An extension of valid syllogisms to valid categorical arguments, and a reduction of the latter to only Barbara, Darapti, Darii and Disamis. To the memory of my sister Cristina Popa, 1948-2018

TL;DR

This paper introduces a Set Theory Model (STM) that extends valid syllogisms to a broader class of valid categorical arguments, reducing them to four main types and providing a unified framework for their analysis.

Contribution

It presents a novel STM that generalizes valid syllogisms, allowing negative terms, expanding premise types, and reducing VCAs to four fundamental forms, with a group-based re-labeling approach.

Findings

STM encompasses more valid arguments than classical syllogisms.

VCAs can be categorized into four types with interchangeable forms.

The model simplifies the analysis of categorical arguments using group re-labelings.

Abstract

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, non-S, P, non-P, M, non-M, and is thus partitioned into eight subsets: SPM, SP(non-M),...,(non-S)(non-P)(non-M), where the intersection was denoted by adjacency. In STM all superfluous syllogistic figures are disregarded, and both the positive terms, S, P, M, and the negative terms, non-S, non-P, non-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 (resp. S) premises from the six formulable via only positive terms, to eight, and the number of distinct PCPs from the 36 appearing in the Classical 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. These VCAs, (and the VS), split into four types. Each type contains eight VCAs. Any VCA can be re-written as any other VCA of the same type, via a group G of eight re-labelings. Thus the VCAs Barbara, Darapti, Darii, Disamis, can be chosen as type representatives for both VCAs and VS, and via a term relabeling, any VCA or VS can be re-written, without changing their PCP or LC contents, as either a Barbara, Darapti, Darii, or Disamis. One discusses the VCAs and their LCs, 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). Note that CCS restricts the VS to only LCs of the formats A(S,P), E(S,P), I(S,P), O(S,P).