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The Logician © Avi Sion All rights reserved
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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 7.
EDUCTION.
Immediate inference is the process of discovering another proposition
implicit in a given proposition, without use of additional information. It
differs from syllogistic reasoning, in that the latter draws a new proposition
from two or more previous ones. We have come across one sample of such inference
in the foregoing text, namely opposition. Here we will deal systematically with
another, which may be called eduction.
What eduction does is to change the position and/or polarity of the
terms; this often results in a proposition of different polarity or quantity.
The original proposition is the premise, the educed proposition an implication
of it.
Often, to fully understand a proposition, we have to restate it in
another way, some hidden character of it is thereby revealed, facilitating
further thought. The structural change we effect in the given form yields new
information, although a simple process.
Starting from an S-P format, we may be able to obtain propositions
through transposition and/or negation of terms, in the following ways. Table
7.1
Eductive Processes.
The source proposition is then called obvertend, convertend,
contraponent, invertend, and so on, while the target proposition is called
obverse, converse, contraposite, inverse, as the case may be.
Whereas such processes are generally possible with one or both of the
universals, they are not always feasible in the case of singulars or
particulars, as we shall see. Note also that some processes are reversible,
and some are not: only in some cases may the source proposition be educed again
from its implication (by the same or any other eductive process).
We shall now list the implications of the various plural forms, and then
validate the processes involved. Although these may tedious details, they do
constitute an important training for the mind. a.
Obversions (S-P to S-nonP).
Thus, all forms are obvertible, and so reversibly. b.
Conversions (S-P to P-S).
Thus, affirmatives yield a particular. Only I's
and E's conversions are reversible. G
and O propositions are not convertible. c.
Obverted Conversions (S-P to P-nonS).
Thus, affirmatives yield a particular. Only I's
and E's obverted conversions are
reversible. G and O
propositions lack an obverted converse. d.
Conversions by Negation (S-P to nonP-S).
Thus, negatives yield a particular. Only A's
and O's conversions by negation are reversible. R and I propositions have
no converse by negation. e.
Contrapositions (S-P to nonP-nonS).
Thus, negatives yield a particular. Only A's
and O's contrapositions are reversible. R and I propositions are
not contraposable. f.
Inversions (S-P to nonS-nonP).
Only universals are invertible, and that irreversibly, to particular
form. R, G,
I, and O propositions are
not invertible. g.
Obverted Inversions (S-P to nonS-P).
Only universals may be subjected to obverted inversion, and that
irreversibly, to particular form. Process not applicable to R,
G, I,
and O propositions.
We note at the outset that while quantity may be lost, it cannot be
gained. A universal or singular proposition may yield a particular, but a
singular or particular cannot produce a universal. It is also clear that, with
the exception of obversion, the processes applicable to singulars are so only by
virtue of the corresponding particulars implicit in them by opposition. This is
true also of A in conversion and
obverted conversion, E in conversion
by negation and contraposition. Universality plays an active role only in
conversion and obverted conversion of E,
in conversion by negation and contraposition of A, and in inversion and obverted inversion.
We can validate all these processes by working on two: obversion and
conversion, for the others follow. a.
Obversion. 'S is P' to 'S is not nonP'. The
negation of a term normally signifies the absence of some phenomenon. In the
absence of a phenomenon, other phenomena necessarily exist: there is a world out
there, be it real or illusory; appearances constantly occur. Furthermore, by the
law of contradiction, a phenomenon S cannot both be and not-be something called
P. Thus, the phenomenon P cannot be both present and absent in the thing called
S. Just as is and is-not are mutually exclusive, so are the affirmation and
negation inconsistent.
To say S is P posits that P is found in S; to say S is-not nonP means
that the absence of P is absent from S. Which is not to imply, in either case,
that S is not simultaneously other things than P or nonP — Q, R, etc. So, S
can be P and something other than P, although it cannot both exhibit and
not-exhibit P. These arguments thus define the copula is-not and the term nonP
more precisely.
What is true here in the case of singular propositions, can be argued
equally for plural propositions, since the latter subsume the former. That is,
they collect them together as a unit while at the same time dealing with them
each one singly; so that the statement does not concern them as either a count
of individuals or as a collective unity, but is merely an abbreviated statement
being distributed out to its instances equivalently.
Thus S-P merely means S1-P1, S2-P2, S3-P3, etc. Here again, this doctrine
provides an opportunity to more precisely define formal concepts. b.
Conversion. Here each
quantitative is considered separately. (i)
For I:
'Some S are P' and 'Some P are S' each means 'Some things are both S and P'; we
are seeing S and P together, we may attribute either to the other; this defines
the generality of our copula is, and proceeds from the law of Identity. (ii)
For E:
likewise, 'No S is P' and 'No P is S' each means 'Nothing is both S and P'; S
and P never appear together, have no instances in common. This clarifies our
copula is-not, telling us that S is-not P is the same as P is-not S; and also
reminding us that 'No X is Y' means 'All X are-not Y'. (iii)For
A: 'All S are P' by subsumption
implies that 'Some S are P', and therefore also that 'Some P are S' as shown
above. However, it could-not imply 'All P are S', although in some cases such
mutual inclusion occurs, because there are cases where it does not. Here again,
we are better defining our form, in accord with its common usage. (iv)For
O: from 'Some S are not P' we cannot
infer 'Some P are not S', for it happens that 'All P are S'; that is, it happens
that only S are P, i.e. that P does not occur elsewhere; our form is intended as
that broad and inclusive of possible circumstance. Other
approaches to these validations are possible. But the intent here was to show
that these need not be viewed as 'proofs', so much as focusing more precisely on
the forms our consciousness naturally uses, and inspecting every aspect of their
selected meanings to delimit the extent of their application to phenomena as
they appear to us. With
regard to the other types of eduction,
they can be reduced to combinations of the above two processes, and thus
validated. Thus: c.
Obverted Conversion. Convert,
then obvert. d.
Conversion by Negation.
Obvert, then convert. e.
Contraposition. Obvert, then
convert, then obvert. f.
Inversion. For A:
contrapose, then convert. For E:
convert, then contrapose. g.
Obverted Inversion. Invert,
then obvert.
Note lastly, one can say 'some nonS are nonP' (or the converse) for just
about any S and P chosen at random, with the exception of certain very broad
terms, like 'existence', which have no real negatives. So processes which yield
such conclusions are not very informative.
Addendum: "No S is P" means that S and P are incompatible - if one of them is present, the other one cannot also be present. "No nonS is nonP" means that S and P are exhaustive - if one of them is absent the other cannot also be absent. To affirm both these propositions is to say the two terms S and P (or nonS and nonP) are contradictory. To affirm the first and deny the second is to say S and P are contrary. To deny the first and affirm the second is to say S and P are subcontrary. To deny both is to say some S are P and some nonS are nonP - i.e. they are compatible and inexhaustive.
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