Figure 2 (iv) Putting n = total number of S’s, PaM, SoM, ∴ SoP (Baroco); putting n = 1, PaM, SeM, ∴ SeP (Camestres).
403 (v) Putting n = 1, PeM, SaM, ∴ SeP (Cesare).
(vi) Putting n = 1, PeM, SiM, ∴ SoP (Festino).
AEO and EAO follow à fortiori.
Figure 3. (vii) Putting n = total number of M’s, MoP, MaS, ∴ SoP (Bocardo); putting n = 1, MeP, MiS, ∴ SoP (Ferison).
(viii) Putting n = 1, MaP, MiS, ∴ SiP (Datisi).
(ix) Putting n = 1, MiP, MaS, ∴ SiP (Disamis).
Darapti and Felapton follow à fortiori.
Figure 4. (x) Putting n = 1, PiM, MaS, ∴ SiP (Dimaris).
(xi) Putting n = 1, PaM, MeS, ∴ SeP (Camenes).
(xii) Putting n = 1, PeM, MiS, ∴ SoP (Fresison).
Bramantip, AEO, and Fesapo follow à fortiori.
EXERCISES.
346. “Whatever P and Q may stand for, we may shew à priori that some P is Q. For All PQ is Q by the law of identity, and similarly All PQ is P ; therefore, by a syllogism in Darapti, Some P is Q.” How would you deal with this paradox? [K.]
A solution is afforded by the discussion contained in section [342]; and this example seems to shew that the enquiry—how far assumptions with regard to existence are involved in syllogistic processes—is not irrelevant or unnecessary.
347. What conclusion can be drawn from the following propositions? The members of the board were all either bondholders or shareholders, but not both; and the bondholders, as it happened, were all on the board. [V.]
We may take as our premisses:
No member of the board is both a bondholder and a shareholder,
All bondholders are members of the board;
and these premisses yield a conclusion (in Celarent),
No bondholder is both a bondholder and a shareholder,
that is, No bondholder is a shareholder.
348. The following rules were drawn up for a club:—
(i) The financial committee shall be chosen from amongst the 404 general committee; (ii) No one shall be a member both of the general and library committees, unless he be also on the financial committee; (iii) No member of the library committee shall be on the financial committee.
Is there anything self-contradictory or superfluous in these rules? [VENN, Symbolic Logic, p. 331.]
Let F = member of the financial committee,
G = member of the general committee,
L = member of the library committee.
The above rules may then be expressed symbolically as follows:—
(i) All F is G ;
(ii) If any L is G, that L is F ;
(iii) No L is F.
From (ii) and (iii) we obtain (iv) No L is G.
The rules may therefore be written in the form,
(1) All F is G,
(2) No L is G,
(3) No L is F.
But in this form (3) is deducible from (1) and (2).
Hence all that is contained in the rules as originally stated may be expressed by (1) and (2); that is, the rules as originally stated were partly superfluous, and they may be reduced to
(1) The financial committee shall be chosen from amongst the general committee;
(2) No one shall be a member both of the general and library committees.
If (ii) is interpreted as implying that there are some individuals who are on both the general and library committees, then it follows that (ii) and (iii) are inconsistent with each other.