Participants
The database was composed by 19 young right-handed subjects (10 males and 9 females), with a mean age of 24.2 years (standard deviation 3.3). One anomalous register was excluded from the final analysis. All participants were recruited during the months of March and May 2019 from the University of xxx (Spain) and received academic credit for their participation. Participants did not report any significant neurological or psychopathological conditions, or any psychoactive drug intake during EEG recordings. Each participant went through two experimental tasks sequentially. First, they performed a LOGICALLY INVALID DEDUCTION paradigm task; afterwards they performed a LOGICALLY VALID DEDUCTION paradigm task. The responding hand for each condition was counterbalanced across subjects. All participants signed an informed consent form before their participation in this study, following the guidelines of the Declaration of Helsinki. The project was approved by the University of xxx and received the approval of the Ethics Committee (the code of ethics for research is 0-181, dated 11-06-2019). After the pandemic stop, raw data have been analyzed in a trial-by-trial approach at the University of xxx (Spain).
Stimuli
The items in the study were trios of cards from the game SET (Set Enterprise, 2019). Each card has a variation of the following four features: figure (diamond, ovoid, or squiggle), color (green, red, or black), cardinality (1 or 2), filling (filled or empty). None of the participants was familiar with the game or its rules. The experiment included the same 200 randomly ordered trials in both conditions. Presenting the same stimuli ensures that the relational complexity of both conditions is exactly the same; that is, they include the same set of cards and are described with the same nomenclature and lexical card descriptions. On the other hand, instructions for the valid condition (SET definition and logical constants) ensure that the valid task has a measurable logical complexity.
Any given trio of cards either does or does not share the same relevant features: figure, color, number, and filling. Figure 1 displays examples of trios which do not share any features (case 1), share one feature (case 2), share two features (case 3), or share three features (case 4).
Time chart
The beginning of the trial was signaled by a cross (+) presented in the center of the screen for 300 ms, which was then followed by the appearance of the items on the screen for 3500 ms. Then, the items disappeared, and the central dot reappeared for 450 ms. Participants were asked to respond quickly (within 3000 ms). The time chart is presented in Fig. 2.
Experimental Design Paradigm Task
The experiment contrasts two inferential tasks which contain the same stimuli (i.e. the same relational variables with the same content and properties). The valid task includes explicit deductive rules as instructions, while the invalid task has no logically valid rules to follow.
In the invalid task, the subject does not receive any specific deductive rule; instead, they are shown a set of visual stimuli (SET game cards) and informed about the cards’ features (figure, color, number, and filling). The instructions for the invalid task were: “If an item follows a rule based on color, figure, number, or filling, press the ‘Ctrl’ key, otherwise press the space-bar”. The invalid task is deductive in the sense that no additional information to the contents given in the premisses (SET features) are used to infer conclusions. Examples of “uncertain deduction” are common in the literature, such as (Evans et al., 2015).
In the valid task, subjects must validly deduce their answer given the definition of what makes up a SET after being shown a trio of cards (i.e. an item). The logical properties of a SET allow one to determine by deduction if any given trio is (or is not) a SET, exclusively applying tools from propositional elementary logic. Any given trio is a SET if all the cards have two or more properties in common. The deductive instruction is: “Press the ‘Ctrl’ key if the presented trio is a SET, otherwise press the space-bar”.
The deduction of the answer (is a SET / is not a SET) is stated without any previous training. It is an integrable inference in all trials and crucially depends on the definition of SET, hence excluding any non-deductive heuristics. It is essential to recognize that the valid task, even if it is simple, has a non-null logical complexity. In summary, the experimental design presents two tasks with the same relational complexity (item’s features) but distinct logical complexity (logical operators). The design does not allow the researchers to describe the precise inference pattern followed by any subject in any trial. For cases (3) and (4) (see Fig. 1), positive propositional inferences are enough, particularly connectives and the Modus Ponens rule (deduce B from {A, if A then B}). Cases (1) and (2) can be negatively treated with connectives and the Modus Tollens rule (deduce not A from {not B, if A then B}). The point of the experiment is not to follow the neural processing of a specific pattern, but to study any deductively valid inference. This task is ecological and user-friendly since it is inspired and presented as a game.
The experiment was programmed and administered using E-PRIME software. The screen has a sampling rate of 60 Hz, and a resolution of 1024 × 768 pixels. Items were presented against a black background. In both tasks, the index fingers of both hands pressed the keys on a computer keyboard to answer. Participants were sitting 60 centimeters in front of the screen in a quiet dimly lit environment.
Recording and preprocessing of EEG signals
The EEG was recorded with a 64-channel amplifier (Neuronic System, Cuba) and specific acquisition software (Neuronic EEG/Edition EEG Software) with a sampling rate of 200 Hz. Reference electrodes were placed on the earlobes. In addition, electrooculography (EOG) signals were acquired using three pairs of sensors in order to acquire the horizontal and vertical movement of the eyes. Electrode impedance was kept below 5 kΩ. Extracephalic channels were removed for the subsequent analyses thus keeping 58 EEG channels according to 10–10 system: : FP1, FP2, F3, F4, C3, C4, P3, P4, O1, O2, F7, F8, T3, T4, T5, T6, FZ, CZ, PZ, F1, F2, P1, P2, AF3, AF4, P5, P6, FC5, FC6, C5, C6, TP7, TP8, PO7, PO8, FPZ, FCZ, CPZ, POZ, OZ, PO3, PO4, CP1, CP2, CP3, CP4, C1, C2, F5, F6, FC3, FC4, FC1, FC2, CP5, CP6, TP9, and TP10. Furthermore, EEG channels were grouped in 13 regions of interest (ROIs) according to Table 1.
Table 1
Correspondence between ROIs and EEG channels
Channels
|
ROI
|
FP1, FP2, AF3, AF4, FPz
|
Prefrontal
|
F4, F8, FC6, F6, FC4
|
Right frontal
|
F3, F7, FC3, F5, FC5
|
Left frontal
|
Fz, F1, F2, FCz, FC1, FC2
|
Medial frontal
|
T4, T6, TP8, TP10
|
Right temporal
|
T3, T5, TP7, TP9
|
Left temporal
|
C4, C6, CP4, CP6
|
Right central
|
C3, C5, CP3, CP5
|
Left central
|
Cz, CPz, CP1, CP2, C1, C2
|
Medial central
|
P4, P6, PO8, PO4
|
Right parietal
|
P3, P5, PO7, PO3
|
Left parietal
|
Pz, P1, P2, POz
|
Medial parietal
|
O1, O2, Oz
|
Occipital
|
The preprocessing consisted on 4 steps: (i) application of bandpass (1–70 Hz) and notch (49.8–50.2 Hz) Finite Impulse Response (FIR) filters with a Hamming window to limit noise bandwidth and to remove powerline noise, respectively; (ii) artifact rejection by means of independent component analysis (with special care to remove eye-derived artifacts, see Figures S1 and S2 in Supplementary Material); (iii) selection of 1.5-second useful trials; and (iv) thresholding to remove noisy trials (Gomez-Pilar et al., 2015). The useful trial selection consisted of localizing a stimulus followed by a correct response and another stimulus, thus discarding stimulus with more (or less) than one response. The trial length was 1.5 seconds comprising two intervals: 0.5 seconds before the stimulus, acting as baseline, and 1 second after the stimulus. Furthermore, the trials whose response time (i.e. time elapsed between the stimulus and the response) was less than 1 second were discarded to minimize the influence of the motor responses in the event-related potentials (ERPs).
Evoked potentials: synchronized averaging of trials
Firstly, the evoked potentials were analyzed to study the electrophysiological response of both experimental deductive conditions (valid and invalid). For this task, all the trials of each condition were averaged to study P3, N4, and P6 components, as they are related with deductive processing (premise integration in P300, semantic analysis in N400 and second processing in P600). Furthermore, the topographic distribution of the potentials was also analyzed.
Time-frequency analyses: trial-by-trial approach
The properties of EEG signals are not stationary but they vary over time (Miñambres et al., 1996). Thus, methods as Fourier transform that require stationarity should not be used to analyze the time-varying properties of ERPs. In line with that, continuous wavelet transform (CWT) provides a framework to study the dynamical properties of the frequency content of the ERP signals. A wavelet is a function with zero-mean with a localization in time and frequency (Torrence & Compo, 1998). We have chosen as mother wavelet the complex Morlet, as it has been proven to provide a good fit with biological data (Gomez-Pilar et al., 2015). To carry out the analysis, the complex Morlet is dilated and translated to generate a wavelet family that is able to capture the fast time-varying properties of the signal with a high frequency resolution (Mallat, 2009). The dilation factor sweeps from 1 Hz to 70 Hz with equally spaced intervals of 0.5 Hz (Gomez-Pilar et al., 2015). Wavelet analysis offers a solution to analyze signals with a high temporal resolution while keeping the frequential resolution also high (Mallat, 2009). It is possible because of its variable time-frequency resolution, with shorter time windows used for higher frequencies and longer windows used for lower frequencies. Deeper insights about wavelets can be found in (Mallat, 2009).
From the wavelet decomposition obtained after applying the previous analysis, the wavelet scalogram was computed for each ERP trial. It summarizes the energy distribution of the wavelet in the time-frequency plane. This time-frequency representation of the energy of each trial can be used to identify the spectral content associated to specific frequency ranges. In this study, we have considered the conventional EEG frequency bands: delta (δ, 1–4 Hz), theta (θ, 4–8 Hz), alpha (α, 8–13 Hz), beta-1 (β1, 13–19 Hz), beta-2 (β2, 19–30 Hz), and gamma (γ, 30–70 Hz).
Statistical analysis
Data distributions were tested in an exploratory analysis to assess normality and homoscedasticity. The first one was tested using Lilliefors test, while the latter with Bartlett test. Response time distributions, peak amplitude distributions, ERPs and wavelets did not meet the normality and homoscedasticity hypotheses; therefore, Wilcoxon signed rank tests were used to assess differences between valid and invalid conditions. Furthermore, to assess the correlation between the time courses of the evoked potentials, Spearman tests were employed, as this method can detect both linear and nonlinear correlations.
Data availability
Scripts to calculate wavelets including raw and preprocessed data are available in:
xxx (2022), “Invalid and Valid Deductive Processes”, Mendeley Data, V2, doi: 10.17632/w95n6rc6fs.2