# br The primary aim of this study is to

The primary aim of this study is to use a set of paraconsistent al-gorithms, denoted as SPA-PAL2v, to distinguish the Raman spectra of premalignant and malignant cutaneous lesions, from the no-tumor sites.

The first application of PAL2v algorithms in 6-NBDG cancer diagnosis was presented in [5], in which discrimination analysis was conducted using a database with 146 samples, comprising patterns of normal skin tissues, basal cell carcinoma (BCC), and melanoma skin tissues. In this work, we used a larger and more complex Raman intensity database, which required greater complexity in structuring the paraconsistent algorithms with diﬀerent matrix arrangements for analysis.

In [19], Raman spectroscopy and discrimination techniques with statistical methods were used to discriminate against non-melanoma skin lesions of human tumor tissues in vivo with an arrangement of groupings similar to those used in this work. Although the database and numbers of samples in [19] are diﬀerent from those of this work and the Raman spectra was obtained ex vivo, the arrangement of the histo-pathological groups used in [19] was replicated in this work, and thus, only the results based on this aspect were used for comparison.

In relation to a previous work [5], we used a more robust para-consistent computational framework that applies the algorithms based on PAL2v to discriminate non-melanoma skin lesions from normal human skin tissues in a database with several ex vivo-obtained Raman spectra samples.

1.4. Paraconsistent annotated logic

PL belongs to the family of non-classical logics, and its main char-acteristic is to accept the contradiction without invalidating the logical conclusions [23,24].

Paraconsistent annotated logic is a type of PL that can be re-presented in a particular way, through a lattice FOUR (Hasse diagram), shown in Fig. 1(A). In an intuitive way, the annotation constants re-presented in their vertices will provide connotations of extreme logical states to propositions [26].

An extended form of PL called paraconsistent annotated logic with annotation of two values (PAL2v) uses the annotation composed of two degrees of evidence [24–26]. In this case, the degrees of favorable evidence (μ) and unfavorable evidence (λ) comprise an annotation that gives a logical connotation to the proposition P [5,25–27]. The μ and λ degrees are normalized values that belong to the set of real numbers ℜ, and by linear transformations in a unitary square on the Cartesian plane (USCP) [26] and the representative PAL2v lattice τ, we can obtain the equations of certainty and contradiction degrees:

Where their values, which belong to the set of ℜ numbers, are in the closed interval of +1 to −1.

A representation of grana is shown in Fig. 1(B).

In the PAL2v notation [26], the two values of Eqs. (1) and (2) re-present a Paraconsistent logical state ετ

Dct = the degree of contradiction computed in functions of the two degrees of evidence, μ and λ.

In the associated PAL2v lattice, we can calculate the distance d between the paraconsistent logical state ετ and one of the horizontal vertices that represents the extreme logical states f (False) or t (true) [5,26,27]. Fig. 2 shows the geometric interpretations and the signals flux for a typical PAL2v application, where distance d is calculated as follows: d = (1 − Dc )2 + Dct2 The DCr real certainty degree value (DC value without the eﬀect of contradiction) is calculated using the fol-lowing conditionals [26]:

For decision making based on paraconsistent analyses, the values of the resulting degrees of evidence (μER) are compared [5,26,27]. In this study, the information analyzed by the algorithms of the PAL2v con-structed with these equations will be the data of the Raman spectrum converted into degrees of evidence of Raman intensity with values between 0 and 1.

The PAL2v equations and their interpretation in the associated lattice allow the creation of algorithms for direct applications. To transform Raman Spectroscopy data into signals that will comprise the values of the paraconsistent pattern, three algorithms of PAL2v are required. The first algorithm is named extractor of degree of evidence [27]. This algorithm calculates the value of the evidence degree of the quantity measured through a mathematical function considered in a discourse universe, or interest interval. Depending on the application, a straight line equation or other adequate equations can be used. The extractor of degree of evidence algorithm is described below [5,26,27]:

1 Enter the maximum (Maxvalue) and minimum ((Minvalue) limit values of the greatness (in its unit of measurement) to form the discourse