Movatterモバイル変換

1. Peano’s arithmetic, Russell and Whitehead’s systems,Gentzen’s natural deductive systems, Hilbert’s programs,and Gödel’s incompleteness theorems are prime examples.

2. According to Peirce’s terminology, there are three kinds ofpredicates, “absolute terms”, “simple relativeterms”, and “conjugative terms” (DNLR [CP 3.63]).These correspond to monadic, dyadic, and ternary predicates, in modernterminology. Also, there is a controversy over “relatives”versus “relations”. Refer to Merrill 1990: 160–163.“I conclude that [Peirce’s] simple relative terms standfor dyadic relations” (1990: 162).

3. Benjamin Peirce placed mathematics before logic (see entry onBenjamin Peirce). Charles Peirce gave full credit to his father’s warning againstnot-so-mathematical philosophical reasoning for steering him away fromhis early ambition to combine philosophy, logic, and mathematics (CP1.560 c. 1905).

4. This occurs as early as his work,Formal Logic (De Morgan1847).

5. “De Morgan’s own outlook remaining essentiallysyllogistic, which …would liable to impede him, unfortunately,from definitively taking a radically novel point of view”(Hawkins 1995: 137).

6. “In his [De Morgan’s] effort to break the bonds oftraditional logic and to expand the limits of logical inquiry, hedirected his attention to the general concept of relations and fullyrecognized its significance. Nevertheless, De Morgan cannot beregarded as the creator of the modern theory of relations, since hedid not possess an adequate apparatus for treating the subject inwhich he was interested, and was apparently unable to create such anapparatus” (Tarski 1941: 73).

7. Emily Michael (1974) argues for Peirce’s independence, whileDaniel Merrill (1978) presents a more nuanced picture by tracking downthe correspondences between De Morgan and Peirce.

8. For more details, see Michael (1974) and Merrill (1978:257–266). The following is one of Peirce’s first examplesfor relational arguments:

Everyone loves him whom he treats kindly.
James treats John kindly.
Therefore, James loves John.

(Peirce 1866: 10 and quoted in Merrill 1978: 258)

9. George Boole’s work (1847) has been considered as the firstsystematic and ambitious project to represent logical concepts interms of algebra. For more details, see section 3 of the entry onGeorge Boole.

10. Houser correctly observes that “[w]hat is most evident in hiswork is the importance Peirce attached to his basic analysis ofrelations” (1997: 14).

11. Some relate the distinction between property and relation in terms toPeirce’s own distinction between corollarial and theorematicreasoning. That is, for monadic inference, simple corollarialreasoning is carried out while the logic of relation involves morecomplicated and ingenious theorematic reasoning (Hintikka 1980 andShin 1997).

12. What does “,” mean here? For a record, the followingcitation is in order:

Thus far, we have considered the multiplication of relative termsonly. Since our conception of multiplication is the application of arelation, we can only multiply absolute terms by considering them asrelatives. Now the absolute term “man” is really exactlyequivalent to the relative term “man that is —,” andso with any other. I shall write a comma after any absolute term toshow that it is so regarded as a relative term. (DNLR [CP 3.73])

This is interesting indeed, as Brady points out: “We see here aninteresting feature of Peirce’s 1870 paper. Peirce takes themultiplication of relative terms as primitive” (Brady 2000: 35).This interpretation is consistent with the citation from DNLR [CP3.68], where Peirce himself starts with the application of a relation,not the relation of a property. A slightly different, but compatible,interpretation was suggested by the reviewer: Peirce’s symbol“,” is an interesting operation which turns a propertyinto a relation.

13. “SomeAs areBs” is an existentialstatement. Peirce called them particular propositions. “As forparticular propositions, Boole could not accurately express them atall” (DNLR [CP 3.138]).

14. “These letters may be conceived to be finite in number orinnumerable” (1882a [CP 3.306]).

15. Even though the basic idea for ordered pairs existed in the 1870paper, we can see more clearly how it is combined with quantifiers andbound variables in the 1883a Note.

16. Using more familiar notation, \(l = \{\langle A, C\rangle, \langle B,D\rangle\}\).

17. The next section “Second-intentional logic” is the firstplace where second-order logic was discussed. “Peirce’s1885 paper was his first attempt to formalize higher mathematicalnotions in the higher order theory of relatives” (Brady 2000:132). The representation of quantifiers led Peirce to explore whetherrelatives may be quantified. For this topic, refer to Putnam 1982 andGoldfarb 1979.

18. 1885a [CP 3.396]. Our modern notation is used here. The expression“\(\phi(x)\)” is a formula whose unbound variable is\(x\). More rules are found in the undated Note (see 1885b).

19. Out of the three volumes Peirce’s influence was clear in Vol. 2and part of Vol. 3, and both of them were published in 1895.

20. “I do believe that the calculus of relations deserves much moreattention than it receives” (Tarkski 1941: 89).

21. Frege does not think these two features are not exclusive to eachother mainly because he locates a difference between propositional andquantified logic. Hence, Frege believed that his own systemBegriffsschrift is bothcalculus ratiocinator anda lingua characterica.

22. Peirce’s letter (1882b) to O.H. Mitchell showed that he hadstarted playing around with the possibility of graphicallyrepresenting relations. See Roberts 1973: 18.

23. Peirce’s theory of signs is complicated. See entry onPeirce’s theory of signs and Short 2007: ch. 8.

24. For example, 1885a [CP 3.359–362], and part of 1903d [CP2.247–249]. Also refer to Dipert 1996.

25. See Grattan-Guinness 2002 and Shin 2012 for a discussion of therelationship between Peirce’s ideas and Kempe’s.

26. Much more interesting work can be done for this claim.

27. For the comparison between these two graphical systems, refer to Shin2002: 48–53.

28. For more details, refer to Shin 2002: §4.3.

29. If we desire to match the syntax and the reading method in an obviousway, we may have the following alternative definition for well-formeddiagrams:

1. An empty space is a well-formed diagram.
2. Any letter is a well-formed diagram.
3. If \(D\) is a well-formed diagram, then a single cut of \(D\)(“\([D]\)”) is a well-formed diagram as well.
4. If \(D_{1}\) and \(D_{2}\) are well-formed diagrams, then all of thefollowings are also well-formed:
   1. \(D_{1}\) \(D_{2}\)
   2. \([D_{1}\ D_{2}]\)
   3. \([D_{1}\ [D_{2}]]\)
   4. \([[D_{1}]\ [D_{2}]]\)
5. Nothing else is a well-formed diagram.

30. For the proof of the legitimacy of the Multiple readings algorithm,refer to Shin 2002, §4.2.2, where the equivalence of theEndoporeutic and the Multiple readings is proven.

31. Roberts’ slightly rearranged version is:

32. An area which is enclosed by an even number of cuts

33. An area which is enclosed by an odd number of cuts

34. As far as Peirce’s pursuit of graphical representation goes,one more EG system, Gamma graphs, was invented by Peirce after Betagraphs. Gamma graphs aim to represent modality by adding two kinds ofsigns to Beta graphs—broken cuts and tinctures (1903c,1906).

35. Although the matter is not yet settled, Dau (2006) has raised aquestion about one of the rules.

36. Examples in Peirce 1903b [CP 4.455] nicely illustrate the visualeffect of an identity line.

37.X is evenly (oddly) enclosed if and onlyX is enclosedby an even (odd) number of cuts.

38. For more complex examples, refer to Peirce 1903b [CP 4.502] and 1906[CP 4.571].

39. According to the Endoporeutic reading, we get the following readingfirst: “It is not the case that something is good but notugly”. If we adopt the Multiple readings, we may directly obtainthe above reading “Everything good is ugly”.

40. Recently, Odland discovered three more pages from Peirce’sLogic Notebook related to his discussion on triadic logic inPeirce’s Triadic Logic: Continuity, Modality, and L(Odland 2020).