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The paper considers the philosophical, rather than mathematical or logical, reasons why MacColl decided to develop a propositional logic.
Acknowledgments I am most grateful to Amirouche Moktefi for kindly providing documentation and advice, and also wish to express my thanks to my two anonymous referees, being very indebted in particular to the minute reading of the one who made so many tremendously accurate corrections and inspiring suggestions. S]uddenly, without any possible historical explanation, modern propositional logic springs with almost perfect completeness into the gifted mind of Gottlob Frege.
As for the absence of any possible historical explanation, it has also been qualified by the study of its sources, in Bolzano and Boole in particular. Can MacColl seriously be held not only as a great logician but also as a great philosopher of logic? Propositional logic is not only a new calculus or a mere study of logical operators, it supposes a new, truth-preserving semantics, a concept of grammar, a clarification of such fundamental notions as inference and substitution, more generally a new approach of the mental, and perhaps even a philosophy of space and time.
The most obvious one is a logic of propositions as opposed to a logic of terms. It maps the distinction between classes of things or events and truth-value bearers. Propositional logic thus presents a more suitable alternative to Peripatetic logic. The axiomatic propositional calculus then opposes syllogistic logic with its intuitive interpretation. Furthermore, to the supporters of an algebra of logic, propositional logic can designate the precedence of equivalence over equality and the central position of the implication relation.
It thus stands in contrast with the algebraic syntax of Boolean logic. There is first a question of chronological priority, which is no doubt minor and need not seriously distract historians.
More interesting is the foundational problem of logical priority: does propositional logic provide a basis for a logic of classes, or does it go the other way round?
The problem is that such a question is biased: looking for a foundation, one is likely to discover logicism as a solution which denies the logical primacy of propositions over sets. Hence, a third question is to be asked: is logic only a convenient tool, or does it reveal some truth about reality? For instance, does a logic based on events require an ontology of events? Does a logic of propositions need a philosophy of propositions?
Whatever be the exact chronology of the articles published, he thought of himself as a pioneer:. Alone, or nearly so, among logicians, I have always held the opinion, and my recent studies have confirmed it, that the simplest and the most effective system of Symbolic Logic is that whose elementary constituent symbols denote—not classes, not properties, not numbers, ratios, regions, or magnitudes, not things of any kind—but complete statements.
It calls attention to the fact that, like Peirce, MacColl played a role in the birth of modern propositional logic not only because of its systematic development, but in constantly arguing for its fundamental place at the base of logic. Indeed, the rather solitary position of MacColl does not mean that other logicians ignored the calculus of propositions, but often rather that they explicitly rejected its relevance.
This early position nevertheless needs to be qualified in order to respond to some obvious objections: symbols and especially letters often stand for numbers, ratios, subjects or predicates. In other words, his logic of propositions would be nothing but a defense of pluralism. Though he claims that the ultimate elements of his system are statements, he nevertheless treats inference as holding between sentences or more atomic terms.
After noticing that the relation between subject and predicate is analogous to the relation between antecedent and consequent, he indeed analyses a syllogism as a complex implication whose premises are expressed in an implicational form.
In short, in , inference is said to take place between statements, but is used between a subject and a predicate or more precisely, between two attributions of predicates. He referred to this temporary abdication from logic as a long pause in his mental life.
MacColl himself stressed the difference of his new, 20 th century views from his period:. That this is no longer the case my recent papers in MIND and in the Proceedings of the Mathematical Society will show; but the new development is still further removed than the old from the allied algebras which it has been the great aim of Mr. Whitehead to unite into one general comprehensive system.
Even his letter-terms differ in meaning from mine, since his letters denote propositions, not things. It is difficult to believe that there is any advantage in these innovations.
I think I may predict that synonyms are destined to play an important part in the future development of symbolic logic. As new needs and new ideas arise with the growth of civilisation and the general advance of humanity, do we not often find that words which were at first synonyms gradually differentiate and, while still remaining synonymous in some combinations, cease to be so in others? The syntactic framework of algebra is too narrow for such a semantics.
It is that for which MacColl reproaches Boole. Boole is indeed mainly concerned with terms, which, treated extensionally, can be reduced to a logic of classes. Perhaps it would be more exact to say that in emphasizing the role of propositions, he in fact aspired to create a calculus of statements that would transcend the traditional partition of various logics. But at least, no ontology is required by pure logic.
Thus, one only has to investigate the relations between signs:. For this reason, in order to mark the analogy, the logic of relations should rather be called the logic of functions.
Hence the failure to give to propositional functions the place they deserve in logic. Of course MacColl defends an unpsychological view of logic. And his formal system also shows evidences of anti-psychologism for instance in embracing the paradoxes of strict implication. As a philosophical and speculative work it is brimful of profound thought and original suggestions, while its style is charmingly lucid and attractive. One can compare Boole and Shakespeare:. Both authors possessed a remarkable analytical insight into the workings of the human mind; the one of its secret motives and passions; the other of the subtle laws of its intellectual operations; yet both—the one judged by his plays, the other by his Laws of Thought —showed but little constructive ability.
Boole indeed alleges that there is an exact adequacy in the laws which conduct the operations of reasoning and of algebra [Boole , 6].
In other words, the actual reasoning does not bear on symbols, be they mathematical or, as we will see, thought-signs; but the major fact Boole clearly understood is that it relies on transitiveness.
Though in purely formal or symbolic logic it is generally best to avoid, when possible, all psychological considerations, yet these cannot be wholly thrust aside when we come to the close discussion of first principles, and of the exact meanings of the terms we use.
Such an apparent contradiction in MacColl is due both to a chronological evolution and, from the start, to his qualified and subtle anti-psychologism.
The priority of the concept is a postulate that separates Boole from some of his most important followers [Vassallo , 89— 91]. The former considered that a logic of terms should be first because the analysis of the laws which govern the conception should precede the analysis of those governing judgment and reasoning.
It corresponds to the first step of the threefold division of logic that has been in use ever since Aquinas at least, namely a doctrine of terms related to the concept, a doctrine of propositions related to judgment, and a doctrine of inferences or syllogism related to reasoning. Nevertheless, many logicians insisted that in contrast to ampliative reasoning, deduction does not lead from a known truth to a truth unknown, because its conclusion is a mere clarification of what was already admitted in the premises [Peirce , ].
Inference is about obtaining knowledge, whereas implication refers to the impossibility that a certain premise does not lead to a certain conclusion. It is noteworthy that, to him, symbolization reflects the inferential process, not the actual reasoning. Writing with symbols and the logical relation of the truth of premisses to the truth of conclusion does not accurately represent the process of implication. The words if and therefore are examples of terms for which we need psychological considerations [MacColl a, 82].
Thus, contrary to Boole, MacColl refuses to view logic as concerned with temporal portions. It is a fundamental reason why MacColl denies that Boole developed a propositional logic or any calculus of pure statements.
The Laws of Thought indeed vacillates between two interpretations of signs, either representative of things or of operations of the human intellect. Therefore, whereas Aristotelian reasoning is a system of inclusion of concepts, reasoning for Chrysippus bears on implications of temporal relations. Not only is the calculus of pure statements not a set of mathematical operations, but it differs from algebra as much as algebra differs from geometry [MacColl , ].
For it is an error to believe that:. Statements, MacColl claims, have sui generis characters, which cannot be reduced to mathematical properties. Having refused to ground logic in time or space, MacColl cannot say that logic is a branch either of arithmetic or of geometry. Propositional logic is not mathematical, as Peirce would perhaps have held when stating that mathematics is logic rather than that logic is mathematics [Peirce ]. Peirce draws the attention of his fellow logician to this difference:.
To this end, a basic theory of signs and language founded on natural evolution is developed. He does not precisely define a sign, and seems to identify it with what he calls a symbol. A statement is what Peirce would simply call a sign, the indication of intelligible information or of a possible meaning for the logician. What meaning do they convey? A statement which, in regard to form, has such a subject-predicate structure, is called a proposition.
I do not say that the same information may be sometimes true and sometimes false, nor that the same judgment may be sometimes true and sometimes false; I only say that the same proposition —the same form of words —is sometimes true and sometimes false. For if in pure logic statements are represented by single letters, then a pure statement is, for instance, S , or P , that is, a subject e. In other words, included among the statements are the terms of a proposition, so that the calculus of equivalent statements should be both a logic of classes and a logic of propositions so as to exclude the possibility of the calculus dealing with classes.
Pure logic is just a classification of propositions:. MacColl nevertheless lays the foundations for a hierarchical typology of propositions based on their order of predication A b being a pure proposition, A bc a modal proposition of the first order, A bcd of the second order, etc.
Unfortunately, MacColl refuses to consider it seriously because he fears they might raise psychological questions. This difficulty is the reason why A. Shearman assumes that MacColl in fact quits the domain of pure logic when developing his modal logic:. Pure Logic can take account of the uncertainties that such data occasion, but the propositions dealt with will then denote not the relation of the respective letters to x , but the relation of the thinker to each implication.
Grammar is concerned with the relations within a proposition, logic with the relations between propositions: that is why a logic of predicates is not a real logic. Logic could focus on the links between propositions rather than on the content of every proposition. The clarification of ordinary language through grammar is thus requisite for propositional logic.
Grammar in this sense is not reduced to the meaning of words in a sentence but implies a whole context:. The mere word, and sometimes the complete grammatical proposition, when considered apart from context, may be meaningless or misleading. His defense of propositional logic remains ambiguous: his attempt at reduction to one formalism conflicts with his hatred for dogmatic symbolization; his claim that symbols are only a question of economy [MacColl , ] 15 contradicts the fundamental role of propositions; and his mathematical use of logic does not accord with the rejection of an algebraic syntax.
Perhaps his best achievement does not consist so much in his logical reduction to a symbolic system of propositions as in his conviction that propositions are the true logical atoms—which would open the road to the traditional problems of logical atomism, such as the unity, objectivity and bipolarity of propositions.
Anellis, Irving H. Castrillo, Pilar — , H. MacColl, C. Church, Alonzo — , Introduction to Mathematical Logic, vol. Peirce Society, 29 1 , 57— Johnson, William Ernest — , The logical calculus, Mind, n. Peirce ed. Peirce, Charles S.
ISBN 13: 9782225840791
Mathématiques pour l’informatique 2