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no syllogism in the fourth figure is valid unless the conclusion can be obtained either by converting a conclusion which can be drawn in the first figure from the same premises, or by reasoning in the first figure from the converse of the premises. In other words, if you have to test a syllogism in the fourth figure, ask first whether it is not merely a syllogism in the first figure with the conclusion converted (or converted and weakened, i.e., O from E as well as I from A). If it is, the reasoning is usually assumed to be valid. If it is not, then convert the premises, if they can be converted (reniembering that A must be converted into I and that O cannot be converted at all), and see whether the conclusion in question will not follow from them according to the principle of the first figure without the violation of any caution. If it will, the syllogism in the fourth figure is assumed to be valid ; but if the syllogism will not stand eíther of these tests, it certainly is not valid.
Here are some examples :
All M's are S's;
According to the rule just laid down this syllogism is valid, because from the premises as they stand we can reason in the first figure to the conclusion All P's are S’s ’, which by conversion gives the conclusion in question “Some S's are P's'.
The following syllogism is not valid:
All M's are S's; .:. All S’s are P's; because (1) the universal conclusion is more than we can get by converting “ All P's are S’s”, and (2) if we convert the premises we get
Some M's are P's;
from which we cannot draw any conclusion whatever without disregarding the caution which says that from a statement ! about some undesignated members of a class we cannot infer anything about any designated member or any one of a designated set of members. (3) No gods are Americans ;
All Americans are mortal ; ... Some mortals are not gods.
To reduce this syllogism to the first figure we must convert the premises, e.g.,
No Americans are gods ;
Some mortals are Americans ; ... Some mortals are not gods.
This reasoning is perfectly valid, and thus our original syllogism is vindicated.
One more example:
No human beings like torture ;
This syllogism can be proved valid like the others by means of the first figure, though when we try to “ reduce' it a difficulty immediately confronts us, for the conclusion of the syllogism is a particular negative proposition which cannot be converted, and when we convert the premises the conclusion will not follow, viz.:
Some human beings are students;
No beings that like torture are human ; .:. Some beings that like torture are not students.
To reason in this way violates this caution : "To say that something is true of certain objects does not imply that it is false of others’, i.e., to say that some (or all) human beings are students does not imply that beings who like torture and are therefore not human are not students.
We cannot convert the conclusion, we cannot prove it by converting the premises, and yet the reasoning is valid ! I
gave this example in order to bring out the difference between converting a conclusion and obtaining that conclusion by converting something else. From the premises in question, | No human beings like torture' and `All students are human', we can reason in the first figure to the conclusion that "No students like torture'. Converting this we get No beings that like torture are students’, and if this is true it is necessarily true also that 'Some beings that like torture are not students'. The fact that this is less (if it really is less) than we might have inferred does not interfere with the validity of the inference. That is to say, the conclusion given in the example can be obtained by converting the conclusion in the first figure, though the conclusion in the first figure cannot be obtained by converting the conclusion given in the example. We were testing the fourth figure by the first, not the first by the fourth !
I have spoken at some length about this reduction to the first figure because it is the traditional method of testing syllogisms not only in the fourth figure but in the second and third as well. But an undiscriminating reduction to the first figure has no more value for thought than a mechanical use of cut-and-dried rules of the syllogism', and probably has much less value than the use of the diagrams. Our thought grows mechanical all too soon, and it is a pity for logic of all studies to hasten the process. If we are really to work in the light in testing syllogisms of the fourth figure, their reduction must be accompanied by their interpretation, and the figure to which we reduce them must be determined by the interpretation--not by mere convenience for formal manipulation.*
* Looking back at example No. 1, let us fill it out as follows:
All persons a hundred miles above the surface of the earth (P's) are organic beings beyond the pressure of the atmosphere (M's).
All organic beings beyond the pressure of the atmosphere (M's) become greatly swollen (S's).
From these premises it is easy enough to draw the conclusion in the first figure that all persons a hundred miles above the surface of the earth (P's) become greatly swollen (S); meaning that if a person should reach such an altitude the intra-organic pressures, not counteracted by pres. sure from without, would cause him to swell up. But the conclusion in the fourth figure, that some things which become greatly swollen (S's) are persons a hundred miles above the surface of the earth (P), looks more like a description of some actually existing swollen objects than like an account of what would happen under certain purely hypothetical circumstances. It cannot be turned into hypothetical form, and the im. plication that such things as swollen persons a hundred miles above the surface of the earth exist is certainly much stronger, to say the least, than in the conclusion drawn according to the first figure. In so far as it involves such an implication the conclusion in the fourth figure is of course misleading and fallacious, for we have no right to confuse hypothetical and real conditions. (See page 109, note.)
Example No. 3 can be reduced as we have seen to the first figure, but if the two universal propositions which compose the premises are inter. preted as stating causal relations no conclusion is possible; that is to say, from the fact that deity involves not being an American and that being an American involves mortality we cannot draw any conclusion. The conclusion follows only if we assume that Americans exist. It is in the third figure that the syllogism is most natural and most significant, e.g.: All Americans are mortal and none of them are gods; therefore “Some mortals are not gods'. The Americans, not gods or mortals as such, are evidently the concrete individuals from a knowledge of whom the con. clusion is drawn.
In the case of example No. 4 the conclusion drawn, 6. Some beings that like torture are not students ", stands the traditional test of reduction toor rather deduction from the first figure; but so far as it implies that beings who like torture exist, it does not follow from the premises. Even the universal conclusion, “ No beings that like torture are students ”, is likely to be misleading, because the causal relation between being a student and disliking torture is so remote that the statement looks a good deal like a description of actually existing beings that like torture. It would have been better to say that if any being likes torture it is not a student. It is this which follows from the premises.
The sum and substance then of what I have said in criticism of this figure is this: It obscures the real relations under discussion, and in doing so is likely to lead to erroneous conclusions besides tempting us to work in the dark by a rule of thumb.
OTHER DEDUCTIVE ARGUMENTS.
We have already distinguished between categorical propositions and those which are hypothetical or disjunctive. The syllogisms discussed so far Hypothetical
syllogisms. involved only categorical propositions, but there are also syllogisms in which hypotheticals and disjunctives have a place. Hypothetical syllogisms run as follows:
If A is B, C is D; If a man is a Christian, he forgives;
J. S. is a Christian;
.: J. S. forgives. If A is B, C is D; If a man is a Christian, he forgives;
C is not D; J. S. does not forgive; .:. A is not B.
... J. S. is not a Christian. A Hypothetical Syllogism is thus one in which the major premise is a hypothetical proposition and the minor a categorical.
The first pair of examples, in which the state of affairs mentioned in the consequent part of the major premise is proved to exist, are said to be constructive or of the modus ponens; the second pair, in which the state of affairs mentioned in the antecedent part of the major is proved not to exist, are said to be destructive or of the modus tollens. According to this a syllogism might be constructive though