CHAPTER 10. SYLLOGISM: VALIDATIONS.
Validation of a syllogism consists in showing its consistency with the axioms of logic. If it is shown that the conclusion follows from the premises, the form of thought is justified. When we encounter a syllogism which results in some antinomy, we obviously reject it; when we reject a sequence of premises and conclusion, we automatically validate that sequence of premises with a contradictory conclusion. Only thus is the balance of consistency restored. This defines validation.
Note well that the conclusion must follow from the premises; mere compatibility between the propositions is not sufficient to imply a connection between them. Thus, invalid syllogisms will display either a conclusion incompatible with the premises somehow; or a conclusion which, though compatible with them, is not more compatible than its contradictory is. Thus, validation could be viewed as the discovery of those forms of thought which satisfy a precondition set by Logic, namely that the premises be shown to imply the conclusion. Invalidity signifies failure of the syllogism to fall under the class so defined.
The validation process itself uses logic; but this circularity does not logically put it in doubt. This apparent paradox can easily be explained as follows. The science of Logic is merely a verbalization of observed fundamental phenomena (identity, inclusion, the need for consistency); these phenomena are out-there and in accordance with themselves; our science’s task is to apprehend the exact extent and limit of their manifestations. If the use by logic, for the validation of its processes, resulted in an inconsistency with any of the apparent controlling principles of our world, we would be justified in questioning it. But so long as no inconsistency is found, it must be trusted. For to say that the validation processes depend on their own conclusions to work, merely confirms how basic this science is. Whereas, the attempt to cast doubt on logic itself appeals to our logical instincts for its credibility, and therefore constitutes an inconsistency, that between the primary denial and hidden dependence on logic. Of the two theses only the former, then, is self-consistent. We conclude that validation is meaningful as a process of clarifying the consistency of valid logical processes with the axiomatic basics.
Many approaches to validation have been developed by logicians. From the start, Aristotle was aware that each figure had its own character, and was able to identify the method of validation most appropriate to each. However, other methods are always worth exploring, to obtain further confirmation, to be exhaustive, and to train the mind.
a. The First Figure.
This is the most basic figure, and essentially defines the nature of subsumption and inclusion. It is validated by ‘exposition’. Aristotle formulated the Law of Identity as “What is, is what it is”. If a thing exists, it has certain attributes. If according to our perceptions and insights it appears that anything which is X is Y, then anything which appears to be X must appear Y. Our suppositions are justified, until otherwise proven, and we must submit to the reality of our experiences and to the meaning of our words. This reflects the self-evidence of the world around us, and attaches our words to their intention.
Thus, if one says ‘All M are P’, then indeed anything which is M, is likewise P, so that the (all or some) S which are M must also be P. If any S were not P, this would signify that some M are not P, and contradict our original assumption that all M are P. Therefore, granting the two premises, the conclusion follows, and AAA, AII, and ARR, are valid. The same can be argued in the case of a negative major, or we can reduce the forms EAE, EIO, and ERG, to the affirmative form, by obverting the major.
Subaltern forms of course follow by eduction. Syllogism with the contradictory conclusions to the above, are proved invalid, by opposition. No conclusion logically follows from all the remaining pairs of premises, so they have no validity as syllogistic processes.
b. The Second Figure.
Reduction consists in demonstrating that the validity of one inference proceeds from that of another, already established. Reductio ad absurdum shows that a major (A) and minor (B) premise together imply a conclusion (C), because if A were asserted together with the negation of C, they would together imply the negation of B, through an already validated process. This method, with the major premise kept constant while the rest is tested, is used to validate the second figure by reference to the first.
Thus, ‘All P are M and No S is M’ imply ‘No S is P’, for granting that all P are M, if some S were P, then some S would be M, which contradicts our original minor premise that no S is M. In this way, AEE in the second figure is reduced to AII in the first figure, through a syllogism involving the original major term as middle term. We can proceed likewise to validate the other valid moods of the second figure.
Although so provable, the second figure should also be viewed as reasonable on its own merit, by exposition. Because essentially it defines for us the mechanics of exclusion, just as the first figure reflected more those of inclusion. If we consider two things one of which is excluded and another is included in a third, they cannot reasonably be visualized as contiguous.
c. The Third Figure.
Validation by exposition seems to be the method most suited for the third figure, although again other approaches are possible, because the conclusion is always particular. We proceed by showing that in certain instances under scrutiny, two events are contiguous (because the whole includes the part), so that the conclusion holds. This is a positive approach, which can be buttressed by reductions.
Thus, we could take AII in the third figure, and reduce it ad absurdum through EIO in the first figure. We test the effect of contradicting the conclusion while holding on to the minor premise, this time. The resulting syllogism has the original subject as its middle term, and its conclusion contradicts the original major premise. We can likewise validate the other valid moods of the third figure.
Of course, we could use similar methods to reduce the third figure to the second, or vice versa by changing our constant.
d. The Fourth Figure.
Here, direct reduction is the most natural treatment. The two premises of EIO in the fourth figure are each converted, to yield EIO in the first figure, which results in the same conclusion. Alternatively, convert the minor premise only, and reduce to the second figure; or convert the major only, to obtain the third figure.
Reductio ad absurdum is an indirect form of reduction, which we use quite often in everyday thinking. Another approach, just sampled, is direct reduction. This is more formal minded, in that one or both premises are subjected to an eductive process to reduce the syllogism to a first figure mood with the same conclusion or one implying it. This method is not restricted to the fourth figure, but can equally be practised in the second and third. For example, for AEE second figure, the major is converted by negation, and the minor obverted, to obtain EAE in the first figure. Again, for EIO in the third figure, the minor is converted, to yield EIO in the first. The full list of such processes is easy to develop and well established, and available in most logic text books, so we will not here belabor the reader with excessive detail.
e. Secondary Syllogisms.
Though subalterns could be analyzed independently, once a subalternant syllogism is established, its subalterns are easily seen to follow by eduction.
With regard to the imperfect syllogism, combinations of premise which do not yield a normal S-P conclusion, but nevertheless can be wrung-dry to yield a nonS-nonP conclusion, they are dealt with by direct reduction through valid third figure arguments. For the first figure, use obversion of the major and obverted conversion of the minor; for the second, contrapose the two premises if positive, or use obverted conversion if negative; for the third figure, obvert both premises; for the fourth figure, draw the obverted converse of the major and obvert the minor.
As already indicated, this is the process of invalidation, and of course should be applied to each and every invalid mood systematically. The method is similar, since a mood which concludes something contradictory or contrary to our valid forms must be rejected. More broadly, forms which are not established as valid somehow, are automatically kept apart: the onus of proof can be left to them, as it were.
The science of Logic has, as above, analyzed validation and invalidation processes used to establish the general truth of the reasoning processes described in the previous chapter. Whereas it works in formal terms, we normally do not refer to formal logic in practise to verify our thinking or spot fallacies in it. We repeat the expository or reductive processes, every time we need to understand or convince ourselves of an argument, with the specific contents of our propositions. Going through such a process serves to integrate our knowledge, comparing its elements and checking their consistency.
Once, however, one is trained in logic, one may well refer to the science’s findings to unravel some argument. In this context, the rules and the canons of Logic may be appealed to intellectually. Analysis of the quantities and polarities involved, consideration of the distribution of terms, are then valuable tools, if one has them well in mind.
A popular way to verify that arguments are kept in accord with logical rigor, is through application of the fallacy tests developed by Aristotle and logicians since. These warn of common pitfalls which one may encounter. They reveal how one may, through hidden equivocation (the Four Terms), confusing suggestions (as in the Many Questions), self-contradiction (Begging the Question), or other such devices, befuddle ourselves or others. Study of these, found in most text books, is of course valuable training.
We have stated that syllogism involves three, and only three, propositions; and likewise three and, only three, terms. In practise, it may seem that other possibilities exist. But logic shows that such atypical argument is actually either abridged or compound syllogism, which can be reduced to the standard formats.
a. Enthymemes are syllogism a premise or the conclusion of which is left unstated, but which is clearly taken to be understood or implied. This artifice is common in normal discourse, as when we rely on context, and can only be formally validated by bringing the suppressed proposition out in the open, and checking that the argument obeys the rules of logic.
An epicheirema is an argument in which one or both of the premises is supported by a reason. This simply means that the explained premise is itself the result of a prior syllogism.
b. We often have trains of thought: these may be reduced to chains of two or more syllogisms, of any kind. Such an entangling of argumentation is called a sorites. The name is more traditionally applied specifically to certain regular chains of argument in the first figure, which suppress intermediate conclusions. These are as follows:
All (or Some) A are B,
All B are C,
All C are D,
All D are E,
All (or No) E are F,
therefore All (or Some) A are (or are not) F’.
We move from a universal or particular, but always affirmative, minor premise, through one or more intermediate universal affirmative premises, to a final affirmative or negative, but always universal, major premise, to obtain a conclusion with the quantity and subject of the minor premise and the polarity and predicate of the major.
There are thus four valid moods. AAAA, AAEE, IAAI, IAEO, for each set of three or more premises. The validation of these is achieved by listing a series of syllogism with the same result. For instance:
A is B and B is C, therefore A is C;
A is C and C is D, therefore A is D;
A is D and D is E, therefore A is E;
A is E and E is F, therefore A is F.
The conclusion of each syllogism is used as premise in the next, if any. Clearly, the middle terms must all be distributive.
The name ‘sorites’ could equally be applied to any complex of arguments, in any combination of figures, instead of just to such a regular series of first figure syllogism. Irregular sorites takes the conclusion of any unit of argument, and transfers it to another argument where it serves as a premise.
Thus, sorites in the widest sense is simply the multiple branching of thought in all directions. Each unit argument within this network may be indicated by only a highlight — a premise or two, or a conclusion — the most significant or controversial part. A sorites is a collection of such highlights, an abridged argument.
c. Certain arguments called immediate inference by added determinants or by complex conception, seem like immediate inference, but are really mediate inference. This refers to arguments like ‘since X is Y, then ZX is ZY’. If the qualifying Z is an adjective, the argument is valid, since if some X are Y, and all X are Z, we may infer, in a third figure syllogism, that some Y (those which are X) are indeed Z. But if the Z clause does not fit in such a valid syllogism, it in some cases cannot be passed on.
In practise, such argument can easily be fallacious, as a result of double meanings (as in ‘science is fun, so scientists are funny’), or the use of terms in inappropriate ways (as in ‘horses are fast, so the head of a horse is the head of a fast’).
Such rough logic is not very reliable, and should not be considered a part of formal logic. It is better to insist on strict conformity to formal processes. If a specific kind of content allows for special logical rules, then these may be clarified explicitly in a small field of logic all of their own.