional support for this hypothesis is the fact that, in general, confidence reached higher values than correlation when averaged with respect to the nodes of G, as well as with respect to the phrases they were associated to (Figure 8). Despite these differences, both correlation and confidence degrees seem to be ruled by the same logical structure, since there was no statistical difference between the distribution of the logical operators in the mean graphs of the two experimental groups.

if

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The fuzzy logic of text understanding

355

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Fig. 6. The graph of the mean text composed by T 3 and R 3. Here, numbers at the left of the nodes are the degree of correlation between the information and the text.

3.3. The rules for fuzzy logical operators The truth values were also averaged with respect to the logical operator handling them. In the case of the operator OR, the mean truth value calculated for the proposition was equal to the truth values averaged for the arguments, both when confidence and correlation were considered (Table 3). This seems to support the classical max-rule introduced by Zadeh for this operator (Mandami [5], Zadeh [20, 21]). The results were somewhat different in the case of the operator AND, because the mean truth value for the proposition was greater than the minimum value o~

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Fig. 7. The distribution of the logical operators AND (A), O R (O), I F ( I ) and IF AND ONLY IF (II) in the entire population at the left, and at the first (1), second (2) order level and the root (R).

356

G. Greco, A.F. Rocha

0.9

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Fig. 8. The averaged degrees of correlation and confidence for each phrase of the original text.

averaged for the arguments, and smaller than their maximum (Table 3). These findings do not support Zadeh's min-rule for the AND, but point to a power mean [5, 20,211. This was confirmed, since the truth value for the proposition was a linear function of the truth values of both arguments (Table 3). One of the differences observed when the two experimental groups were compared, was that the linear coefficient was different from zero in the case of correlation, but not in the case of confidence. However, this is in agreement with the finding that correlation - - but not confidence - - increases from the terminal nodes up to the root. Also, the angular coefficients were the same (0.40) for both arguments when confidence was considered, but they pointed out an influence of the first argument (0.45) greater than that of the second (0.16) in the case of correlation. The analysis of the operator IF revealed that the mean truth value for the proposition reached the maximum value calculated for the arguments in the case of confidence, but was greater than these latter values when correlation was considered (Table 3). Linear dependences were once again established between the truth values, and the linear coefficient for correlation - - but not for confidence - - was different from zero. Once again, the results did not support the theoretical constructs proposed by others (Lesmo et al. [3], Mizumoto [6]), but stressed that truth values may depend on the predictability of the model or text (Rocha and Franqozo [11], Rocha [12]). The fact that Q is deductable from P may strengthen the membership of P to the set of beliefs considered. The operator ~E AND ONLY IV differed from the above operators, since the proposition truth value tended to be equal to the minimum value calculated for the arguments (Table 3). This was statistically true for confidence. Linear dependences were responsible for at least a 60% correlation between truth values

357

The f u z z y logic of text understanding

Table 3. The logical operators rules The operator oR

Group I Group II

P

OR

Q

83 86

83 84

72+ 80

P 87 84 45 39

AND 81+ 77+ 30 0

Q 72+ 67+ 16 40

Mean

The operator AND

Group Group Group Group

I II I II

Mean r = 56 r = 72

The operator tF

Group Group Group Group

I II I II

The operator

P

IF

Q

76 81 35 49

80+ 78 32 0

70+ 73+ 25 40

I II I II

r = 63 r = 75

IF A N D O N L Y IF

P

Group Group Group Group

Mean

78 88 38 26

IF ONLY IF

77 82+ 22 44

Q 77 80+ 31 25

Mean r = 63 r = 69

For mean values: numbers under the arguments and the operator are the values calculated for P, Q and the Proposition, respectively. For linear functions: numbers under the arguments and the operator are the angular coefficients for P, Q, and the linear coefficient, respectively, r is the correlation coefficient. Numbers marked with + are statistically different from the means calculated for P.

h a n d l e d b y this o p e r a t o r ( T a b l e 3). T h e a n g u l a r coefficients w e r e different f r o m z e r o for b o t h c o n f i d e n c e a n d c o r r e l a t i o n . A l l t h e s e results s h o w t h a t c o n f i d e n c e a n d c o r r e l a t i o n u n d e r w e n t different c a l c u l a t i o n s on t h e s a m e chain o f logical d e d u c t i o n s . T h e fuzzy g r a p h s of v e r b a l d e c o d i n g for t h e t w o e x p e r i m e n t a l g r o u p s w e r e the s a m e , as w e r e t h e distributions of the logical o p e r a t o r s for t h e s e g r a p h s . This i m p l i e s t h a t t h e fuzzy logical s t r u c t u r e s of t h e fuzzy v e r b a l d e c o d i n g w e r e the s a m e w h e n p e o p l e w e r e i n q u i r e d a b o u t different c o n c e p t s o f truth. A l s o , the fuzzy logical o p e r a t o r s use the s a m e g e n e r a l rules to h a n d l e t h e s e two c o n c e p t s . H o w e v e r , t h e y p e r f o r m the c a l c u l a t i o n s using distinct r e l a t i o n s . A m o n g t h e m o s t striking d i f f e r e n c e s in calculus is the e n h a n c e m e n t of t h e c o r r e l a t i o n ( b u t n o t the c o n f i d e n c e ) d e g r e e with t h e o r g a n i z a t i o n of t h e recall. This stresses t h e d e p e n d e n c e o f c o r r e l a t i o n on the s t r u c t u r e o f t h e text, a n d the i n d e p e n d e n c e o f c o n f i d e n c e . This c o u l d e x p l a i n w h y l i n e a r coefficients w e r e

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G. Greco, A.F. Rocha

different from zero for the correlation degree, but not for the confidence degree. The linear coefficients may have quantified the influence of the structure of the text upon the evaluation of the correlation between each piece of information and the text itself.

4. Discussion Fuzzy set theory provided the tools necessary for the development of a method, used here and by Theoto et al. [18], to analyse the process of verbal understanding. It is now possible to obtain the 'mean comprehension' of a text shared by a given population, and the logical structure of this understanding. The structure of text understanding is given by a set of fuzzy graphs describing its skeleton, and a set of fuzzy logical operators distributed over these graphs. This structure is the history of the chains of logical deductions performed during text decodification, to provide the conclusion for a set of themes and rhemes. The same logical network handles two different sets of truths, using the same set of rules, but performing different calculations with each of these truth values. Many authors have suggested that speech is a partially closed activity, depending on one hand on the language structure itself, and on the other hand on the local, individual, etc., contexts ([4, 7, 8, 14, 16], Sgall et al. [17], Washabaugh [19]). The first dependence is measured by a fuzzy quantifier, here called correlation degree, and the second is evaluated by another, called confidence degree. People may have a personal degree of confidence when first receiving the pieces of information composing a text, but have to wait until they have at least a minimum picture of the subject of the text, before evaluating the correlation of each of these pieces of information with the theme and rheme of the text. First, they must attempt to construct the scheme of the text organization, assembling all the pieces of information into a logical network of meanings, guided by their confidence in the veracity of this information [4, 11, 12]. Once the initial picture of the text is thus arrived at, one may try to verify text consistency [12] through the evaluation of the correlation between the phrases chosen to compose the decoded text. Because of this, the logical networks dealing with both correlation and confidence degrees are expected to be the same, as observed here. However, since correlation - - unlike confidence - - is sensitive to the structure it helps building, its actual values associated to each piece of information have to depend on text cohesion - - which is not the case for confidence. Besides this, correlation has to increase as text organization is improved. This was actually the case here, because on one hand the correlation degree depended on a non-zero linear coefficient which could be accepted as a measure of the evaluated text cohesion, and on the other hand it increased from the terminal nodes up to the root. The distinction between the concepts of correlation and confidence was stressed by the simplified calculus used to deal with the confidence degree.

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5. Conclusion The method developed here and its results constitute the foundations of a paradigm, permitting us to begin the study of brain activity correlated with the process of logical verbal decodification, since these results were correlated to the statistics of certain measures of the electroencephalogram (EEG) recorded during the text perception. Although these findings will be the subject of a separate paper (Greco and Rocha, in preparation), a summary of the principal conclusions is presented below. The principal measurement of the EEG analysed here was the area of the EEG curve recorded during the listening of each phrase of the text. The EEG was first rotated around its mean value in order to calculate the area of its curve, because in it both positive and negative voltages were recorded. Then the mean averaged area was calculated as the mean of the units of area composing an entrie EEG epoch. Each unit of area was the area integrated for successive samples of the EEG, measured through and A/D converter at the rate of 100 Hz. The EEG was recorded at six different skull positions, three (FL, CL and PL) of them on the left hemisphere, and the other three (FR, CR and FR) on the right side of the skull. Coefficients of dominance between the left and right side, as well as between areas of the same hemisphere were calculated as relations between FL, CL, PL, FR, CR and PR, because it well established some brain dominances for language processing (e.g. [4, 9, 10]). The frequency of occurrence of the phrases (F) in the recalled texts was a linear function of the dominance of the left brain at the frontal (F) central (C) and parietal (P) areas: F = 345 - 2FL/FR + 0.36CL/CR - 0.9PL/PR, r = 88%,

Significance at level 3%.

In other words, F was dependent on a global activity of the brain. The confidence degree (Cf) was a linear function of the activity in the frontal areas during the first but not during the second hearing session: Cf = 448 - 3.7 FL/FR,

r = 76%,

Signif. = 0.8%.

On the other hand, the correlation degree (Cr) was a linear function of the activity in these frontal areas during the second but not during the first hearing session: Cr = 89 + 0.57 FL/FR,

r = 55,

Signif. = 13%.

Dealing with logical relations and with logical organization of verbal information are assumed to be the most important tasks processed by frontal areas [4]. The degree of confidence was dependent on the activity of these areas during the first but not during the second listening of the text, whereas the correlation was related to the activity during the second hearing session, when people had already constructed a clear picture of the theme and rheme. Thus, the experimental data recorded at the laboratory support the hypothesis, discussed above, that

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G. Grew, A.F. Rocha

confidence and correlation degrees undergo different calculations during the logical processing of verbal decoding. We are now able to conclude that fuzzy set theory and fuzzy logic provide the foundations for the development of paradigms to guide the experimental approach of human reasoning at physiology laboratories.

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