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Atomism and the reasoning by a Non-Classical Logic

Antonino Drago
Group of History of Physics - Dept. of Phys. Sciences, University of Naples

Romina Oliva
Dept. of Chemistry, University of Naples

Abstract
More than a century ago, classical logic was formalized in mathematical terms. Next, some "deviant" mathematical logic's were born, whose status stood for a long time dubious. Finally, some decades ago mathematical logic was recognized not to be unique, as generally believed. At a formal level what discriminates the classical logic from the non-classical ones (say absolute, positive, minimal and intuitionist logic's), is the axiom A ->A (read: "not-not-A implies A", where A is any statement), which holds true for classical logic, whilst it fails for non-classical ones (1). In other words, the basic difference between classical and non-classical logics may be confined to the law of double negation: if classical logic is applied, a double negated sentence implies its related positive, otherwise we are in the presence of what we call a DNS, i.e. a double negated sentence, which (and in that is the fundamental point) is not equivalent to the related positive one. By an historical investigation one may recognize the theoretical origin of these DNSs in Leibniz' principle of sufficient reason; even the sentence of this principle is a DNS: "… rien n'est sans raison, ou que toute verité a sa preuve a priori tirée de la notions des termes, quoyqu'il ne soit pas toujours en notre pouvoir de parvenir à cette analyze" (2); where in a first time Leibniz states the principle as a A, then he tries to draw the corresponding A, yet he has to recognize it impossible in most cases.
Let us remark that the failure of the double negation logical law may be plainly manifested by a double negation of a single statement, really a common linguistic scheme -'litote'- expressing the thought of a layman; whereas, for instance, the failure of the excluded middle law - -n formal terms AÚØA (read: A or not A) - has to be deliberately emphasized by an author through a whole argument. Thus the occurrence of such DNSs in a scientific text may point out that the writer is not applying the double negation law since he is arguing by a non-classical logic. An accurate analysis of the original writings of a scientist can recognize his employing of a logical arguing belonging to a non-classical logic. Of course, we don't claim that people deliberately argued by means of a non-classical logic before it was recognized and properly formalized. No doubt each of these scientists overtly committed himself to classical logic (it was simply 'the logic' for what was manageable by the scientific reason of the time, i.e. syllogistic and then propositional sentences). Rather, we suggest that in the past some scientists were motivated by some uncommon situations - we will define in the following -, for searching new ways of arguing, and therefore they made an ante litteram use of non-classical logic. Hence, if interested in investigating one's way of reasoning, we may begin by analyzing his writings to mark all recurring DNSs. Here we apply this research method to writings pertaining to 'hard' sciences. (All double negated sentences we will report in the following are DNSs, i.e. double negates sentences not reducible to the related positive ones.) In science a DNS generally points out a problem, i.e.: there are not sufficient experimental proofs to assert A, while, on the other side, several evidences and/or logical conclusions - often an ad absurdum argument (3) -, make apparent that 'it is not possible that A is not true'. At the same time a DNS offers a methodological principle inasmuch as it advises that the subsequent theory never will draw the sentence A; for instance the impossibility of a motion without an end prevents the theory from stating a perpetual motion.
In the following we begin by listing some instances of DNSs in science. Then we will focus our attention upon chemistry, in particular on the texts pertaining to its early historical phase, chemical atomism, which was developed in the first half of 19th century. Finally, we analyze in detail the study-case of Amedeo Avogadro, in order to get a deeper insight in the way he first formulated the well-known hypothesis about gases constitution; actually he made use of several DNSs and ad absurdum proofs. As a result we show that arguing by DNSs was a characteristic feature of classical chemistry.

Drago A and Oliva R.: Atomism and the reasoning by a Non-Classical Logic, Hyle, 1995


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