Triadic Logic
Robert Lane
Peirce was the first logician to define three-valued logical connectives. In 1909, he defined four one-place three-valued connectives and six two-place three-valued connectives, all of which were rediscovered by later logicians. Peirce's motivation was to accommodate within formal logic a specific, narrow range of propositions he took to be neither true nor false, viz. propositions that predicate of a breach in mathematical or temporal continuity one of the properties that is a boundary-property relative to that breach.
Peirce's work on many-valued 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 ukasiewicz (1920, 1930) and Emil Post (1921) had developed the first operators for three-valued logic.[3] But Peirce is now recognized as the first to use the truth-table method to define three-valued operators. Subsequent to the publication of Fisch and Turquette's paper, the formal aspects of Peirce's three-valued connectives were explored extensively by Turquette (1967, 1969, 1972, 1973, 1976, 1978, 1981/4). In conducting his triadic experiments, Peirce defined four different one-place connectives and six different two-place connectives. Peirce's three-valued one-place connectives are:
As Fisch and Turquette point out, all four of these connectives were rediscovered by later logicians:
Nx, as well as to Halldén's and Körner's negation operators.Peirce's corresponds to Supecki's "tertium function" Tx.Peirce's and correspond respectively to Post's negations and .[4] Peirce's three-valued two-place connectives are as follows:
It is unclear from what Peirce wrote how satisfied he was with this work. On one page he wrote that "Triadic Logic is universally true"; but on another, he wrote that "All this is mighty close to nonsense." Whatever Peirce's own opinion of his work, his operators were sufficient for a functionally complete system of three-valued logic. All possible operators definable by three-valued matrices can be defined in terms of just two of Peirce's operators: and
In experimenting with many-valued 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) 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 Peirce's Logic of Vagueness; Principles of Excluded Middle and Contradiction] By "vague proposition" Peirce meant object-indefinite 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 non-application of PC to vague propositions did not motivate the development of Peirce's three-valued connectives. Still others have assumed that Peirce intended his third value to be taken by modal propositions. 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 boundary-property relative to that breach. I call such propositions 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 On one of the pages of the logic notebook in which he defined his three-valued connectives, Peirce gave an example involving an ink-blot. He seems to have intended that example as an illustration of an object-singular, non-modal proposition that takes "L" as its value: neither." Again, Peirce described an L-proposition "S is P" as follows:continuity-breach; it is a line in an otherwise uninterrupted surface. Peirce intended "L" to value propositions that predicate of a mathematical or temporal continuity-breach one of the properties that is a boundary-property relative to that breach. Such propositions are boundary-propositions.
This might seem strange at first. Why, after all, would Peirce take boundary-propositions to be interesting or important enough to motivate him to introduce three-valued connectives? The answer lies in the fact that the notion of continuity was itself of supreme philosophical importance for Peirce. That the question of continuity-breaches and their boundary-properties 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 continuity-breaches and their boundary-properties changed as well. (Lane 1999)
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" (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: boundary-propositions. Thus, Peirce was guilty of blocking, not a wide avenue of inquiry, but only a narrow alley-way. |