John Corcoran

(1937 - 2021)

Prior Analytics by the Greek philosopher Aristotle and Laws of Thought by the English mathematician George Boole are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle's system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss many other historically and philosophically important aspects of Boole's book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole's contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology.… using mathematical methods … has led to more knowledge about logic in one century than had been obtained from the death of Aristotle up to … when Boole's masterpiece was published.Paul Rosenbloom 1950.

The syllogistic mnemonic known by its first two words Barbara Celarent introduced a constellation of terminology still used today. This concatenation of nineteen words in four lines of verse made its stunning and almost unprecedented appearance around the beginning of the thirteenth century, before or during the lifetimes of the logicians William of Sherwood and Peter of Spain, both of whom owe it their lasting places of honor in the history of syllogistic. The mnemonic, including the theory or theories it encoded, was prominent if not dominant in syllogistics for the next 700 years until a new paradigm was established in the 1950s by the great polymath Jan Łukasiewicz, a scholar equally at home in philosophy, classics, mathematics, and logic. Perhaps surprisingly, the then-prominent syllogistic mnemonic played no role in the Łukasiewicz work. His 1950 masterpiece does not even mention the mnemonic or its two earliest champions William and Peter. The syllogistic mnemonic is equally irrelevant to the post-Łukasiewicz paradigm established in the 1970s and 1980s by John Corcoran, Timothy Smiley, Robin Smith, and others. Robin Smith’s comprehensive 1989 treatment of syllogistic does not even quote the mnemonic’s four verses. Smith’s work devotes only 2 of its 262 pages to the mnemonic. The most recent translation of Prior Analytics by Gisela Striker in 2009 continues the post-Łukasiewicz paradigm and accordingly does not quote the mnemonic or even refer to the code—although it does use the terminology. Full mastery of modern understandings of syllogistic does not require and is not facilitated by ability to decode the mnemonic. Nevertheless, an understanding of the history of logic requires detailed mastery of the syllogistic mnemonic, of the logical theories it spawned, and of the conflicting interpretations of it that have been offered over the years by respected logicians such as De Morgan, Jevons, Keynes, and Peirce. More importantly, an understanding of the issues involved in decoding the mnemonic might lead to an enrichment of the current paradigm—an enrichment so profound as to constitute a new paradigm. After presenting useful expository, bibliographic, hermeneutic, historical, and logical background, this paper gives a critical exposition of Smith’s interpretation.

This expository paper on Aristotle’s prototype underlying logic is intended for a broad audience that includes non-specialists. It requires as background a discussion of Aristotle’s demonstrative logic. Demonstrative logic or apodictics is the study of demonstration as opposed to persuasion. It is the subject of Aristotle’s two-volume Analytics, as its first sentence says. Many of Aristotle’s examples are geometrical. A typical geometrical demonstration requires a theorem that is to be demonstrated, known premises from which the theorem is to be deduced, and a deductive logic by which the steps of the deduction proceed. Every demonstration produces knowledge of its conclusion for every person who comprehends the demonstration. Aristotle presented a general truth-and-consequence theory of demonstration meant to apply to all demonstrations: a demonstration is an extended argumentation that begins with premises known to be truths and that involves a chain of reasoning showing by deductively evident steps that its conclusion is a consequence of its premises. In short, a demonstration is a deduction whose premises are known to be true. Aristotle’s general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining theory of deduction was meant to apply to all deductions: any deduction that is not immediately evident is an extended argumentation that involves a chaining of immediately evident steps that shows its final conclusion to follow logically from its premises. His deductions, both direct and indirect, were rule-based and not tautology-based. The idea of tautology-based deduction, which dominated modern logic in the early years of the 1900s, is nowhere to be found in Analytics. Rule-based deduction was rediscovered by modern logicians. To illustrate his general theory of deduction, Aristotle presented a prototype: an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. With reference only to propositions of the four so-called categorical forms, he painstakingly worked out exactly what those immediately evident deductive steps are and how they are chained to complete deductions. In his specialized prototype theory, Aristotle explained how to deduce from a given categorical premise set, no matter how large, any categorical conclusion implied by the given set. He did not extend this treatment to non-categorical deductions, thus setting a program for future logicians. The prototype, categorical syllogistic, was seen by Boole as a “first approximation” to a comprehensive logic. Today, however it appears more as the first of the dozens of logics already created and as the first exemplification of a family that continues to expand.

This exposition of ockham's theory of (common, Personal) supposition involves the logical form of the four descent/ascent conditions and the logical relations of these with the three main modes of supposition. Central theses: each condition is a one-Way entailment, Each mode is a truth-Functional combination of conditions, Two of the three modes are not even coextensive with the two-Way entailments commonly taken as their definitions. Ockham's idea of "the singulars" of a general proposition is vague and problematic and the entailment used in the descent/ascent conditions probably cannot be taken to be logical consequence in any strict sense

An information recovery problem is the problem of constructing a proposition containing the information dropped in going from a given premise to a given conclusion that folIows. The proposition(s) to beconstructed can be required to satisfy other conditions as well, e.g. being independent of the conclusion, or being “informationally unconnected” with the conclusion, or some other condition dictated by the context. This paper discusses various types of such problems, it presents techniques and principles useful in solving them, and it develops algorithmic methods for certain classes of such problems. The results are then applied to classical number theory, in particular, to questions concerning possible refinements of the 1931 Gödel Axiom Set, e.g. whether any of its axioms can be analyzed into “informational atoms”. Two propositions are “informationally unconnected” [with each other] if no informative (nontautological) consequence of one also follows from the other. A proposition is an “informational atom” if it is informative but no information can be dropped from it without rendering it uninformative (tautological). Presentation, employment, and investigation of these two new concepts are prominent features of this paper.