DIAGRAM 4
We suggested a general formula for the first three (actually, four) of the hermeneutic principles which begin with the phrasekol davar shehayah bikhlal veyatsa…
Given the three premises, common to the four Rules: 1.Major premise:All S1 are P1. 2.Minor premise:All S2 are P2. 3.Subjectal premise:All S2 are S1, but not all S1 are S2. and, the fourth premise, as applicable in each Rule: 4.Predicatal premise: The relation between P1 and P2. What are resulting relations (conclusions)? ·Between S1 and P2 (main issue). ·Between S2 and P1. ·Between S1 and P1, other than the above given. ·Between S2 and P2, other than the above given. |
The first three premises can be individually depicted as follows:
Diagram 4a
Note that the first two premises leave open the possibility that subject and predicate may be co-extensive, so that the circles labeled S1 and P1 might be equal in size, and likewise the circles labeled S2 and P2 might be one. On the other hand, the relation between S2 and S1 can only be as above depicted, with S2 smaller than S1.
As for the remaining (predicatal) premise and the conclusion(s), we shall consider each case each in turn.
But first, let us consider what general conclusions can be drawn from the common premises of all such arguments.
Given the major and subjectal premises, we can at the outset, without resort to the other premises, make the following syllogistic inferences and graphic presentation:
1/AAA | 3/OAO |
All S1 are P1 | Some S1 are not S2 |
All S2 are S1 | All S1 are P1 |
So, all S2 are P1 | So, some P1 are not S2 |
Diagram 4b
Note: I did not mention the above 3/OAO syllogism in my original treatment (Judaic Logic, p. 147).
It should, however, be pointed out that in the case of Rule 10, since the major premise is particularized in an effort to restore consistency, these initial inferences become annulled.
Similarly, given the minor and subjectal premises, we can at the outset, without resort to the other premises, make the following syllogistic inference and graphic presentation:
3/AAI |
All S2 are P2 |
All S2 are S1 |
So, some S1 are P2 |
Diagram 4c
This conclusion is an indefinite particular, note – i.e. in some cases, we may find “All S1 are P2”; and in others, “Only some S1 are P2”.[1]
[1]Quite incidentally, I notice while writing this that inFuture Logic(p. 37), I state that the mood 3/AAI is a derivative of 3/AII; but it could equally be derived from 3/IAI. Similarly, 3/EAO could be derived from either 3/EIO (as stated) or 3/OAO.