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The Logician © Avi Sion All rights reserved |
FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
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.
f.
Rejection.
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. |
