Ben U., List: It depends on what exactly we mean by "mathematics of logic." Peirce used that specific term only once in CP/EP (1.185/2:259, 1903) and twice in NEM (1:256, no date; 3:333, 1903). In all three cases, he was referring to the very first branch of the very first science, Mathematics--i.e., what he elsewhere called "mathematical logic" or "formal logic," distinguishing this as "the science which *draws *necessary conclusions" from deductive logic as "the science of *drawing *necessary conclusions" (CP 4.239-240; 1902). To me, that seems directly parallel to the distinction that he drew elsewhere between *logica utens* and *logica docens*.
CSP: But mathematics performs its reasonings by a *logica utens* which it develops for itself, and has no need of any appeal to a *logica docens*; for no disputes about reasoning arise in mathematics which need to be submitted to the principles of the philosophy of thought for decision. (CP 1.417; c. 1896) I take "the philosophy of thought" in this context to mean what Peirce later identified as the Normative Science of "logic proper" or "formal semeiotic." Regards, Jon S. On Tue, Sep 3, 2019 at 8:41 PM Ben Udell <[email protected]> wrote: > Jon A.S., list, > > I agree, Peirce doesn't say or imply that there is or ought to be a > discipline of formal semiotic between phaneroscopy and the normative areas > esthetics, ethics, logic. > > However, mathematics of logic in Peirce's sense seems *logica docens*, > not as you call it *logica utens*. *Docens*, because it's explicit and > theoretical, not to mention rather formal. > > Best, Ben > On 9/3/2019 9:31 PM, Jon Alan Schmidt wrote: > > Ben U., List: > > Thanks for this correction; I need to remember to include NEM whenever I > undertake terminological searches in the future. Fortunately for me, these > earlier passages (1902) are consistent with what I have been saying all > along--Formal Semeiotic is simply how Peirce defined Logic, which is the > third branch of Normative Science in his 1903 classification. It certainly > *does > not* fall under Phenomenology. > > No one is claiming that "the mathematics of logic" or "mathematical logic" > is limited to *deductive *logic. It is simply the *logica utens* that is > required for *every *science, including the other branches of > Mathematics, Phenomenology, Esthetics, and Ethics. Logic proper--Formal > Semeiotic--provides a *logica docens* by studying the process of > reasoning (semeiosis), including the relation between signs and the end of > truth, which is why it is a *Normative *Science. > > Regards, > > Jon Alan Schmidt - Olathe, Kansas, USA > Professional Engineer, Amateur Philosopher, Lutheran Layman > www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt > > On Tue, Sep 3, 2019 at 7:58 PM Ben Udell <[email protected]> wrote: > >> Jon A.S., John F.S., list, >> >> On 9/3/2019 1:53 PM, Jon Alan Schmidt wrote: >> >> JAS: [...] there is no passage whatsoever where he employed the term >> "Formal Semeiotic," [....] >> >> *Au contraire* (variant spellings notwithstanding), >> >> http://www.iupui.edu/~arisbe/menu/library/bycsp/L75/ver1/l75v1-05.htm >> >> QUOTE: >> Final Version of the Carnegie application - MS L75.363-364 >> MEMOIR 12 >> ON THE DEFINITION OF LOGIC >> >> Logic will here be defined as formal semiotic. A definition of a sign >> will be given which no more refers to human thought than does the >> definition of a line as the place which a particle occupies, part by part, >> during a lapse of time. Namely, a sign is something, A, which brings >> something, B, its interpretant sign determined or created by it, into the >> same sort of correspondence with something, C, its object, as that in which >> itself stands to C. It is from this definition, together with a definition >> of "formal", that I deduce mathematically the principles of logic. I also >> make a historical review of all the definitions and conceptions of logic >> and show not merely that my definition is no novelty, but that my >> non-psychological conception of logic has virtually been quite generally >> held, though not generally recognized. >> >> From Draft D - MS L75.235-237 >> I define logic very broadly as the study of the formal laws of signs, or >> formal semiotic. I define a sign as something, A, which brings something, >> B, its interpretant, into the same sort of correspondence with something, >> C, its object, as that in which itself stands to C. [....] >> END QUOTE >> >> These passages are in New Elements of Mathematics, which includes a >> passage (absent from Joe Ransdell's version of the Carnegie application) >> that Jon Awbrey likes to quote (e.g., he put it into the Peirce Wikipedia >> article on Peirce): >> >> No. 12. *On the Definition of Logic* [Earlier Draft] >> >> QUOTE: >> Logic is *formal semiotic.* A sign is something, A, which brings >> something, B, its *interpretant* sign, determined or created by it, into >> the same sort of correspondence (or a lower implied sort) with something, >> C, its object, as that in which itself stands to C. This definition no >> more involves any reference to human thought than does the definition of a >> line as the place within which a particle lies during a lapse of time. It >> is from this definition that I deduce the principles of logic by >> mathematical reasoning, and by mathematical reasoning that, I aver, will >> support criticism of Weierstrassian severity, and that is perfectly >> evident. The word “formal” in the definition is also defined. (NEM 4, 54). >> END QUOTE. >> >> As regards mathematics of logic, it's been unclear to me just what it >> consists of. It's not enough to say, it's all and only the deductive >> logic. There is deductive math applied in philosophy, according to Peirce, >> e.g. applied as the doctrine of chances (probability theory). Are the >> existential graphs logic applied in philosophy, or are they in Peirce's >> first part of math, called mathematics of logic? Is maths of logic just an >> algebra of two values *v, f,* that could stand for Caesar, Pompey, >> (Peirce said something like that), just as well as for true (*verum*) >> and false? There's a passage about that where Peirce goes on to discuss >> triadic mathematics, which I didn't understand, I'm no mathematician. There >> is to keep in mind is that the mathematics of logic is not necessarily >> fully the selfsame thing as the logic of mathematics. Peirce often >> discusses how mathematics USES diagrammatic reasoning, but usually says >> that mathematics needs, and anyway has taken, no help FROM logic except in >> a few cases, involving infinities if I recall aright. >> >> Best, Ben >> >
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