The Mathematical Works of Lewis Carroll. Ð›ÑŒÑŽÐ¸Ñ ÐšÑрролл
In order to answer this question, let us begin with the smaller Proposition, “Some Members of the Subject are Members of the Predicate,” and suppose that this is all we have been told; and let us proceed to inquire what else we need to be told, in order to know that “All Members of the Subject are Members of the Predicate”.
[Thus, we may suppose that the Proposition “Some bankers are rich men” is all the information we possess; and we may proceed to inquire what other Proposition needs to be added to it, in order to make up the entire Proposition “All bankers are rich men”.]
Let us also suppose that the ‘Univ.’ (i.e. the Genus, of which both the Subject and the Predicate are Specieses) has been divided (by the Process of Dichotomy) into two smaller Classes, viz.
(1) the Predicate;
(2) the Class whose Differentia is contradictory to that of the Predicate.
[Thus, we may suppose that the Genus “men,” (of which both “bankers” and “rich men” are Specieses) has been divided into the two smaller Classes, “rich men”, “poor men”.]
Now we know that every Member of the Subject is (as shown at p. 6) a Member of the Univ. Hence every Member of the Subject is either in Class (1) or else in Class (2).
[Thus, we know that every banker is a Member of the Genus “men”. Hence, every banker is either in the Class “rich men”, or else in the Class “poor men”.]
Also we have been told that, in the case we are discussing, some Members of the Subject are in Class (1). What else do we need to be told, in order to know that all of them are there? Evidently we need to be told that none of them are in Class (2); i.e. that none of them are Members of the Class whose Differentia is contradictory to that of the Predicate.
[Thus, we may suppose we have been told that some bankers are in the Class “rich men”. What else do we need to be told, in order to know that all of them are there? Evidently we need to be told that none of them are in the Class “poor men”.]
Hence a Proposition of Relation, beginning with “All”, is a Double Proposition, and is ‘equivalent’ to (i.e. gives the same information as) the two Propositions
(1) “Some Members of the Subject are Members of the Predicate”;
(2) “No Members of the Subject are Members of the Class whose Differentia is contradictory to that of the Predicate”.
[Thus, the Proposition “All bankers are rich men” is a Double Proposition, and is equivalent to the two Propositions
(1) “Some bankers are rich men”;
(2) “No bankers are poor men”.]
§ 4.
What is implied, in a Proposition of Relation, as to the Reality of its Terms?
Note that the rules, here laid down, are arbitrary, and only apply to Part I of my “Symbolic Logic.”
A Proposition of Relation, beginning with “Some”, is henceforward to be understood as asserting that there are some existing Things, which, being Members of the Subject, are also Members of the Predicate; i.e. that some existing Things are Members of both Terms at once. Hence it is to be understood as implying that each Term, taken by itself, is Real.
[Thus, the Proposition “Some rich men are invalids” is to be understood as asserting that some existing Things are “rich invalids”. Hence it implies that each of the two Classes, “rich men” and “invalids”, taken by itself, is Real.]
A Proposition of Relation, beginning with “No”, is henceforward to be understood as asserting that there are no existing Things which, being Members of the Subject, are also Members of the Predicate; i.e. that no existing Things are Members of both Terms at once. But this implies nothing as to the Reality of either Term taken by itself.
[Thus, the Proposition “No mermaids are milliners” is to be understood as asserting that no existing Things are “mermaid-milliners”. But this implies nothing as to the Reality, or the Unreality, of either of the two Classes, “mermaids” and “milliners”, taken by itself. In this case as it happens, the Subject is Imaginary, and the Predicate Real.]
A Proposition of Relation, beginning with “All”, contains (see § 3) a similar Proposition beginning with “Some”. Hence it is to be understood as implying that each Term, taken by itself, is Real.
[Thus, the Proposition “All hyænas are savage animals” contains the Proposition “Some hyænas are savage animals”. Hence it implies that each of the two Classes, “hyænas” and “savage animals”, taken by itself, is Real.]
§ 5.
Translation of a Proposition of Relation into one or more Propositions of Existence.
We have seen that a Proposition of Relation, beginning with “Some,” asserts that some existing Things, being Members of its Subject, are also Members of its Predicate. Hence, it asserts that some existing Things are Members of both; i.e. it asserts that some existing Things are Members of the Class of Things which have all the Attributes of the Subject and the Predicate.
Hence, to translate it into a Proposition of Existence, we take “existing Things” as the new Subject, and Things, which have all the Attributes of the Subject and the Predicate, as the new Predicate.
Similarly for a Proposition of Relation beginning with “No”.
A Proposition of Relation, beginning with “All”, is (as shown in § 3) equivalent to two Propositions, one beginning with “Some” and the other with “No”, each of which we now know how to translate.
[Let us work a few Examples, to illustrate these Rules.
(1)
“Some apples are not ripe.”
Here we arrange thus:—
“Some” Sign of Quantity.
“existing Things” Subject.
“are” Copula.
“not-ripe apples” Predicate.
or thus:—
“Some | existing Things | are | not-ripe apples.”
(2)
“Some farmers always grumble at the weather, whatever it may be.”
Here we arrange thus:—
“Some | existing Things | are | farmers who always grumble at the weather, whatever it may be.”
(3)
“No lambs are accustomed to smoke cigars.”
Here