Abstract
Charles Peirce incorporates modality into his Existential Graphs (EG) by introducing the broken cut for possible falsity. Although it can be adapted to various modern modal logics, Zeman demonstrates that making no other changes results in a version that he calls Gamma-MR, an implementation of Jan Łukasiewicz's four-valued Ł-modal system. It disallows the assertion of necessity, reflecting a denial of determinism, and has theorems involving possibility that seem counterintuitive at first glance. However, the latter is a misconception that arises from overlooking the distinction between the intermediate truth values (ITVs) that are assigned to possibly true propositions as either X-contingent or Y-contingent. Any two propositions having the same ITV are possible together, while any two propositions having different ITVs, including those that are each other's negation, are possible individually yet not possible together. Porte shows that Ł-modal can be translated into classical logic by defining a constant for each ITV such that its implication of another proposition asserts the latter's possibility, while its conjunction with another proposition asserts the latter's necessity. These are expressed in the Alpha part of EG without broken cuts, simplifying derivations and shedding further light on Łukasiewicz's system, as long as graphs including either of the constants are properly interpreted. Ł-modal and Gamma-MR thus capture the two-sided nature of possibility as the limit between truth and falsity in Peirce's triadic conception.
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Notes
Consistent with standard practice in Peirce scholarship, his writings are cited herein as CP with volume and paragraph number(s) for (1931–1958), NEM with volume and page number(s) for (1976), and LF with volume number for (2020–2022), followed by the year of original composition or publication. Most existed only in manuscripts when he died in 1914, but the title is also given where the reference is to an article that appeared in print during his lifetime.
By omitting excluded middle, intuitionistic logic treats the universe of discourse as general rather than individual, and philosophical intuitionism also treats it as mind-dependent rather than real. Since double negation elimination is invalid, a double cut is no longer equivalent to a scroll. Oostra (2010) and Ma and Pietarinen (2019) have developed intuitionistic versions of EG for propositional logic with this restriction, and Oostra (2011; 2012) has also described adaptations for intuitionistic first-order predicate logic and modal logic. Pietarinen and Chiffi (2018) observe that the blank sheet now signifies each graph’s “relation to the possibility of its illative transformations, its verifications or proofs,” such that “intuitionistic graphs are strictly speaking not species of existential graphs.”
Gamma also incorporates additional extensions including “second-order (higher-order) logics, abstractions, [and] logic of multitudes and collections” (Ma & Pietarinen, 2018). Lewis (1912) is generally credited with originating modern modal logic, but Peirce’s manuscripts associated with his 1903 Lowell Lectures (LF 2/1–2) show that he was on the verge of doing so years earlier; unfortunately, he was unable to get them published during his lifetime. For a more thorough summary of EG, especially Alpha and Beta, see (Roberts, 1992). For detailed expositions of all three parts, see (Zeman, 1964), (Roberts, 1973), and Pietarinen’s extensive introductory material in all the volumes of LF.
An anonymous reviewer rightly observed that Zeman's pioneering and extensive exposition of all three parts of EG is not as well-known nor as widely studied as it deserves to be. However, discussing all his interesting results and conclusions in greater detail is not within the scope of this paper.
According to an anonymous reviewer, Zeman became acquainted with Ł-modal through A. N. Prior, who worked with him while visiting Chicago in 1962.
Łukasiewicz was skeptical of intensional modal logics that rely on possible worlds semantics to maintain bivalence: “The analogy of possibility and necessity with the particular and universal quantifier is, in my opinion, superficial and misleading” (1953a). By contrast, Peirce quite presciently anticipated such an approach, including strict implication: “The quantified subject of a hypothetical proposition is a possibility, or possible case, or possible state of things” (CP 2.347, c. 1895). “But an ordinary Philonian conditional [if A then B] is expressed by saying, ‘In any possible state of things, i, either Ai is not true, or Bi is true’” (CP 3.444, “The Regenerated Logic,” 1896). He even briefly experimented with extending the Alpha part of EG to incorporate such quantification by using a heavy line to denote some possible state of things where each proposition attached to it would be true (LF 1, 1909). For more historical and theoretical background, see (Copeland, 2002).
This aspect of Ł-modal explains another initially puzzling result that Łukasiewicz himself highlights (1954; 1970, pp. 397–400) and Prior later addresses (1962): There exists a positive integer such that it is both Δ-possible that it is equal to one and ∇-possible that it is greater than one. The two predications are possibly true individually but not together, so the theorem properly evaluates one of them as X-contingent and the other as Y-contingent.
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The author is grateful to Brent Odland for providing helpful feedback on a draft version of this paper, and to the anonymous reviewers for their suggested enhancements.
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Schmidt, J.A. Peirce and Łukasiewicz on modal and multi-valued logics. Synthese 200, 275 (2022). https://doi.org/10.1007/s11229-022-03755-2
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DOI: https://doi.org/10.1007/s11229-022-03755-2