Boolean Subtraction and Division with Application in the Design of Digital Circuits

The subject of Boolean subtraction and division dates back for over a century to the work of George Boole. Nevertheless, this subject is unfamiliar to us because it has been banished from Boolean algebra. In fact, some authors claim that there is no such thing as Boolean subtraction and division. The purpose of this work, however, is to present with clarity the subject of logical subtraction and division and its practical application in the design of digital circuits.

The Laws of Thought

Chapter 16 tells the story of George Boole, his wife Mary Everest Boole, and their five daughters. Boole was largely self-taught and was appointed the first Professor of Mathematics at Queen’s College Cork, now University College Cork, in 1849. When he died, his wife Mary and young daughters returned to London, where Mary made the acquaintance of James Hinton and his circle of literary friends. Mary developed methods of teaching mathematics that she passed on to her daughters, who all lived remarkable lives. Ethel Lily Boole was a very successful novelist, author of The Gadfly. She married Wilfrid Michael Voynich, famous for discovering the Voynich manuscript. Mary Ellen Boole married Charles Howard Hinton, who would popularize the notion of higher dimensions.

The Birth of Semantics

We attempt here to trace the evolution of Frege’s thought about truth. What most frames the way we approach the problem is a recognition that hardly any of Frege’s most familiar claims about truth appear in his earliest work. We argue that Frege’s mature views about truth emerge from a fundamental re-thinking of the nature of logic instigated, in large part, by a sustained engagement with the work of George Boole and his followers, after the publication of Begriffsschrift and the appearance of critical reviews by members of the Boolean school.

An extension of valid syllogisms to valid categorical arguments and a reduction of the latter to only Barbara, Darapti, Darii and Disamis

One presents a simple Set Theory Model (STM) of the valid categorical arguments (VCAs) - a proper superset of the valid (categorical) syllogisms (VS). The main STM initiator was George Boole, who worked with a “universe of discourse”, U, which contains the pairwise complementary sets, or categorical terms, S,S'(non-S),P,P'(non-P),M,M'(non-M), and is thus partitioned into eight subsets: SPM:= S∩P∩M, S'PM,...,S'P'M'. In STM all superfluous syllogistic figures are disregarded, and both the positive terms, S,P,M, and the negative terms, S',P',M', are allowed to appear in the pairs of categorical premises (PCPs) and their entailed logical consequences (LCs). This increases the number of distinct P (and S) premises from the six formulable via only positive terms, to eight, and the number of distinct PCPs from the 36 appearing in the Classic Categorical Syllogistic (CCS), to the 64 appearing in the STM. Out of the latter 64 PCPs, only 32 PCPs entail LCs and thus generate VCAs. The PCPs, VCAs, and the VS, split into four types. In short, one may say that each type contains eight VCAs, which can be re-written, via the term relabelings, p:= P↔P', s:=S↔S', m:=M↔M', and their compositions, ps, pm, sm, psm, as any other VCA of the same type. Thus the VCAs Barbara, Darapti, Darii and Disamis, can be chosen as type representatives for both VCAs and VS, and, via one of the above term relabelings, any VCA or VS can be re-written, without changing their PCP or LC contents, as either a Barbara, Darapti, Darii, or Disamis. Besides the VCAs and their precise LCs, (out of which the middle term was not eliminated), one discusses simple/biliteral VCA sorites, empty set constraints (ESC), “VCA distribution conservation”, and other Rules of Valid Categorical Arguments (RofVCA) which are “VCA generalized versions” of some of the Rules of Valid Syllogisms (RofVS). Both STM and CCS follow, for PCP classification purposes, the convention that the P term has to appear in the firstly listed premise. One compares the CCS, which defines the VS as PCPs formulable via only positive terms, whose entailed LCs are restricted to only the statements A(S,P), E(S,P), I(S,P), O(S,P), with the STM, whose VCAs also entail LCs of these other formats: A(P,S), E(S',P'), I(S',P'), O(P,S).

An extension of valid syllogisms to valid categorical arguments

One presents the Set Theory Model (STM) of the valid categorical arguments (VCAs) as an improvement on the Classic Categorical Syllogistic (CCS) approach to the valid (categorical) syllogisms (VS) – a proper subset of the VCAs. The STM was initially developed by George Boole and Lewis Carroll, who worked with a “universe of discourse”, U, which contains the pairwise complementary sets, or categorical terms, S,S'(non-S),P,P'(non-P),M,M'(non-M), and is thus partitioned into 8 subsets: SPM:= S∩P∩M, S'PM,...,S'P'M'. In STM both the positive terms, S,P,M, and the negative terms, S',P',M', are allowed to appear in the pairs of categorical premises (PCPs) and their entailed logical consequences (LCs). One thus counts 64 distinct PCPs, out of which only 32 PCPs generate VCAs, and 32 PCPs do not entail any LC. By comparison, CCS admits PCPs worded only via the positive terms S,P,M, and accepts as VS LCs only the statements E(S,P), I(S,P), A(S,P), or O(S,P). It is easier to see on the VCA set, than on its VS proper subset, that there are only three distinct types of VCAs, (and of VS), and that all the VCAs of the same type are equivalent modulo the term relabelings p:= P↔P', s:=S↔S', m:=M↔M' and their compositions. Besides the VCAs and their LCs, one discusses VCA (or categorical) sorites, “VCA distribution conservation”, empty set constraints (ESC), and “VCA generalized versions” of the Rules of Valid Syllogisms (RofVS).