**Abstract**

In the present paper, a novel approach is introduced for comparing and classifying recorded ERP signals from subjects applying valid (Aristotle’s) and paradox (Zeno’s) syllogisms. In fact, the authors conceived and realized a corresponding experiment, as well as a new method for processing, fitting and classifying the corresponding captured ERP signals into groups according to their similarity. Subsequently, for each such group, an ideal curve that represents all signals of the group has been evaluated for valid and paradox reasoning separately. These ‘‘ideal representatives’’ manifest essential statistical differences per subject for a considerable number of electrodes (5 electrodes with 99% level of confidence, 14 electrodes with 95% level of confidence, 17 electrodes with 90% level of confidence). These results support the assumption that the obtained ideal representatives may indeed reflect essential differences in the underlying brain functions which generated the obtained ERPs. Equivalently, one may claim that the performed experiment and the associated results manifest statistically essential differences between the mental functions during valid and paradox reasoning.

**Introduction**

One of the most prominent intellectual abilities of the human brain, which so far characterizes the human kind, is the process of logical reasoning/syllogism. Historically, one of the first philosophers, if not the first one, who tried to analyze logical reasoning by the means of deduction was Aristotle. According to his analysis, the reasoning process starts with two statements, i.e. ‘‘All men are mortal’’; ‘‘All Athenians are men’’. These two statements, in Aristotle’s approach, lead to the conclusion that ‘‘All Athenians are mortal’’ with absolute certainty [18]. We note that this Aristotle’s syllogism, also known as valid reasoning, has been a very active, open field of research. Recently, a number of models have been proposed in order to encode and elucidate the underlying brain functions associated with this type of reasoning; however, one may safely say that these brain functions are, by no means, fully understood [7,19]. On the other hand, about 2500 years ago, Zeno the Eleatic, extending ideas of his teacher Parmenides, conceived some paradoxes in order to elucidate contradictions of ‘‘Common Sense’’, as well as inconsistencies in the Pythagorean ideas of multiplicity and change. Zeno employed 3 major steps in order to manifest these contradictions and inconsistencies: (a) he temporarily adopted a thesis that he opposed, (b) he tried to deduce an absurd conclusion or contradiction in connection with it, (c) thus, he undermined the originally adopted thesis [5]. This approach is most commonly called ‘‘a Zeno’s paradox’’. These paradoxes have always intrigued and puzzled philosophers and mathematicians and they largely influenced subsequent research [1,6,20]. The nature of the mental processes induced by the paradoxes has not been extensively studied and it is still an open research subject. The eventual results of such a research may be proven to be of considerable importance both from the Academic and Clinical point of view. More specifically, studying the brain functions which correspond to Zeno’s paradoxes in contrast to mechanisms associated with Aristotle’s valid reasoning may contribute in the understanding of the fundamental operation of reasoning into its extreme form [22]. In the present paper, the results of such a study are presented; the associated analysis employed, among others, principles and tools of Signal Processing, Pattern Recognition and Biostatistics. The recorded and processed electro-physiological activity is associated with event-related potential (ERP) techniques. These techniques are commonly used as they appear to be sensitive to subtle neuropsychological changes [11,12,16,15,4,3,23]. For this reason, the research presented here is based on an experiment during which the ERP signals of various healthy adult participants are recorded when these subjects applied valid and paradox reasoning. Special effort has been taken to induce the working memory (WM) of each participant during this experiment. Contemporary neuropsychological views define WM as the capacity of the human subject to keep information ‘on-line’ necessary for an ongoing task [2,6]. Accordingly, WM is not for ‘memorizing’ per se; it is rather in the service of complex cognitive activities, such as reasoning, monitoring, problem solving, decision making, planning and searching/shifting the initiation or inhibition response [13,8].

The main goal of the present work is to determine if different patterns of electro-physiological activity exist during Aristotle’s valid reasoning in one hand and Zeno’s paradoxes on the other. A comparative study of these activated patterns in Aristotelian and paradox-related reasoning could reveal critical aspects of reasoning processing, associated with perception, attention and cognitive behavior. We emphasize that these aspects are unobservable with behavioral methods alone.

A brief description of the introduced approach A set of forty-five healthy subjects participated in an experiment, where each one of them was asked to verify or not the validity of a number of presented syllogisms. During this process, 30 scalp Ag/AgCl electrodes have been attached to each subject’s scalp in order to record the electroencephalo- graphic (EEG) activity in accordance with the International 10–20 system of electroencephalography [10]. These Event Related Potential signals (ERPs) were recorded for a 2000 ms interval and they have been digitalized at a sampling rate of 1 kHz. We have limited the obtained digital signal for each subject to the time interval (100,400] ms for reasons that we will present in the following Section 4, step 1. This restricted digital ERP signal is symbolized by RX k;q;j where subscript k runs through the electrodes, q through the number of questions and j through the subjects; in order to explain superscript X, we must report that the aforementioned syllogisms are divided into two classes, one representing Aristotle’s valid reasoning, where X = V and another expressing Zeno’s paradoxes, where X = P. The basic ideas upon which the present study is based may be described as follows: suppose that there is indeed a class of common mental processes, which are activated when a person is asked to verify or not a valid or paradox syllogism. Then, one may expect that this causality will reflect in the form of the digital signal RXk;j. Thus, we make the fundamental assumption that for each group of persons manifesting the same mental behavior, if any, in respect to ‘‘valid reasoning’’ and/or ‘‘paradoxes’’, there is a common underlying prototype curve PXk;j, where, this time, j runs through the different groups of persons. Moreover, we make the additional assumption that the various signals RX k;j, corresponding to subjects with similar mental response to valid or paradox reasoning, are noisy versions of the related PX k;j. We further assume that the distortion of PX k;j that generates RXk;j is due to two substantially different factors: (a) a causal one and (b) an erratic noise. The causal component of the distortion is associated with brain functions that do not affect the general shape of the signal PX k;j; in fact, we assume that the most important such functions are (i) the intensity – amplitude of the emitted electromagnetic wave, reflected on the ERP amplitude and (ii) the speed of the subject’s reaction. In order to account for these causal components, we apply suitable transformations on the ERP signal, while, in order to suppress the erratic component, we have developed a new approach that generates a good estimation of PX k;j. Therefore, in consistence with these assumptions, we have developed a method for classifying subjects according to their ‘‘valid reasoning’’ or ‘‘paradox understanding’’, consisting of the following steps: Step 1 – A first stage processing of the data Step 2 – We have defined a kind of amplitude scaling and time dilation or contraction, applied to each signal RX k;j (Section 4, step 2), in order to suppress the aforementioned causal discrepancies among signals, corresponding to specific differences in the various subjects’ mental functions. Step 3 – In order to test similarity of two curves, we have defined a proper error function presented in Section 4, step 3. This error function takes into account the transformations defined in step 2 and it is practically independent of the prototype curve energy. Step 4 – We formed subgroups of similar curves by optimally fitting curves RX k;j using the results of steps 2 and 3. More specifically, we have momentarily set each RXk;j to play the role of a prototype signal and we have optimally fit all other ERPs of the same class X (valid or paradox separately) to it on the basis of the transformations and error defined in steps 2 and 3. As far as the fitting error remained smaller than a properly chosen threshold, we tentatively classified all optimally fit curves to the same subgroup GX k;1;j. Thus, with each ERP RX k;j, that played the role of the prototype curve, we have associated a group GXk;1;j of other ERPs that optimally match to the prototype one with a fitting error smaller than the selected threshold (Section 4, step 4). Step 5 – In each such subgroup, a kind of ‘‘ideal representative’’ MX k;i;j has been produced, by proper averaging of the optimally fit curves that form group GX k;1;j. This process suppresses the erratic noise in the various versions of PX k;j, thus offering a good estimation of the ideal ERP signal of the group of the subjects in hand (Section 4, step 5). Step 6 – We repeat the process described in steps 4 and 5, using this time as prototype curve MX k;i;j instead of RX k;j. As far as the fitting error remains smaller than the threshold described in step 3, we assume that the matched curves belong to the same group. In this way, we obtain the final estimation of groups GXk;i, which is invariant in the choice of the prototype curve when this is an element of the group. Hence, we have ended up with a particularly small number of groups GX k;i, for each brain function X, where index i now indicates the cardinal number of each distinct group, instead of the subjects’ cardinal number, previously represented by index j (Section 4, step 6). For example, concerning ‘‘valid reasoning’’, j runs from 1 to 45, i.e. the subjects’ number, while after completion of these 5 steps, i usually runs from 1 to 6, since all 45 subjects’ ERPs have been classified into 6 groups. For each group GXk;i we choose the ideal representative that offered the smallest overall fitting error per sample and we let this play the role of the ideal representative of group GX k;i; naturally we symbolize it with PXk;i. Step 7 – The final decision about possible distinction of ‘‘valid reasoning’’ and ‘‘paradox syllogism’’, is performed as follows: First, to each subject separately we attribute both the ideal representative of the ‘‘valid reasoning’’ group, PV k;i, and the ‘‘paradox syllogism’’ group, PP k;i, to which the subject belongs. Next, we assume that the brain functions’ V and P differentiations substantially reflects on the difference of the ideal representatives PV k;i and PP k;i, where the comparison of these two curves is made exactly as in steps 2 and 3. Eventually, the decision on the possible difference between ‘‘valid reasoning’’ and ‘‘paradox syllogism’’ is made by properly introduced statistical criteria (Section 5).

An analytic presentation of the performed experiment

3.1. The group of subjects/participants and the experimental setup This study was approved by the Ethics committee of University Mental Health Research Institute (UMHRI). The performed experiment involved forty-five healthy subjects, aged 33.1 years on average with a standard deviation of 9.2; each one of them had normal vision and no one had neurological or psychiatric history. All these subjects gave written consent, after being analytically informed about the process of the experiment. Each subject was comfortably seated in front of a computer monitor and at a distance of approximately 1 meter from it. In accordance with the International 10–20 system of electroen-cephalography [10], on the scalp of each subject, thirty (30) Ag/AgCl electrodes have been attached in order to capture his/her ERPs (the associated electrodes’ map is shown in Fig. 1). Two additional electrodes were attached to the subject’s ear lobes, so that the corresponding recordings could be used as reference signals. The resistance of the electrodes was kept constantly below 5 kV. Recordings with EEG higher than 75 mV were excluded and, in addition, an electro-oculogram (EOG) has been used in order to record the subject’s eye movement. The entire setup, together with the seated participant, was inside a Faraday cage, which achieved suppression of external electromagnetic interferences, up to 30 dB.

3.2. The experimental process A number of questions (39) were set to each subject via the computer’s monitor; the content and the purpose of these questions will be analytically presented in the next sub- Section. Each question appeared on the monitor for a time period proportional to the number of characters of it. For example, a question consisting of 92 letters remained on the monitor for about 11 s. At the end of this time period, the question disappeared and a blank screen took its place on the monitor for a period of 1000 ms. Subsequently, two sound warning stimuli, of 100 ms duration each, were heard at a time difference of 900 ms. After the second sound warning stimulus, the subject was asked to verify or not the validity of the question that preceded. In order to circumvent habituation with the conditions of the experiment, the onset of the presentation of the next question on the monitor fluctuated from 4 to 9 s after completion of the previous subject’s oral response. We emphasize that the obtained signals were recorded for a total of 2000 ms interval and more specifically, for 1000 ms before the first sound stimulus (EEG) and for 1000 ms after that (ERP). The aforementioned sequence of actions, together with the corresponding duration, is shown in Table 1.

3.3. The questionnaire of valid and paradox syllogisms Two sets of thirtynine (39) questions each, were presented to each subject separately, via the computer’s monitor. These sets of questions were designed to validate the human brain behavior in association with two mental functions, namely (a) syllogisms that may be characterized as ‘‘valid’’ and (b) syllogisms that may be characterized as ‘‘paradox’’. Frequently on refers to the valid syllogisms as ‘‘Aristotle’s reasoning’’ and to the paradox ones as ‘‘Zeno’s paradoxes’’.

Two indicative examples of such syllogisms are presented below:

(i) Concerning the class ‘‘valid’’, an example shown to each participant is: ‘‘All men are animals. All animals are mortal. Hence, All men are mortal.’’

(ii) Concerning the class ‘‘paradox’’, an example shown to each participant is: ‘‘A moving arrow occupies a certain space at each instant. But, when an object occupies a specific space, it is motionless. Therefore, the arrow cannot simultaneously move and be motionless.’’ [18].

Defining an ‘‘ideal representative’’ for each class of subjects with similar ERPs

In this section, we will give an extensive analysis of steps 1–7, introduced in Section 2.

4.1. Step 1 – A first stage processing of the data As it has already been stated, for each question and for each electrode separately, 2000 samples (expressed in mV) have been recorded in 2 s; the employed sample period was 1 ms. We will employ for this subsequence of the data the symbol SX k;q;j where subscript k runs through the electrodes, q through the 39 questions, j through the subjects and X determines the class; thus, X 2 {V, P} where V stands for ‘‘Valid reasoning’’ and P for ‘‘Paradoxes’’. In order to optimize the signal-to-noise-ratio (SNR) for each subject, for each channel and for each class of questions we have proceeded as follows, following a rather standard method:

(a) For each question separately, we have averaged the values of the EEG, namely the data acquired in the 1000 ms before the first sound stimulus. Thus, we have obtained quanti- ties aX k;q;j.

(b) We have subtracted quantity aX k;q;j from SX k;q;j, thus obtaining a translated version of SX k;q;j for which we will employ the same symbol.

(c) At each time instant separately, we have averaged the translated SX k;q;j over all 39 questions, thus obtaining a mean curve sXk;j.

(d) We have evaluated the eventual offset of the EEG signal by averaging the first 1000 values of sX k;j, thus obtaining the offset aX k;j.

(e) We have calculated the sequence SX k;j ¼ sX k;j_aX k;j. Therefore, in the end of this step, we have generated the offset free signals SX k;j for each electrode, each subject and each type of question (V or P).

(f) We have restricted the obtained digital signal for each subject to the time interval (100,400] ms. We have decided to study this restricted sequence of electrode recordings, since the interval [1, 100] ms refers to the EEG recordings previous to the first sound stimulus, while in the interval [301, 1000] ms the Contingent Negative Variation (CNV) [21,14] is dominant. The latter could obscure the micro-scopic analysis we have developed. Therefore, for the present study we reckon that the ERPs signals in the interval (100,400] ms best expresses the brain functions associated with Valid and Paradox reasoning. We will employ for this restricted signal the symbol RX k;j, where, as always, X belongs to the set {V, P} indicating the class of questions, k indicates the electrode number, except the ones attached on the ear lobes, and j the subject’s cardinal number. All these ERPs RX k;j form a group which we will symbolize EX k;0.

4.2. Step 2 – Defining and applying proper transformations to the restricted signals RXk;j In order to circumvent an eventual latency in the human response, we have applied time scaling in the domain of RX k;j. This is accomplished by properly transforming the arbitrary signal x(t), which can be seen as a generic symbol of the restricted signal RX k;j, as follows: at each time instant t, we correspond another time instant lt, where l is a real scaling factor. If x(t) were an analog signal, then we could generate a time shifted signal y(t) = x(lt) for each t in the domain sequence. However, the restricted signals we study are digital, say x(ti); hence, the values of the signal in between the samples are unknown. Consequently, the values of y(ti) = x(lti), in practice, are unknown. To bypass this intrinsic difficulty, we first interpolate the signal by ensuring continuity of it and its first derivative at the data points. Next, we round number lti and, if the obtained rounded number is ti, we let y(ti) be the value of the interpolated signal at lti. We would like to point out that we round up lti, so that both the transformed and the reference curve correspond sample-wise, where the sampling of the reference curve has always a period of one millisecond. It is logical to assume that similar ERP signals correspond- ing to the same brain function may manifest, perhaps substantial differences in the amplitude, namely in their y-value. To deal with these differences, we perform scaling along the y-axis. This transformation applied to a signal x(t) yields signal ax(t). The joined action of both these transformations to a signal x(t) yields signal ax(lt). We assume that all the aforementioned transformations are applied to each restricted signal RX k;j, thus obtaining a transformed version of it; we will employ the same symbol RX k;j for this transformed version of the restricted signals. Now, we will compare the approach introduced in the present work with the previous methods (e.g. [3,14,15]). Indeed, in the previous published work, one usually proceeds as follows:

(a) One defines four time intervals in the domain (100,400], namely, the I50 = [130, 180] ms, I100 = [170, 250] ms, I200a = [250, 350] ms and I200b = [280, 400] ms.

(b) One evaluates the maximum of RX k;j in the interval I50; its value is frequently denoted by P50 and the point where the maximum occurs by T50.

(c) One evaluates the minimum of RX k;j in the interval I100, obtaining its value N100 and its position T100.

(d) One evaluates the minimum of RX k;j in the interval I200a, obtaining value N200a and its position T200a.

(e) One evaluates the maximum of RX k;j in the interval I200b, getting value P200b and its position T200b.

(f) One performs statistical tests for comparing the peaks amplitudes, among subjects, for each electrode separately and

(g) One also performs statistical tests for comparing the peaks positions, among subjects, for each electrode separately. In the present work, before applying statistical tests, we classify signals RX k;j into different groups, according to the signals similarity. This similarity is judged as it will be shown in step 3 below. Subsequently, using the restricted signals of each group, we evaluate a smoother curve, which optimally represents the signals of the group in hand; we will employ for this smooth curve the term ‘‘ideal representative’’ of the group. Then, statistical hypothesis and associated tests are performed on these ideal representatives.

4.3. Step 3 – Defining a proper fitting error e Suppose that a signal x(t) is subject to the transformations described in step 2 and another one, y(t), is considered to be the reference curve. We assume that signal y(t) extends from time instant t1 to t2, where both t1 and t2 are considered fixed; we stress that the independent variable t of the signal y(t) takesvalues in the time interval [t1, t2] and it is not subjected to scaling. On the contrary, the argument of signal x lt ðÞ indicates that we associate the value of signal y(t) with the value of signal x(t), where t = lt, for a certain scaling factor l. Therefore, one may compare signals y(t) and the transformed x(t) by introducing and using the following fitting error e:

When the signals are digital, then the integral is trans-formed to summation.

4.4. Step 4 – Determination of the optimal transformations’ parameters

We optimally fit curves y(t) and the transformed x(t), by evaluating the scaling factors l* and a*, which minimize the aforementioned error function e(a, l).

The optimal time and magnitude scaling factors are obtained by the analytic solution of the error function’s gradient when set equal to zero. Their analytic determination is given in Appendix A. Using the definition of the internal product of two signals in a common domain, we let hx; yi _R t2 t1xðtÞyðtÞdt be the internal product of signals x(t) and y(t) under the transformation (3) that corresponds [t1, t2] to [t1, t2]. Then the optimal scaling factors computed in Appendix A are given by the following stationary conditions:

where evidently, x0ðtÞ ¼ d dt xðtÞ, [t1, t2] is the domain of signal x (t) that corresponds to the given domain [t1, t2] of the reference signal y(t). These formulas offer all pairs of time, l*, and magnitude, a*, scaling factors that render the fitting error e stationary for each selected pair of (t1, t1). Actually, for l* < 1, t1 = 0 and t1 2 [0, t2 _ t2 + t1], while for l* > 1, t1 = 0 and t1 2 [0, t2 _ t2 + t1]. For each such l*, we obtain the corresponding optimal a*, through (5), and only when a* > 0, we calculate the fitting error e, since a* < 0 yields mirroring of the signal, which is not an acceptable transformation for the ERP signals. The pair (t1, t1), which satisfies the aforementioned criteria and offers the minimal e, simultaneously determines the optimal relative placement and transformation of a signal x(t), so as to fit the reference signal y(t). This version of x(t), together with the corresponding fitting error, are used in the subsequent classification process in order to obtain groups of signals, which are congruent modulo amplitude and time scaling.

We let any one of the digital curves RXk;j play the role of the reference curve as described previously. Moreover, consider all other sequences RX k;i for the same set X 2 {V, P} and the same electrode k. We transform and optimally fit all these curves to the reference signal, by the methods described in the previous analysis. It is logical to assume that the corresponding fitting error follows a Snedecor (F) distribution with (ny _ 1, ny _ 1) degrees of freedom, where ny is the number of points of the digital ERP curve yi(n), whose interpolation generated y(t); this assumption has not been rejected by the performed related Kolmogorov–Smirnoff test (a = 0.01). Suppose that two ERPs signals recorded from different subjects have been generated by two quite analogous brain functions. Then, it is logical to assume that these two ERP signals are noisy versions of the same ideal curve. In this case, one may expect that, statistically, their fitting error (no formula 1) will be pretty close to zero. Consequently, we adopt the hypothesis that the upper point eT of the 5% left tail of the above Snedecor distribution is a satisfactory upper bound for this error. Thus, we classify the ERP signals into groups, by letting two such signals belong to the same group if their fitting error is smaller than eT. Via application of this method, we have associated a group of corresponding data sequences to each reference curve RX k;j. We emphasize that, at this stage, these groups of signals may overlap; however, in the subsequent steps, we will generate disjoint groups of similar ERP signals by properly excluding intersections of the various groups generated in the present step.

4.5. Step 5 – Estimating a first representative of all signals of the same group

First, we evaluate an initial representative for the group containing the greatest number of optimally fit signals; we employ the adjective ‘‘initial’’ to point out that, later on, in the next step 6, we obtain a better estimation of the representa- tive of each group of signals. The initial representative is a curve derived from the averaging of all curves that belong to the group of optimally fit signals. This curve is a smoother version of every curve of the group. In other words, it is a first, less noisy estimation of the curve which represents the common behavior of the signals of the group. We use the symbol YXk;1 for the reference ERP signal that generated this group, where, superscript X 2 {V, P}, subscript k is the electrode number and subscript 1 stands for the group’s cardinal number. For the ensemble of curves, optimally fit to YXk;1, we employ the symbol XX k;m;1, where the additional subscript m indicates the corresponding transformed curve. At this point, for each sample in the interval (100,400] ms, we average the values of XX k;m;1 and YXk;1 simultaneously. The sequence of the averaged curves at each time instant in the interval (100,400] ms defines a mean curve denoted by MX k;1. Taking into consideration the hypothesis we made that the members of this group manifest common brain behavior, this averaging process leads to a reduction of the overall erratic noise. In this way, one may claim, for example, that the digital sequence MV k;1 better represents the causal underlying process of valid reasoning for all members of the considered group (see Fig. 2).

4.6. Step 6 – Obtaining the final classification of subjects’ ERPs into groups and their final ideal representative

Next, we apply the previous fitting process to all RV k;j, using MV k;1 as the prototype curve, thus obtaining a more ‘‘safe’’ group of well-fit curves, GVk;1. We repeat this process, for all elements of group GV k;1. In other words, if an ERP curve RV k;j belongs to GV k;1, we let RV k;j play the role of the prototype curve and we optimally fit all other ‘‘valid reasoning’’ ERPs to it; in this way, we obtain a group of optimally fit ERPs, say GV k;1;j. We use this group to obtain another version of the ideal representative MVk;1;j of this group. We fit all curves RV k;j to MV k;m;1, thus obtaining another version of group GV k;1;j. In practice, groups GV k;m;1 coincide with GV k;1, a fact that indicates an internal consistency of the introduced approach. In any case, we redefine GV k;1 as the intersection of all groups GV k;m;1 to avoid any ambiguity in the definition of the corresponding group. We attribute to the final group GV k;1, an ideal representative PV k;1 of all its elements; in fact, we let PV k;1 be the average curve MV k;m;1 to which all other elements – ERPs of GV k;1 have optimally matched with a smaller fitting error per sample, as described in steps 1, 2 and 3. Next, from the initial group EV k;0, we exclude all of them that belong to GV k;1, thus obtaining a proper subgroup EV k;1 ¼ EV k;0_GV k;1. We apply to all ERPs of this restricted group EV k;1 the same procedure described above. Therefore, we may end up with another group GV k;2 of ERPs, having a signal, say PV k;2, as their common ideal representative. Excluding the curves of GV k;2 from EV k;1, we obtain a proper subset of ERPs EV k;2 ¼ EV k;1_GV k;2; application of the introduced methodology may offer a new group GV k;3 of really well-fit ERPs having PV k;3 as their common ideal representative. We continue this process until all elements of EV k;0 are exhausted or until no further ERPs’ grouping is acceptable. At the same time, an ideal representa- tive PV k;i is associated with each obtained group GVk;i. For example, concerning valid reasoning and in association with electrode k = 1, the aforementioned method offered a first group of ERPs GV 1;1, consisting of twelve (12) curves; after excluding these signals from EV 1;0, we have obtained EV1;1 consisting of thirty-three (33) ERPs from which a group GV1;2 of eight (8) well-fit ERPs has been extracted; then, an equi-numerous group GV 1;3 of eight (8) elements has been derived from subset EV 1;2; then, a disjoined group GV 1;4 of six (6) curves has been obtained; subsequently another distinct group of five (5) and finally a group of three (3) ERPs have been obtained, while three (3) ERPs have not been grouped. Evidently, each group GV 1;i, i = 1, 2, . . ., 6 has had an ideal representative PV 1;i, i = 1, 2, . . ., 6, estimated as described previously; these ideal representatives when compared in a pair-wise manner by means of the previous method, proved to be dissimilar. For consistency of the developed method, we let each one of the three unclassified ERPs be the ideal representative of itself. The very same process has been repeated for all other electrodes concerning ‘‘valid reasoning’’, thus obtaining groups GV2;i; GV 3;i; . . .; GV30;i as well as their corresponding ideal represen-tatives PV 2;i; PV 3;i; . . .; PV30;i. Next, also concerning ‘‘paradox reasoning’’ ERPs RP k;i, following the same procedure, the electrode groups GP k;i, and their ideal representatives PP k;i have been obtained for each electrode separately.

- Conclusions

6.1. The main results of the present work and their interpretation

In the present study, we have first established an experimental setup and proper questionnaires in order to capture the ERP signals associated with each subject’s valid and paradox reasoning. Next, we have classified these ERPs into groups by introducing a novel methodology including the following sub-procedures:

(a) time domain and amplitude scaling of the digitized ERP curves

(b) minimization of a properly chosen error function to optimally determine the corresponding scaling factors

(c) grouping the ERP signals according to the value of their fitting error, per electrode, and per subject

(d) evaluating a reliable representative, called ‘‘ideal’’ repre-sentative, for each such group separately

(e) and performing and testing corresponding statistical hypotheses.

Application of this methodology indicated statistically significant differences of the ideal representatives in 5 electrodes with 99% level of confidence and 14 electrodes, with 95% level of confidence. The introduced approach manifested that valid reasoning ERPs optimally fit the corresponding ideal representatives with very low error per group. The same also holds true for each group of the paradox reasoning ERPs. These results support the authors’ assumption about an underlying common mental behavior of the subjects of each such group. In addition, it seems that the ideal representative attributed to each group well expresses the underlying common brain activity, as Figs. 2–5 indicate. At the same time, the ideal representatives manifest significant statistical differences for a considerable number of electrodes per subject; this result underpins the hypothesis that corresponding differences exist in the underlying mental processes of ‘‘valid syllogism’’ and ‘‘paradox reasoning’’ (see Fig. 3). Considering the obtained results from a psychophysio-logical point of view, we have demonstrated that two different arguments engage different brain regions involving an anterior network (prefrontal leads), as well as a rather widespread posterior network (parieto-temporal and occipital leads), suggesting that these dissociations are due to the type of deductive argument (deductive vs. paradoxes). Such a notion appears to be compatible with studies focusing on the anatomy of reasoning [9]. However, it is also possible that the observed patterns of activity, point towards a fractionated system, dynamically configured in response to specific task and environmental cues, consistent with the assumption provided by recent neuroimaging research [17].

6.2. Future considerations

Thus, future research will aim at (a) a more precise determination of these causal behavioral functions, (b) the probable relation of the ERPs valid and paradox ideal representatives with each subject’s mental state and (c) extension of the introduced methodology to include other mental functions, too, such as the invalid syllogisms and illusions.