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  • From Time to IterabilityThe Synthetic Operativity of Traces in Logical Forms
  • Juan Manuel Garrido Wainer (bio)

I. Logical Form as Operative Form

In his theory of the syllogism, Aristotle uses letters of the alphabet as algebraic (or proto-algebraic) symbols to represent the subject- and the predicate-terms that appear in premises and conclusions. How do these letters function? If I represent the proposition "all Greeks are mortal" with the symbols "all G are M," the letters "G" and "M" are not abbreviations for the subject-term "Greeks" and the predicate-term "mortal," respectively. They help to reveal the logical form of the proposition "all Greeks are mortal." They do this by allowing arbitrary transformations of proposition's meaning. "All Greeks are mortal," "all men are animals," and "every A is B" correspond to the same logical form. Letters are meant to express the arbitrariness of syllogism's meaning.1 This arbitrariness is not obtained by conceptual abstractness. The letter "G" is neither the concept of the subject-term "Greeks" nor the concept [End Page 199] of subject-terms in general. The letter "G" is the subject-term itself, the function of which is determined not by its referent but by a system of rules for its use and combination with other letters and by some properties derived from the nonphonetic character of its inscription, such as its position in the proposition and in the syllogism. Even the word "all" (or variants such as "every," "for all," and the like), a whole word and not a letter or a symbol, functions as a nonphonetic operative mark, the use and meaning of which, in Aristotle's logic, are related to a system of rules that makes it different from the operator "some." Logical forms are operative devices.2 They have, so to speak, a generative efficiency. The proposition "all G are M," which in Aristotle's terminology is called "universal affirmative," represents a pattern or scheme for producing an infinite number of other universal affirmative propositions. And combining the different forms of propositions (universal affirmative, universal negative, particular affirmative, particular negative), we can produce sets of propositions called syllogisms.3

Under the heading of a "grammar of pure logic" or a "theory of pure forms of significations," Husserl studies what he calls the "fundamental forms" (Grundformen) at play in the construction of terms, judgments, and sets of judgments. Despite Husserl's characterization of such forms as generic universality (Gattungsallgemeinheit), they function as rules for the construction of logical objects. Logical forms are not only abstract forms that subordinate specified forms, as when "Sp is q" is subordinated to or contained in the generic predicative form "S is p." In "S is p," one should recognize the copulative pattern that produces the judgment "Sp is q" or any other predicative judgment, even "S is p" itself. "S is p" is a fundamental form in the sense that it refers to the copulative operation that produces predicative judgments. The "generic universality" of "S is p" must therefore be understood in a "wholly different sense," namely, as primitive form (Ur-form), that is, a form "from which one can derive particularizations and modifications," in sum, a rule for the construction of other predicative judgment (Husserl 1973 § 13 b, 51).4 Thus, the judgment "Sp is q" is a transformation operated by the copulative scheme "S is p," which takes as its substrate a whole assertion. "Finally one will be able to take the point of view of operation so broadly that one regards even the fundamental form'S is p' as an operation: the operation of determining a determinable substrate, S" (Husserl 2013, § 13 c, 52). Along [End Page 200] with copulation, logical connectives are also understood as schemes for producing judgments. In any composed judgments, like "no S is p," "S is p implies Q is r," "S is p and Q is r," or "S is p or Q is r," logical connectives act as fundamental forms. In sum, the fundamental forms are formation-forms (Bildungsformen) and fundamental kinds (Grundarten) of operations for the production of judgments and sets of judgments: "Such universal formation-forms as the...

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