Symbolic Logic

[The Reader had better provide himself with 4 Red Counters and 5 Grey ones.]

[For example, “Some x exist”, “No y′ exist”, &c.]

[For example, “Some xy′ exist”, “No x′ are y”, &c.]

[Thus, “Some xy′ exist”, “No x′ are y”, &c., are said to be in terms of x and y.]

[The Reader should make out all these for himself. In the “books” example, these Propositions would be “Some new books exist”, &c.]

This tells us that there is nothing in the North Half; that is, that the North Half is empty; that is, that the North-West Cell and the North-East Cell are both of them empty. And this we can represent by placing two Grey Counters in the North Half, one in each Cell.

[The Reader may perhaps think that it would be enough to place a Grey Counter on the partition in the North Half, and that, just as a Red Counter, so placed, would mean “This Half is occupied”, so a Grey one would mean “This Half is empty”.

This, however, would be a mistake. We have seen that a Red Counter, so placed, would mean “At least one of these two Cells is occupied: possibly both are.” Hence a Grey one would merely mean “At least one of these two Cells is empty: possibly both are”. But what we have to represent is, that both Cells are certainly empty: and this can only be done by placing a Grey Counter in each of them.

In the “books” example, this Proposition would be “No old books exist”.]

[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No new books exist”, &c.]

This tells us that there is at least one Thing in the North-West Cell; that is, that the North-West Cell is occupied. And this we can represent by placing a Red Counter in it.

[In the “books” example, this Proposition would be “Some old English books exist”.]

[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “Some old foreign books exist”, &c.]

This tells us that there is nothing in the North-West Cell; that is, that the North-West Cell is empty. And this we can represent by placing a Grey Counter in it.

[In the “books” example, this Proposition would be “No old English books exist”.]

[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No old foreign books exist”, &c.]

[In the “books” example, this Proposition would be “No old books exist”.

Hence this is a Double Proposition, and is equivalent to the two Propositions “No old English books exist” and “No old foreign books exist”.]

This tells us that at least one Thing, in the North Half, is also in the West Half. Hence it must be in the space common to them, that is, in the North-West Cell. Hence the North-West Cell is occupied. And this we can represent by placing a Red Counter in it.

[Note that the Subject of the Proposition settles which Half we are to use; and that the Predicate settles in which portion of it we are to place the Red Counter.

In the “books” example, this Proposition would be “Some old books are English”.]

[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “Some old books are foreign”, &c.]

This tells us that at least one Thing, in the West Half, is also in the North Half. Hence it must be in the space common to them, that is, in the North-West Cell. Hence the North-West Cell is occupied. And this we can represent by placing a Red Counter in it.

[In the “books” example, this Proposition would be “Some English books are old”.]

[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “Some English books are new”, &c.]

We see that this one Diagram has now served to represent no less than three Propositions, viz.

(1) “Some xy exist;
(2)   Some x are y;
(3)   Some y are x”.

Hence these three Propositions are equivalent.

[In the “books” example, these Propositions would be

(1) “Some old English books exist;
(2)   Some old books are English;
(3)   Some English books are old”.]

[For example, if we were told to convert the Proposition

“Some apples are not ripe,”

we should first choose our Univ. (say “fruit”), and then complete the Proposition, by supplying the Substantive “fruit” in the Predicate, so that it would be

“Some apples are not-ripe fruit”;

and we should then convert it by interchanging its Terms, so that it would be

“Some not-ripe fruit are apples”.]

(1) “Some xy exist” = “Some x are y” = “Some y are x”.
(2) “Some xy′ exist” = “Some x are y′” = “Some y′ are x”.
(3) “Some x′y exist” = “Some x′ are y” = “Some y are x′”.
(4) “Some x′y′ exist” = “Some x′ are y′” = “Some y′ are x′”.

This tell us that no Thing, in the North Half, is also in the West Half. Hence there is nothing in the space common to them, that is, in the North-West Cell. Hence the North-West Cell is empty. And this we can represent by placing a Grey Counter in it.

[In the “books” example, this Proposition would be “No old books are English”.]

[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No old books are foreign”, &c.]

This tells us that no Thing, in the West Half, is also in the North Half. Hence there is nothing in the space common to them, that is, in the North-West Cell. That is, the North-West Cell is empty. And this we can represent by placing a Grey Counter in it.

[In the “books” example, this Proposition would be “No English books are old”.]

[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No English books are new”, &c.]

We see that this one Diagram has now served to present no less than three Propositions, viz.

(1) “No xy exist;
(2)   No x are y;
(3)   No y are x.”

Hence these three Propositions are equivalent.

[In the “books” example, these Propositions would be

(1) “No old English books exist;
(2)   No old books are English;
(3)   No English books are old”.]

The two equivalent Propositions, “No x are y” and “No y are x”, are said to be ‘Converse’ to each other.

[For example, if we were told to convert the Proposition

“No porcupines are talkative”,

we should first choose our Univ. (say “animals”), and then complete the Proposition, by supplying the Substantive “animals” in the Predicate, so that it would be

“No porcupines are talkative animals”, and we should then convert it, by interchanging its Terms, so that it would be

“No talkative animals are porcupines”.]

(1) “No xy exist” = “No x are y” = “No y are x”.
(2) “No xy′ exist” = “No x are y′” = “No y′ are x”.
(3) “No x′y exist” = “No x′ are y” = “No y are x′”.
(4) “No x′y′ exist” = “No x′ are y′” = “No y′ are x′”.

Let us take, next, the Proposition “All x are y”.

 
Let us take, lastly, the Double Proposition “Some x are y and some are y′”, each part of which we already know how to represent.

Similarly we may represent the three similar Propositions, “Some x′ are y and some are y′”, “Some y are x and some are x′”, “Some y′ are x and some are x′”.

The Reader should now get his genial friend to question him, severely, on these two Tables. The Inquisitor should have the Tables before him: but the Victim should have nothing but a blank Diagram, and the Counters with which he is to represent the various Propositions named by his friend, e.g. “Some y exist”, “No y′ are x”, “All x are y”, &c. &c.

The Diagram is supposed to be set before us, with certain Counters placed upon it; and the problem is to find out what Proposition, or Propositions, the Counters represent.

As the process is simply the reverse of that discussed in the previous Chapter, we can avail ourselves of the results there obtained, as far as they go.

First, let us suppose that we find a Red Counter placed in the North-West Cell.

We know that this represents each of the Trio of equivalent Propositions

“Some xy exist” = “Some x are y” = “Some y are x”.

Next, let us suppose that we find a Grey Counter placed in the North-West Cell.

We know that this represents each of the Trio of equivalent Propositions

“No xy exist” = “No x are y” = “No y are x”.

Next, let us suppose that we find a Red Counter placed on the partition which divides the North Half.

We know that this represents the Proposition “Some x exist.”

Next, let us suppose that we find two Red Counters placed in the North Half, one in each Cell.

We know that this represents the Double Proposition “Some x are y and some are y′”.

Next, let us suppose that we find two Grey Counters placed in the North Half, one in each Cell.

We know that this represents the Proposition “No x exist”.

Lastly, let us suppose that we find a Red and a Grey Counter placed in the North Half, the Red in the North-West Cell, and the Grey in the North-East Cell.

We know that this represents the Proposition, “All x are y”.

[Note that the Half, occupied by the two Counters, settles what is to be the Subject of the Proposition, and that the Cell, occupied by the Red Counter, settles what is to be its Predicate.]

Red in North-East, Grey in North-West;
Red in South-West, Grey in South-East;
Red in South-East, Grey in South-West;
Red in North-West, Grey in South-West;
Red in South-West, Grey in North-West;
Red in North-East, Grey in South-East;
Red in South-East, Grey in North-East.

Once more the genial friend must be appealed to, and requested to examine the Reader on Tables II and III, and to make him not only represent Propositions, but also interpret Diagrams when marked with Counters.

The Questions and Answers should be like this:—

Q. Represent “No x′ are y′.”
A. Grey Counter in S.E. Cell.
Q. Interpret Red Counter on E. partition.
A. “Some y′ exist.”
Q. Represent “All y′ are x.”
A. Red in N.E. Cell; Grey in S.E.
Q. Interpret Grey Counter in S.W. Cell.
A. “No x′y exist” = “No x′ are y” = “No y are x′”.
&c., &c.

At first the Examinee will need to have the Board and Counters before him; but he will soon learn to dispense with these, and to answer with his eyes shut or gazing into vacancy.

[The Reader is strongly advised, in reading this Chapter, not to refer to the above Diagrams, but to make a large copy of the right-hand one for himself, without any letters, and to have it by him while he reads, and keep his finger on that particular part of it, about which he is reading.]

[Thus, in the “books” example, we might say “Let m mean ‘bound’, so that m′ will mean ‘unbound’”, and we might suppose that we had subdivided the Class “old English books” into the two Classes, “old English bound books” and “old English unbound books”, and had assigned the N.W. Inner Cell to the one, and the N.W. Outer Cell to the other.]

[Thus, in the “books” example, we might suppose that we had subdivided the “new English books” into the two Classes, “new English bound books” and “new English unbound books”, and had assigned the S.W. Inner Cell to the one, and the S.W. Outer Cell to the other.]

[Thus, in the “books” example, we have assigned the Inner Square to “bound books” and the Outer Border to “unbound books”.]

This tells us that there is nothing in the Inner portion of the North Half; that is, that this Compartment is empty. And this we can represent by placing two Grey Counters in it, one in each Cell.

These sixteen Propositions of Existence are the only ones that we shall have to represent on this Diagram.

“Some x are m” = “Some m are x.”

We know that each of these is equivalent to the Proposition of Existence “Some xm exist”, which we already know how to represent.

“No x are m” = “No m are x.”

We know that each of these is equivalent to the Proposition of Existence “No xm exist”, which we already know how to represent.

These thirty-two Propositions of Relation are the only ones that we shall have to represent on this Diagram.

The Reader should now get his genial friend to question him on the following four Tables.

The Victim should have nothing before him but a blank Triliteral Diagram, a Red Counter, and 2 Grey ones, with which he is to represent the various Propositions named by the Inquisitor, e.g. “No y′ are m”, “Some xm′ exist”, &c., &c.