Antonino Drago

Often, in the original scientific writings, a double negated statement (DNS) is not equivalent to his corresponding positive one; that means the inferring law 'non non A -> A' does not apply. Recent studies recognized in the failure of this logical law the borderline between classical and non-classical logics. Original writings by classical chemists dealing with the problem of atomism are particularly characterized by the occurrences of DNSs. An historical case, Avogadro's contribution to atomism (i.e. the well-known hypothesis about the constitution of gases), is here analyzed in such terms. It turns out that, in order to support his ideas, Avogadro suggested several ad absurdum proofs, indeed a way of reasoning typically linked to the use of DNSs

Motivated by the alarming recurrence of wars, Lanza del Vasto became Gandhi’s disciple. From a social interpretation of biblical texts he derived a characterization of non-violence as the conversion from not only personal negative drives, but also from the influences of negative social institutions and the entire civilization. His intellectual categories comprise an ethical conception of four essentially different models of development. This pluralism subsists inasmuch as the representatives of each model do not search the destruction or the suppression of the adversaries, but reconciliation through non-violent means.

Whereas “Science does not think" (Heidegger), the "new historiography of science" - mainly Koyré's and Kuhn's ones - has addressed our minds to think about science in a new way. Recently Heelan suggested taking this new viewpoint for conceiving hermeneutic method in a more adequate way to present scientific practice. By an interpretative analysis of the categories of the above two historians, the present paper suggests a new way to conceive the foundations of science. Two basic dichotomies result: on the kind of logic and on the kind of mathematics. They generate four models of scientific theory, sharply severed by the radical differences in their respective choices; that offers an accurate definition of incommensurability and even of an alternative scientific theory to a dominant one, which can be properly called a paradigm. In the past hermeneutic scholars gave negative appraisals on Western science; according to the new viewpoint recognising pluralism in the foundations of science, these appraisals concern the dominant paradigm only. In the light of the basic dichotomies a new way to define hermeneutics is suggested; it can be qualified in short by the following words: ‘The understanding of science is the science of understanding’.

I introduce a new method for investigating those original textswhich illustrate a theory which is aimed at solving a general problem. Themethod comprises two investigations. The first investigation is aimed atrecognising, within the original texts, all the occurrences of double negationsand ad absurdum arguments. These occurrences mean that the author argueswithin non-classical logic, his logical thread develops through a chain of doublenegated sentences which compose units of argument, each ending with an adabsurdum argument. More generally, the theory is organised in an alternativeway to the classical deductive organisation. The second investigation takes intoaccount the recent formalization of a hierarchy of degrees of infinity, through ahierarchy of types of mathematics. The mathematics of almost the least degreeof actual infinity relies upon the use of one only quantifier. Hence, the secondinvestigation within the original text is aimed at recognising the occurrences ofquantifiers, as an evidence of the introduction of this degree of actual infinity.I apply this method to the first chapter of De Docta Ignorantia byCusanus, a scholar belonging to the school of “negative thinking”. I obtainevidence showing an exceptional capability of innovating his method ofarguing, through a clever use of both double negated sentences and adabsurdum arguments. But he was unsuccessful in organizing the entire theoryin this way, because he mixed logical arguments with theological suggestions.Moreover, from the direct reading of his text one may surmise that his appeal tothe intellectualis visus refers to geometrical intuition, which is the basis of amathematics involving the lower degree of actual infinity. Instead, Cusanuscorrectly introduced the basic notion, i.e. the infinitesimal, of an higher degreeof mathematics; which, two centuries after him, was intuitively introduced inmathematics and in the last century was eventually defined in a formal way.As a general appraisal, Cusanus’ effort to obtain knowledge of divinesubjects appears much more consistent in its use of non-classical logic than inits use of mathematical objects, although, in the past, it was the latter thatbrought him celebrity. However, although anticipating methods and notionswhich were to be formalised by subsequent science, his method, to transferre mathematical properties to divine properties, is largely an allusive method.In conclusion, the application of the above-mentioned method of analysisto a first part of Cusanus’ De Docta Ignorantia was successfully inasmuch as itwas capable to circumscribe both the logical content and the mathematicalcontent of this text

I will present paraconsistent logic in three different contexts: i) the experimental context, in which a statement is considered as concerning an experimental datum ; ii) the theoretical context, in which a statement is considered in a provisional way, as a candidate for playing inside a theoretical framework the role of a principle; iii) the foundational context, where a same statement leads to compare two different semantics according to two different formalizations of a same theory; in other terms, when these formalizations are mutually incommensurable. The case-study of the original Goe-del's proof is analysed as an instance of the last use of paraconsistent logic