One shows that a set theoretical approach to categorical syllogisms is much more natural than the“figures, moods and rules of the syllogism approach”. (The latter satisfies “Aristotle's requirementthat the middle term M should not appear in the conclusion. Striker, 2009. ) In the “set approach”one deals with the eight 3-set intersections that span a universal set U: U=MSP+MS'P+MSP'+MS'P'+M'SP+M'S'P+M'SP'+M'S'P', where S,P,M are the usual categorical terms (interpreted nowas sets) appearing in the wording of the pairs of categorical premises (PCPs) and of the logicalconclusions (LCs) which the PCPs might entail; the union of disjoints sets is denoted by a + sign;MSP:= M∩S∩P, etc.; S',P',M' are the categorical terms non-S, non-P, non-M,, now interpreted asthe complementary sets in U of the S,P,M, respectively. In this model of categorical syllogisms,when using a cylindrical Venn diagram, (Marquand 1881), (Veitch 1952), (Karnaugh 1953), it isself-evident that if a PCP entails an LC at all, (and thus generates a valid categorical argument(VCA), then the LC singles out one and only one of the 8 subsets of U, and affirms about it eitherthat it is non-empty, or, that one of the sets S,P,M,S',P',M' is empty except, possibly, the subset whichthe LC singled out. Thus the middle term M, or its complementary M', are very much part of theLC. (Of course, to satisfy Aristotle's requirement that M should not appear in the LC, an LC of thefirst type, e.g., SPM≠Ø, may be re-written, with some loss of “intuitive information”, as I(S,P), andan LC of the second type, e.g., S=SPM, may be re-written as A(S,P).) The valid syllogisms (VSs) arethose VCAs whose LCs can be re-written in the “(S,P)-format”, i.e., one of the categoricaloperators A,O,E,I is applied to the ordered pair (S,P). After the “middle term elimination”, the LCsof the VCA\VS set are of the I(S',P'), A(P,S), or O(P,S) type. It is easy to see that there are fiveclasses of PCPs – two do not entail LCs, and three do, thus generating three distinct VCA classes.Inside each VCA class, via a relabeling transformation of the sets S,P,M, S',P',M', any of the VCA(or VS) can be recast (or reformulated) as any other VCA from the same class. This “namingcovariance” suggests that, at least from a set theoretical point of view, (i) the (S,P)-format LCrestriction is not meaningful, and (ii) one may consider that there are only three distinct VCAs (andVSs), chosen as any one representative per VCA class; for example, one may choose asrepresentatives Darapti, Darii and Barbara – all the other VCAs (not only VSs) maybe written,using appropriately chosen terms, as either a Darapti, Darii or Barbara VS. There are alwaysrelabelings transforming any VCA from the VCA\VS set into a VS. The role of VCA relabelings issimilar to the role of “reduction of syllogisms” (the latter was applied only to the VSs).
The Use of Film in Geography and History Classes: A Theoretical Approach
