Peirce's Search for a Graphical Modal Logic (Propositional Part)

Abstract

This paper deals with modality in Peirce's existential graphs, as expressed in his gamma and tinctured systems. We aim at showing that there were two philosophically motivated decisions of Peirce's that, in the end, hindered him from producing a modern, conclusive system of modal logic. Finally, we propose emendations and modifications to Peirce's modal graphical tinctured systems and to their underlying ideas that will produce modern modal systems.

Notes

¹Although Peirce consistently uses the terms Alpha, Beta and Gamma (most of the time capitalized), we prefer using the Greek letters for brevity.

²For our purposes, it is important to know that at different stages of the development of his systems, Peirce has different concepts of what operation actually makes up a cut, that is, negation. These range from the cut being a simple closed line separating true statements from statements that are false to sophisticated operations such as first writing false propositions at the bottom surface of the sheet (Peirce calls it the verso) and then making them visible by cutting them out and turning around the cut out part of the sheet.

³Note that this clause makes the grammatical structure of a given graph ambiguous, but this does not matter for our purposes.

⁴Roberts considers it corresponding ‘roughly, to second (and higher) order functional calculi, and to modal logic’ (Roberts 1973, p. 64). Zeman concentrates on γ’s modal power (Zeman 2002).

⁵Were this not the case, the rule would license the transition from ¬ ◊ ¬ P, that is, from □ P, to ¬ ◊ ¬ (P ∧ Q), that is, □ (P ∧ Q).

⁷The derivation will start with the assertion of ¬ ◊ ¬ P, which in γ notation is a P within a broken cut, enclosed by a standard cut. Now the inner cut, that is, the inner negation, will be enclosed by an odd number of cuts (i.e. by one cut), licensing our filling it up to become a standard cut. This operation leaves us with ¬ ¬ P, which is the EG equivalent of P.

⁸For a discussion of the philosophical impact of Peirce's tinctured graphs, cf. Ramharter (forthcoming).

⁹Actually, the sl verso usually is of bluish grey, but it may as well be yellow, or rose, or green, and the recto is of cream white – according to CP 4.573.

¹⁰Of course, one could express ‘possible’ as ‘not-necessary not’ and use a different tincture for necessity, but then there is no connection between the tincture that is used for ‘not-possible’ and that which is used for ‘not-necessary’.

¹¹Ms 280, pp. 21–22, cit. Roberts (1973, p. 27). They are sometimes considered equivalent to the much later Laws of Form system, most directly stated by Robert Burch (2006): ‘A version of the entitative graphs later appeared in G. Spencer Brown's Laws of Form, without anything remotely like proper citation of Peirce’.

¹²‘Any entitative graph may be converted into the equivalent existential graph by, first, enclosing each spot separately and, second, enclosing the whole graph’ (Ms 485, p. 1, cit. Roberts 1973, p. 28).

¹³We discuss this example in Appendix 2.

¹⁴ Roberts (1973, p. 96) finds this astonishing because Peirce had already made several diagrams for similar examples in his manuscripts. Roberts offers quite a simple explanation: in the weeks before the publication of the ‘Prolegomena’, the development of Peirce's logical theory was progressing so rapidly that he did not manage to complete the examples in time. Roberts (1973) shows the missing diagram (p. 97), but it differs from the one Peirce gives later in ‘Prolegomena’. Roberts’ version is the one coinciding with the one we suggested as being ‘what Peirce has actually meant or should have meant at least’.

¹⁵Concerning ‘real possibility’, see Kent (1997, p. 448).

¹⁶We are concerned with concepts involving logical properties only. Interpretative terms such as ‘physical necessity’ are beyond the scope of our investigations.

¹⁷In the technical part of his γ system, Peirce very strongly takes the view that possibility and necessity are subjective concepts and that they are ‘relative to the state of information’ (CP 4.517 from the Lowell Lectures of 1903) of a certain individual, the Graphist.

¹⁸For our purposes, problems concerning the epistemic status, reliability, etc., of necessary reasoning may be put aside; it is just the role it plays in logic that is important for us.

¹⁹That is, sign.

²⁰The axiom we call (K) here is usually called (R) – a combination of (M) and (C) (cf. Chellas 1995, p. 114) – but it does the job of (K) in our context.

²¹From now on, we will substitute axioms stating implications by corresponding transformation rules.

²²If you take the usual version of (K) using→, you do not get a self-evident scheme, either.

²³In the entitative system, the version of (K) using possibility would look like (K _P ) in the EGs.

²⁴If the tincture expresses necessity, inserting would be wrong; if the tincture expresses possibility, an axiom is needed.

²⁵There are, of course, different such mappings depending on the calculus used to formulate the logical system. Zeman (1964) uses two of them.

²⁶To be precise: to get an expression of this form, you have to transform the graph by (K _P ) before the translation so that any two elementary propositions are separated, which means that they lie on different pieces of red sheet. Another proof by induction is required, but since there is neither a problem in principle nor any doubt in the correctness of the result of this sub-proof, we omitted it here. It would use the same sort of arguments as that follow in the next lines.

²⁷It is not clear whether Peirce wants to say that representation by tinctures at all is not iconic or if he just sees no way for making the special choices of tinctures iconic. But most likely he means the former. If you consider the relation between ‘possibility’ and ‘icon’ the other way around, his position is beyond doubt: an icon is a possibility (see Peirce 1983, p. 73).

²⁸Neither did we take the game theoretic aspects of Peirce's logic into consideration, though we see that it is an important motivational factor for Peirce's logic.

²⁹At an earlier stage, in his 1903 A Syllabus of Certain Topics of Logic, CP 4.415, Peirce makes an exception to this rule by adding that an empty cut must not be deleted from the sheet of assertion. We do not know why. He does not always make this exception, though – cf., for example, CP 4.377 (‘anything written down may be erased’) or, in a way, CP 4.489 (‘[a ny partial graph may be erased’).

³⁰The exception to the rule is explicitly stated only by Roberts (1973, p. 42 sq.), not by Peirce himself. Omitting it would falsely license, for example, the derivation of P(Q(Q)) from (P(Q)), that is, the derivation of (P→(Q ∧ ¬ Q)) from P→Q.

³¹Peirce's wording of this rule in CP 4.566, ‘[a ny Graph […] (if already Iterated) may be Deiterated’, might be understood this way. That this was not his intention shows his comment that ‘[t]o deiterate a Graph is to erase a second Instance of it’ (loc. cit.) and, even more clearly, the remark that ‘[t]he operation of deiteration consists in erasing a replica which might have illatively resulted from an operation of iteration’ (CP 4.506, italics by us).

³²In the much simplified spirit of Roberts’ R5, p. 44.

³³ Edgington (2006) gives an overview of the discussions on material implication.