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The Logician © Avi Sion All rights reserved |
FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 51.
ELEMENTS AND COMPOUNDS.
Our inquiry must now turn to a new doctrine, which may be called
factorial analysis. This doctrine is to some extent an offshoot of that of
opposition, and interesting for its own sake. Its essential value, however, is
to prepare us for the investigation of modal induction, although some
information of relevance to deduction is to be found in it. This doctrine is
new, because modal logic involves a lot more forms than the traditional logic,
and so an issue which was obvious and minor now looms large.
The various categorical propositions, A,
E, I,
O, and their modal counterparts, were
presented as the building blocks or elements of knowledge. Elementary
propositions are relatively abstract items of knowledge, which intersect in
various combinations.
The conjunctions of two or more such elementary propositions, concerning
the same subject and predicate, may be referred to as compounds. A compound is
in a sense a unit of information too, although it is expressed by us as a sum of
elements. Knowledge could conceivably have been constructed by giving each
compound a distinct form, but then the elements of data they contain in common
would have remained hidden. We wisely, even if instinctively, chose to limit the
number of forms in our thoughts, and deal with compounds in terms of their
constituent elements.
Not all elementary propositions may be conjoined, of course; some are
incompatible, for example 'A and O'. Some conjunctions are redundant, as when a proposition is
conjoined with another which is in any case implicit in it; for instance, A
and I together mean no more than A
alone. However, some compounds are significant, and our task will now be to
identify these.
At any given stage in the development of knowledge we may have no or
partial or complete information, concerning the relation between a specific
subject and predicate pair. The sum of information available may be called a
formula. A formula may consist of one or more elementary propositions. The
elementary propositions taken individually may all be formulas, if they happen
to summarize the state of knowledge at that point. Their combinations in
distinct consistent compounds, summing up the known without redundancies, are
also possible formulas.
Any information not included in a formula is to be considered unavailable
in the context of knowledge; thus a formula must contain all known data
concerning the two terms in question.
We will express compound formulas in the briefest way, e.g. 'AIn'
signifying 'A and In',
without use of extraneous words or symbols for conjunction; it being understood
that propositions so fused concern the same subject and predicate, of course.
This study will concern itself only with plural propositions, although
some comments about singulars will be made when useful. This is done for the
sake of simplicity and clarity, but also in recognition that science is
primarily interested in broad statements, and only incidentally in minutiae.
Within the closed system of actual propositions, that covered effectively
by classical logic, only five formulas were conceivable: A,
I, E,
O, and IO. This in a sense resolves the issue of formulas with regard to
extensional modality taken in isolation.
When the other types of modality are introduced, the issue becomes less
obvious and more complex. The following table shows methodically what
combinations, of the 20 elementary propositions (singulars ignored), can occur
consistently and without redundancy. It results that there are a total of 195
distinct formulas, 20 of which are of course elementary, and the rest compounds
of up to 4 propositions (75X2 + 79X3 + 21X4).
Note that compounds are expressed in their most compressed form (e.g. AI
is included in A). In practise,
we do not always compress compound statements; sometimes we prefer to stress an
implication. For instances, 'can never be' stresses that 'cannot be' implies 'is
never'; 'can sometimes be' suggests a compound of 'is sometimes' and 'can'. Table
51.1 Consistent
Conjunctions of Categoricals. (Click to see table.)
Note in passing that if we considered either natural or temporal modality
as a closed system, we would find ourselves in each case with a total of 49
formulas, 12 of which were elementaries, and the remaining 37 were compounds of
up to 4 propositions. Formulas involving actual propositions only are 5 in
number, and formulas which mix modality types number 102.
Now although this list of formulas is complete in itself, it will become
apparent that it does not in fact exhaust the possible states of knowledge. We
shall see that formulas of this kind are gross assertions, which do not clarify
all the issues involved.
Once we view a compound as a unit, one complex proposition, we may ask
what oppositional relations exist between compounds. Consider, for example, the
affirmative compounds AIn, ApIn, ApI. They may be
placed in a hierarchy relative to each other and to the cognate elements, as
follows: Diagram
51.1 Hierarchy of
Compounds.
Looking at the arrows of subalternation, we see a gradual softening of
position, ranging from An to Ip. There is a continuum of affirmative statements, in which
temporal modality could also be inserted. A similar hierarchy may be developed
for the analogous negatives. More complex, bipolar compounds also have their
inter-oppositions, including many such subalternations.
The contradictory of any compound is a disjunctive proposition, note
well; it disjoins the contradictories of the various elements involved, in an
'and/or' manner. Thus, for examples: AIn
is contradicted by 'O and/or Ep', ApIn
is
contradicted by 'On and/or Ep', ApI
is
contradicted by 'On and/or E'.
If AIn is false, then one of O
or EpO or Ep must be true; each of the latter is by itself only contrary to AIn:
it is the disjunction as a whole which is contradictory.
Similarly for all other compounds. Note that some 'ands' yield impossible
combinations; these are as such automatically eliminated. For example: the
contradictory of ApIOc is 'On and/or E and/or At',
in which any combination of On with At is rejectable at once, meaning that only the alternatives 'On
or EOn or E or AtE
or At' are viable.
There is no need for us to work out all the interrelationships in
advance. The work can be done ad hoc, as specific need arises.
Once we regard a compound proposition as a single whole in its own right,
we are enticed to ask whether there are corresponding compound syllogisms.
Consider, for example, the closed system of actuals. Here, we have one
conjunctive formula, 'I and O'; its
contradictory is 'A or E',
since not-{I and O} means notI
and/or notO, which means E and O
(= E) or I and A
(= A) or E and A
(impossible).
With regard to the conjunctive compound 'I
and O'. Compound syllogism is impossible in the first figure, since we
would need both an A and an E
major premise with the same terms. It is also impossible in the second figure,
since this figure only yields negative conclusions. However, in the third
figure, we have the following valid double syllogism, merging 3/IAI
and 3/OAO:
Some M are P and some M are not P,
and All M are S,
therefore, Some S are P and some S are not P.
It must follow that the disjunctive compound 'A
or E' (which contradicts IO)
also has a valid mood of the syllogism. It must be in the first figure,
disjoining 1/AAA and 1/EAE,
so that denial of its conclusion causes denial of its major premise, by reductio
ad absurdum to the above one:
All M are P or No M is P,
and All S are M,
therefore, All S are P or No S is P.
This shows that the compound IO,
and its contradictory, have a deductive life of their own. These are the only
Aristotelean syllogisms capable of processing compounds.
The same can be done with modal compounds. I will not go into detail but
simply give a pair of examples:
Some M can be P and some M can not-be P,
and All M must be S,
therefore, Some S can be P and some S can not-be P.
All M must be P or No M can be P,
and All S must be M,
therefore, All S must be P or No S can be P.
Other quantities and modalities than these can similarly be processed.
The reader is encouraged to try and evolve a full list of compound syllogisms,
as an exercise, with reference to the full list of compound propositions given
earlier. Are there tandems involving triple or quadruple compounds?
To fully understand how any two terms, S and P, are related, we must know
their relations in both directions: from S to P and from P to S. These may be
called the front and reverse side of the overall relation. The S-P side alone
can only provide us with a 'flat' picture of the intersection of the terms; the
reality is 'stereoscopic', and to express it entirely we need to specify the P-S
side as well.
The S to P and P to S relations may be called complementary. The possible
complements of any S to P relation are the propositions compatible with its
converse. Thus, the doctrine of complements is an offshoot of the doctrines of
eduction and opposition.
Consider actual categoricals. Since 'All/This/Some S are P' are
convertible to 'Some P are S' only, the possible complements of A,
R, or I (in S-P), are A or O
(in P-S). Since 'No S is P' is convertible fully to 'No P is S', the latter is
the only possible complement of the former. Lastly, since G
and O are not at all convertible,
they are compatible with any of A, E,
or IO, on the reverse side.
Similarly for modals. Since An,
Rn, In, Ap,
Rp, Ip, are all
convertible to Ip only, their
possible complements are all the propositions compatible with this converse,
namely any form but En. For En,
which converts fully to En, the only
possible complement is En. Lastly,
since Gn, On, Ep,
Gp, Op, are none of them
at all convertible, any form may complement them. As with naturals, so with
temporals.
Just as we developed a list of possible gross formulas for the S to P
relationship, we could additionally work out the compatible P to S gross
formulas for each S to P gross formula. This would provide us with more complex,
'two-way' gross formulas, yielding a fuller picture of reality than heretofore
available.
(If the complement is identical in form to the original proposition, then
the relation may be said to be reciprocal; otherwise, it is nonreciprocal. Thus,
for instance, if 'all S are P' and 'all P are S' are both true, the relation of
S and P is reciprocal; in contrast, A complemented
by IO is nonreciprocal.)
Note in passing that we could go a step further, and consider not only
P-S relations as complements to S-P, but also relations involving the antitheses
of one or both of the terms. In that case, obversion, obverted conversion,
conversion by negation, contraposition, inversion, and obverted inversion, all
become significant, telling us more about the possible combinations of S and P
in all their facets.
To get deeper still, we would perhaps have to take transitives into
consideration, looking into their possible conjunctions, as 'supplements', on
the S-P and P-S sides, and indeed, on every other side.
However, all these complications will be ignored in this treatise. |
