The given premisses may all be summed up in the proposition: Everything is AbC or AbD or aBCd or abcd or BcD. From this, the above special results and others follow immediately.
497. Given that everything is either Q or R, and that all R is Q, unless it is not P, prove that all P is Q. [K.]
The premisses may be written as follows: (1) All r is Q, (2) All PR is Q.
By (1), All Pr is Q, and by (2), All PR is Q ; but All P is Pr or PR ; therefore, All P is Q.
498. Where A is present, B and C are either both present at once or absent at once; and where C is present, A is present. Describe the class not-B under these conditions. [Jevons, Studies, p. 204.]
The premisses are (1) All A is BC or bc, (2) All C is A.
By (1) All b is a or c, and by (2) All b is A or c, therefore, All b is c.
499. It is known of certain things that (1) where the quality A is, B is not; (2) where B is, and only where B is, C and D are. 514 Derive from these conditions a description of the class of things in which A is not present, but C is. [Jevons, Studies, p. 200.]
The premisses are: (1) All A is b ; (2) All B is CD ; (3) All CD is B.
No information regarding aC is given by (1). But by (2), All aC is b or D ; and by (3), All aC is B or d.
Therefore, All aC is BD or bd.
500. Taking the same premisses as in the previous section, draw descriptions of the classes Ac, ab, and cD. [Jevons, Studies, p. 244.]
By (1), Everything is a or b, and by (2), Everything is b or CD. Therefore, Everything is aCD or b ; and by (3), Everything is B or c or d. Therefore, Everything is aBCD or bc or bd.
Hence we infer immediately All Ac is b, All ab is c or d, All cD is b.
501. There is a certain class of things from which A picks out the ‘X that is E, and the Y that is not Z,’ and B picks out from the remainder ‘the Z which is Y and the X that is not Y.’ It is then found that nothing is left but the class ‘Z which is not X.’ The whole of this class is however left. What can be determined about the class originally? [Venn, Symbolic Logic, pp. 267, 8.]