Peirce’s Threevalued Connectives
Charles Peirce was the first logician to define logical operators for a manyvalued system of logic.^{1} In February 1909, on three pages of a notebook in which he recorded his thoughts on logic (MS 339), he defined several threevalued connectives using the truthtable, or matrix, method.^{2} The system of triadic logic that Peirce envisioned employs the values “V”, “F”, and “L”. He interpreted “V” and “F” as “verum” (“true”) and “falsum” (“false”), respectively, and he interpreted the third value, “L”, as “the limit.”
Peirce’s work on manyvalued logical connectives was first brought to light by Max Fisch and Atwell Turquette (1966). As Fisch and Turquette describe, it had long been thought that Jan Łukasiewicz (1920, 1930) and Emil Post (1921) had developed the first operators for threevalued logic.^{3} But Peirce is now recognized as the first to use the truthtable method to define threevalued operators. Subsequent to the publication of Fisch and Turquette’s paper, the formal aspects of Peirce’s threevalued connectives were explored extensively by Turquette (1967, 1969, 1972, 1973, 1976, 1978, 1981/4).
In conducting his triadic experiments, Peirce defined four different oneplace connectives and six different twoplace connectives. Peirce’s threevalued oneplace connectives are:
As Fisch and Turquette point out, all four of these connectives were rediscovered by later logicians:
Peirce’s corresponds to Łukasiewicz’s negation Nx, as well as to Halldén’s and Körner’s negation operators.
Peirce’s corresponds to Słupecki’s “tertium function” Tx.
Peirce’s and correspond respectively to Post’s negations and .^{4}
Peirce’s threevalued twoplace connectives are as follows:

Θ resembles the disjunction operator of twovalued, classical logic, in that “x Θ y” takes the maximum of the values taken by “x” and “y” (V > L > F). Θ corresponds to Emil Post’s (1921) “alternation,” V_{3}, and to Körner’s disjunction (e.g., 1966, p. 39).


Z resembles the conjunction operator of twovalued, classical logic, in that “x Z y” takes the minimum of the values taken by “x” and “y” (V > L > F). Z corresponds to Körner’s conjunction (e.g., 1966, p.39).


Υ is similar to the disjunction operator of twovalued, classical logic, in that “x Υ y” takes the maximum of the values taken by “x” and “y” (V > L > F) when each of the two conjuncts has a classical value. The value “L” is “infectious,” however, in that when either of the disjuncts takes “L”, the formula as a whole takes “L”. Υ corresponds to a connective used by Bochvar (1939), to Kleene’s weak alternation (1952, pp. 327336) and to Halldén’s disjunction operator (1949).


Ω is similar to the conjunction operator of twovalued, classical logic, in that “x Ω y” takes the minimum of the values taken by “x” and “y” (V > L > F) when each of the two conjuncts has a classical value. The value “L” is “infectious,” however, in that when either of the conjuncts takes “L”, the formula as a whole takes “L”. Ω corresponds to a connective used by Bochvar (1939), to Kleene’s weak conjunction (1952, pp. 327336), and to Halldén’s conjunction operator (1949).


Turquette holds Φ and Ψ to be more mysterious than Peirce’s other twoplace connectives, since, as he says, “their motivation is not entirely clear and they seem to have played no very important part in later literature on triadic logic” (1967, p. 66). He argues that Peirce may have been motivated to introduce these connectives by considerations of duality and completeness. Parks (1971), on the other hand, points out that Φ and Ψ did in fact play a part in the later development of triadic logic: they occurred as A and K (disjunction and conjunction) in the system developed by Sobocinski (1952), and, subsequent to the publication of Turquette (1967), they occurred as ∨ and ∧ in the “logic of ordinary discourse” developed by Cooper (1968) as well as in work by Belnap (1970).

Philosophical Motivations
Peirce characterized triadic logic as
that logic which … recognizes that every proposition, S is P, is either true, or false, or else S has a lower mode of being such that it can neither be determinately P, nor determinately notP, but is at the limit between P and not P. (MS 339, Feb. 23, 1909)^{5}
In experimenting with manyvalued connectives, Peirce was motivated by the desire to accommodate within formal logic propositions which are neither true nor false; and this means that he believed that some propositions are, indeed, neither true nor false. He thus rejected the Principle of Bivalence (PB), according to which any proposition is either true or else false.^{6}
Commentators disagree about Peirce’s philosophical reasons for rejecting PB and acknowledging propositions that are at “the limit” between true and false. Because of potentially misleading comments Peirce made regarding the principle of excluded middle (PEM) (see Lane, 2001), some have taken him to have intended “L” to value objectgeneral propositions (roughly, universally quantified propositions). This has the odd consequence that, for example, the proposition “All bachelors are unmarried” is neither true nor false. But for Peirce, to say that PEM does not apply to a proposition “S is P” is not to imply that “S is P” is neither true nor false. The nonapplication of PEM to general propositions did not motivate the development of Peirce’s threevalued connectives.
Others have assumed that Peirce meant “L” to be taken by what he called “vague” propositions, presumably because he held that the principle of contradiction (PC) does not apply to such propositions (see Chiasson, 2001; Lane, 2001). By “vague proposition” Peirce meant objectindefinite propositions (roughly, existentially quantified propositions). So the view that “L” values vague propositions has the odd consequence that, for example, the proposition “Some US President is from Texas” is both true and false. But for Peirce, to say that PC does not apply to a proposition “S is P” is not to imply that “S is P” is both true and false. The nonapplication of PC to vague propositions did not motivate the development of Peirce’s threevalued connectives.
Still others have assumed that Peirce intended his third value to be taken by modal propositions. This is because Peirce wrote that PEM does not apply to assertions of necessity and PC does not apply to assertions of possibility. But a correct understanding of Peirce’s “principles of excluded middle and contradiction” shows that these comments do not suggest that a value other than “true” and “false” is needed for modal propositions. Considerations from quantum physics have led others to suggest that Peirce intended “L” to be taken by any proposition containing “scientifically sound predicates” (Jauhari, 1985), including propositions that express natural laws. But this interpretation is not at all supported by the textual evidence.
In fact, Peirce intended his third value to be taken only by propositions that predicate of a breach in mathematical or temporal continuity one of the properties that is a boundaryproperty relative to that breach. I call such propositions boundarypropositions. To see that this is the sort of proposition Peirce intended “L” to value, we must recognize the distinction between saying that a logical principle does not apply to a proposition and saying that it is false with regard to a proposition. Peirce’s view is that a principle can only be false with regard to a proposition if it applies to that proposition. (MS 641:24 2/3  3/4, 1909) So to say that a given principle does not apply to a proposition is to imply that the principle is not false with regard to that proposition.
This distinction is important to a correct understanding of Peirce’s triadic logic because he intended his triadic logic to accommodate propositions with regard to which PEM is false, and thus to which PEM applies. This means, first, that Lpropositions (propositions that take Peirce’s third value, “L”) have individual (nongeneral) subjectterms. It also means that Lpropositions do not express necessity, i.e., they are not of the form “S must be P”. Further, Peirce’s view seems to have been that PC is true with regard to Lpropositions; and this means that PC applies to Lpropositions, and therefore that Lpropositions have definite (nonvague) subjectterms and that they do not express possibility, i.e., they are not of the form “S may be P” or “S can be P”. In sum: Lpropositions have singular (individual and definite) subjectterms and are nonmodal (they express neither necessity nor possibility).
On one of the pages of the logic notebook in which he defined his threevalued connectives, Peirce gave an example involving an inkblot. He seems to have intended that example as an illustration of an objectsingular, nonmodal proposition that takes “L” as its value:
Thus, a blot is made on the sheet. Then every point of the sheet is unblackened or is blackened. But there are points on the boundary line, and those points are insusceptible of being unblackened or of being blackened, since these predicates refer to the area about S and a line has no area about any point of it. (MS 339, February 23, 1909)
The question Peirce found interesting was whether the boundary between the ink blot and the rest of the paper is black or nonblack. His answer, it seems, was “neither.” Again, Peirce described an Lproposition “S is P” as follows:
S has a lower mode of being such that it can neither be determinately P, nor determinately notP, but is at the limit between P and not P. (MS 339, February 23, 1909)
The boundary between the black ink blot and the nonblack paper is neither black nor nonblack, and the (objectsingular, nonmodal) propositions “The boundary is black” and “The boundary is nonblack” are neither true nor false. Each is the sort of proposition that Peirce thought should take the value “L”. The boundary between the black and the nonblack areas of the paper is a continuitybreach; it is a line in an otherwise uninterrupted surface. Peirce intended “L” to value propositions that predicate of a mathematical or temporal continuitybreach one of the properties that is a boundaryproperty relative to that breach. Such propositions are boundarypropositions.
This might seem strange at first. Why, after all, would Peirce take boundarypropositions to be interesting or important enough to motivate him to introduce threevalued connectives? The answer lies in the fact that the notion of continuity was itself of supreme philosophical importance for Peirce. That the question of continuitybreaches and their boundaryproperties was for him not simply an afterthought or a relatively unimportant aspect of the broader issue of the nature of continuity, is indicated by the fact that each time he revised his definition of continuity in a significant way, his position regarding continuitybreaches and their boundaryproperties changed as well. (Lane 1999)
Blocking the way of inquiry?
What are the consequences of Peirce’s rejection of PB for his pragmatic account of truth, i.e., his account of truth as that which would be agreed upon at the hypothetical, ideal limit of inquiry? As noted by Cheryl Misak (1991), Peirce did not intend (at least, not from the 1890s on) to give a biconditional definition of “truth” but instead held that what she calls the Truth to Inquiry Conditional is a regulative principle of inquiry, a hope that must be adopted by an inquirer with regard to the question she is investigating:
The Truth to Inquiry Conditional: If “S is P” is true, then, if inquiry relevant to whether S is P were pursued as far as it could fruitfully go, it would be agreed that S is P.
Peirce’s view of bivalence seems to have been the same. So in rejecting bivalence with regard to a proposition “S is P”, Peirce was in effect giving up the hope that, if inquiry with regard to whether S is P were pursued as far as it could fruitfully go, belief about whether S is P would never be settled.
This seems to be in tension with Peirce’s injunction against blocking the “way of inquiry” (CP 1.135, c.1898)^{7}; after all, one way to block the way of inquiry is to assert, with regard to a given question, that inquiry would never result in consensus regarding the answer to that question. Had he claimed that a broad class of proposition (modal propositions, say, or, propositions containing “scientifically sound predicates”) fails to be either true or false, Peirce himself would have been guilty of blocking a relatively wide avenue of inquiry. But he rejected bivalence only for a very narrow range of propositions: boundarypropositions. Thus, Peirce was guilty of blocking, not a wide avenue of inquiry, but only a narrow alleyway.
References
Belnap, N. (1970). Conditional Assertion and Restricted Quantification. Noûs, 4, 112.
Berry, G. (1952). Peirce’s Contributions to the Logic of Statements and Quantifiers. In P. Wiener & F. Young (Eds.), Studies in the Philosophy of Charles Sanders Peirce. Cambridge, MA: Harvard University Press.
Bochvar, D. (1939). Ob odnom tréhznachom iscislénii i égo priménénii k analizu paradoksov kalssicéskogo rassirénnogo funkcional ‘nogo iscislenija. Matématiceskij sbornik, 4, 287308. [Published in English as: On a threevalued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus (M. Bergmann, Trans.) History and Philosophy of Logic 2: 87112. 1981.]
Chiasson, P. (2001). Peirce’s Logic of Vagueness. In M. Bergman & J. Queiroz (Eds.), The Commens Encyclopedia: The Digital Encyclopedia of Peirce Studies. New Edition. Pub. 1212201909a. Retrieved from http://www.commens.org/encyclopedia/article/chiassonphyllispeirce%E2%80%99slogicvagueness.
Church, A. (1956). Introduction to Mathematical Logic. Princeton, NJ: Princeton University Press.
Cooper, W. (1968). The Propositional Logic of Ordinary Discourse. Inquiry, 11, 295320.
Fisch, M. & Turquette, A. (1966). Peirce’s Triadic Logic. Transactions of the Charles S. Peirce Society, 2, 7185.
Haack, S. (1978). Philosophy of Logics. Cambridge: Cambridge University Press.
Haack, S. (1997). The First Rule of Reason. In J. Brunning & P. Forster (Eds.), The Rule of Reason: The Philosophy of Charles Sanders Peirce. Toronto: University of Toronto Press.
Halldén, S. (1949). The Logic of Nonsense. Uppsala: Uppsala Universitets Arsskrift.
Körner, S. (1966). Experience and Theory. An Essay in the Philosophy of Science. New York, NY: Humanities Press.
Kleene, S. (1952). Introduction to Metamathematics. New York: D. Van Nostrand Company.
Jauhari, J. (1985). Peircean and Quantum Generals. Transactions of the C. S. Peirce Society, 21, 51334.
Lane, R. (1999). Peirce’s Triadic Logic Revisited. Transactions of the Charles S. Peirce Society, 35, 284311.
Lane, R. (2001). Principles of Excluded Middle and Contradiction. In M. Bergman & J. Queiroz (Eds.), The Commens Encyclopedia: The Digital Encyclopedia of Peirce Studies. New Edition. Pub. 1407302107a. Retrieved from http://www.commens.org/encyclopedia/article/lanerobertprinciplesexcludedmiddleandcontradiction.
Lewis, C.I. & Langford, C. (1959). Symbolic Logic. New York: Dover Publications.
Łukasiewicz, J. (1920). O logice trojwartosciowej. Ruch Filozoficzny, 5, 16971. [Published in English as: On ThreeValued Logic. In S. McCall (Ed.), Polish Logic 19201939 (H. Hi, Trans.) London: Oxford University Press.]
Łukasiewicz, J. (1930). Philosophiche Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls. Comptes Rendus des séances de la Société et des Lettres de Varsovie, 23, 5177. [Published in English as: Philosophical Remarks on ManyValued Systems of Propositional Calculus. In S. McCall (Ed.), Polish Logic 19201939 (H. Weber, Trans.) London: Oxford University Press.]
Łukasiewicz J., & Tarski, A. (1930). Untersuchungen über den Aussagenkalkül. Comptes Rendus des séances de la Société et des Lettres de Varsovie, 23, 3050. [Published 1956 in English as: Investigations into the Sentential Calculus. In A. Tarksi, Logic, Semantics, Metamathematics; papers from 1923 to 1938. Oxford: Oxford University Press. Page reference is to the reprint.
Łukasiewicz J., & Tarski, A. (1956). Investigations into the Sentential Calculus. In A. Tarksi, Logic, Semantics, Metamathematics; papers from 1923 to 1938. Oxford: Oxford University Press. (Reprinted from Untersuchungen über den Aussagenkalkül. Comptes Rendus des séances de la Société et des Lettres de Varsovie, 23, 3050)
Misak, C. (1991). Truth and the End of Inquiry: a Peircean Account of Truth. New York, NY: Oxford University Press.
Parks, R. (1971). The Mystery of Phi and Psi. Transactions of the Charles S. Peirce Society, 7, 176177.
Post, E. (1921). Introduction to a general theory of elementary propositions. American Journal of Mathematics, 43, 163185.
Rescher, N. 1969. Manyvalued Logic. New York, NY: McGrawHill.
Rosser, J.B. & Turquette, A.R. (1952). Manyvalued Logics. Amsterdam: NorthHolland Publishing Company.
Słupecki, J. (1936). Der volle dreiwertige Aussagenkalkül. Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III (vol. 29, pp. 911).
Sobocinski, B. (1952). Axiomatization of a Partial System of ThreeValue Calculus of Propositions. Journal of Computing Systems, 1, 2355.
Turquette, A. (1967). Peirce’s Phi and Psi Operators for Triadic Logic. Transactions of the Charles S. Peirce Society, 3, 6673.
Turquette, A. (1969). Peirce’s Complete Systems of Triadic Logic. Transactions of the Charles S. Peirce Society, 5, 199210.
Turquette, A. (1972). Dualism and Trimorphism in Peirce’s Triadic Logic. Transactions of the Charles S. Peirce Society, 8, 131140.
Turquette, A. (1973). Implications for Peirce’s Triadic Logic [Abstract]. International Federation of Philosophical Societies: Abstracts of Communications Presented at the XVth World Congress of Philosophy (no. 408). Varna, Sofia.
Turquette, A. (1976). Minimal Axioms for Peirce’s Triadic Logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 22, 169176.
Turquette, A. (1978). Alternative Axioms for Peirce’s Triadic Logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 24, 443444.
Turquette, A. (19811984). Quantification for Peirce’s Preferred System of Triadic Logic. Studia Logica, 40, 373382.
Turquette, A. (1983). Defining Peirce’s Verum. In V. Cauchy (Ed.), Philosophy and Culture: Actes du XVIIe Con’gres mondial de Philosophie (vol. 2). Montreal: Editions Montmorency.
Endnotes