FUTURE LOGIC

CHAPTER 7.EDUCTION.

1.Definitions.

2.Applications.

3.Validations.

1.Definitions.

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 7.1Eductive 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 arereversible, 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).

2.Applications.

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. OnlyI‘s andE‘s conversions are reversible.GandOpropositions are not convertible.

c.Obverted Conversions (S-P to P-nonS).

Thus, affirmatives yield a particular. OnlyI‘s andE‘s obverted conversions are reversible.GandOpropositions lack an obverted converse.

d.Conversions by Negation (S-P to nonP-S).

Thus, negatives yield a particular. OnlyA‘s andO‘s conversions by negation are reversible.RandIpropositions have no converse by negation.

e.Contrapositions (S-P to nonP-nonS).

Thus, negatives yield a particular. OnlyA‘s andO‘s contrapositions are reversible.RandIpropositions are not contraposable.

f.Inversions (S-P to nonS-nonP).

Only universals are invertible, and that irreversibly, to particular form.R,G,I, andOpropositions 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 toR,G,I, andOpropositions.

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 ofAin conversion and obverted conversion,Ein conversion by negation and contraposition. Universality plays an active role only in conversion and obverted conversion ofE, in conversion by negation and contraposition ofA, and in inversion and obverted inversion.

3.Validations.

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)ForI: ‘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)ForE: 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)ForA: ‘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)ForO: 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 theother typesof 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. ForA: contrapose, then convert. ForE: 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.