MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 5XI]. MacColl invoked the letters ɛ and η to stand for certainty and impossibility,initially describing them as replacements for 1 and 0, and then introduced a thirdletter θ to denote a statement that was neither certain nor impossible, and hencewas “a variable (neither always true nor always false)”. He wrote the equations(a = ɛ), (b = η) and (c = θ) to express that a is a certainty, b is an impossibility,and c is a variable. Then he changed these to the symbols a ɛ , b η , c θ , and wenton to write a τ for “a is true” and a ι for “a is false”, noting that a true statementis “not necessarily a certainty” and a false one is “not necessarily impossible”. Inthese terms he stated that a : b is equivalent both to (a.b ′ ) η (“it is impossible thata and not b”) and to (a ′ + b) ɛ (“it is certain that either not a or b”).Once the step to this superscript notation had been taken, it was evident thatit could be repeated, giving an easy notation for iterations of modalities. MacCollgave the example of A ηιɛɛ as “it is certain that it is certain that it is false that it isimpossible that A”, abbreviated this to “it is certain that a is certainly possible”,and observed thatProbably no reader—at least no English reader, born and brought up inEngland—can go through the full unabbreviated translation of this symbolicstatement A ηιɛɛ into ordinary speech without being forcibly reminded ofa certain nursery composition, whose ever-increasing accumulation of thatsaffords such pleasure to the infantile mind; I allude, of course, to “The Housethat Jack Built”. But trivial matters in appearance often supply excellentillustrations of important general principles. 4There has been a recent revival of interest in MacColl, with a special issue of theNordic Journal of Philosophical Logic 5 devoted to studies of his work. In particularthe article [Read, 1998] analyses the principles of modal algebra proposedby MacColl and argues that together they correspond to the modal logic T, laterdeveloped by Feys and von Wright, that is described at the end of section 2.4below.2.3 The Lewis SystemsMacColl’s papers are similar in style to earlier nineteenth century logicians. Theygive a descriptive account of the meanings and properties of logical operations but,in contrast to contemporary expectations, provide neither a formal definition ofthe class of formulas dealt with nor an axiomatisation of operations in the sense ofa rigorous deduction of theorems from a given set of principles (axioms) by meansof explicitly stated rules of inference. The first truly modern formal axiom systemsfor modal logic are due to C. I. Lewis, who defined five different ones, S1–S5, inAppendix II of the book Symbolic Logic [1932] that he wrote with C. H. Langford.Lewis had begun in [1912, p. 522] with a concern that4 Mind (New Series), vol. 9, 1900, p. 75.5 Volume 3, number 1, December 1998, available athttp://www.hf.uio.no/filosofi/njpl/vol3no1/index.html.