And now what counters will this information enable us to place in the SMALLER Diagram, so as to get some Proposition involving x and y only, leaving out m? Let us take its four compartments, one by one.
First, No. 5. All we know about THIS is that its OUTER portion is empty: but we know nothing about its inner portion. Thus the Square MAY be empty, or it MAY have something in it. Who can tell? So we dare not place ANY counter in this Square.
Secondly, what of No. 6? Here we are a little better off. We know that there is SOMETHING in it, for there is a red counter in its outer portion. It is true we do not know whether its inner portion is empty or occupied: but what does THAT matter? One solitary Cake, in one corner of the Square, is quite sufficient excuse for saying "THIS SQUARE IS OCCUPIED", and for marking it with a red counter.
As to No. 7, we are in the same condition as with No. 5--we find it PARTLY 'empty', but we do not know whether the other part is empty or occupied: so we dare not mark this Square.
And as to No. 8, we have simply no information at all.
The result is
------- | | 1 | |---|---| | | | -------
Our 'Conclusion', then, must be got out of the rather meager piece of information that there is a red counter in the xy'-Square. Hence our Conclusion is "some x are y' ", i.e. "some new Cakes are not-nice (Cakes)": or, if you prefer to take y' as your Subject, "some not-nice Cakes are new (Cakes)"; but the other looks neatest.
We will now write out the whole Syllogism, putting the symbol &there4[*] for "therefore", and omitting "Cakes", for the sake of brevity, at the end of each Proposition.
[*][NOTE from Brett: The use of "&there4" is a rather arbitrary selection. There is no font available in general practice which renders the "therefore" symbol correction (three dots in a triangular formation). This can be done, however, in HTML, so if this document is read in a browser, then the symbol will be properly recognized. This is a poor man's excuse.]
"Some new Cakes are unwholesome; No nice Cakes are unwholesome &there4 Some new Cakes are not-nice."
And you have now worked out, successfully, your first 'SYLLOGISM'. Permit me to congratulate you, and to express the hope that it is but the beginning of a long and glorious series of similar victories!
We will work out one other Syllogism--a rather harder one than the last--and then, I think, you may be safely left to play the Game by yourself, or (better) with any friend whom you can find, that is able and willing to take a share in the sport.
Let us see what we can make of the two Premisses--
"All Dragons are uncanny; All Scotchmen are canny."
Remember, I don't guarantee the Premisses to be FACTS. In the first place, I never even saw a Dragon: and, in the second place, it isn't of the slightest consequence to us, as LOGICIANS, whether our Premisses are true or false: all WE have to do is to make out whether they LEAD LOGICALLY TO THE CONCLUSION, so that, if THEY were true, IT would be true also.
You see, we must give up the "Cakes" now, or our cupboard will be of no use to us. We must take, as our 'Universe', some class of things which will include Dragons and Scotchmen: shall we say 'Animals'? And, as "canny" is evidently the Attribute belonging to the 'Middle Terms', we will let m stand for "canny", x for "Dragons", and y for "Scotchmen". So that our two Premisses are, in full,
"All Dragon-Animals are uncanny (Animals); All Scotchman-Animals are canny (Animals)."
And these may be expressed, using letters for words, thus:--
"All x are m'; All y are m."
The first Premiss consists, as you already know, of two parts:--
"Some x are m'," and "No x are m."
And the second also consists of two parts:--
"Some y are m," and "No y are m'."
Let us take the negative portions first.
We have, then, to mark, on the larger Diagram, first, "no x are m", and secondly, "no y are m'". I think you will see, without further explanation, that the two results, separately, are
----------- ----------- | | | |0 | | | --|-- | | --|-- | | |0 | 0| | | | | | | |--|--|--|--| |--|--|--|--| | | | | | | | | | | | --|-- | | --|-- | | | | |0 | | ----------- -----------
and that these two, when combined, give us
----------- |0 | | | --|-- | | |0 | 0| | |--|--|--|--| | | | | | | --|-- | |0 | | -----------
We have now to mark the two positive portions, "some x are m'" and "some y are m".