Traditional “Immediate Inference”

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Immediate Inference (a.k.a. “eduction“)

An immediate inference contrasts with a mediate inference.

A mediate inference passes from two propositions to derive a third proposition. Most famously, the categorical syllogism is a mediate inference. But in an immediate inference, the inference goes directly from a single proposition to derive a second.

“Immediate Inference” is defined in two different (incompatible) ways.

A traditional textbook’s definition is broader in scope: it is where we can take a given categorical proposition and then derive a new categorical proposition from it. The truth of the former implies the truth of the latter (but not necessarily vice versa).

A contemporary textbook’s definition is often narrower in scope: it is where we take a given categorical proposition and then derive a new categorical proposition that is equivalent. Thus, the truth of the former implies the truth of the latter and the truth of the latter implies the truth of the former. And since this is the case, the newly derived proposition is the same as the given proposition that has been stated differently in a different form.

The advantage of the contemporary approach is that, despite a few additional steps in working on some propositions, with the help of “oppositional inference,” it can replicate the results of the traditional approach.

A More Traditional Approach to Immediate Inference. . .

With the traditional approach, there are 7 kinds of eductions:
(1) Conversion,
(2) Obversion,
(3) Obverted Conversion,
(4) Contraposition,
(5) Obverted Contraposition,
(6) Inversion,
and (7) Obverted Inversion.

Like in the “contemporary” approach, distribution must be preserved.
Review the page on distribution, if need be.

(Traditional) Conversion. . .

Strictly speaking, “conversion” is not the same as “simple conversion.” The latter concerns itself only with equivalencies (which the “contemporary” approach to immediate inference is all about). They are extremely similar, however, since in a “conversion” the subject and predicate terms are interchanged.

Question: Can SAP be converted? Yes!
But only if there is a change in the proposition’s quality.
Clearly, we can’t validly go from “all men are mortal” to “all mortal beings are men.”

That’s because “mortal beings” is not distributed in the original. It says nothing about “all mortal beings.” Thus, we cannot derive a new proposition that speaks of “all mortal beings.” That goes (invalidly) beyond what the original proposition reveals to us.

We can, though, convert SAP to PIS.
“All men are mortal” validly converts to “some mortal beings are men.”

Notice that SAP and PIS are not equivalent, though.
We can validly go from SAP to PIS, but not from PIS to SAP.

This type of conversion has been called “conversion by limitation.”

A More “Contemporary” Approach Can Reach This Conclusion. . .

We can reach the same result by using the Square of Opposition (“oppositional inference”) and an Equivalency Circle. It takes noting that the truth of SAP implies the truth of SIP. From there, we can derive that PIS is equivalent from SIP. Ergo, given SAP, the proposition PIS follows.

Question: Can SEP be converted? Yes!
SEP validly converts to PES.
“No theist is a denier of God” converts to “No denier of God is a theist.”

Both subject and predicate in SEP are distributed and they remain exactly so in the conversion to PES. Thus, it’s a valid immediate inference.

Question: Can SIP be converted? Yes!
SIP validly converts to PIS.
“Some walks are enjoyable activities” converts to “some enjoyable activities are walks.”

Nothing is distributed in the original or the new proposition. Since the derived proposition doesn’t go “beyond” the original, it’s a valid immediate inference.

Question: Can SOP be converted? No!

There’s no way to validly convert this. Note that only P is distributed.

For one thing, SOP cannot be validly converted to POS because S is not distributed in SOP but is in POS. Nor can we change the quality. PIS certainly doesn’t follow from SOP (since the truth of SOP doesn’t demand the truth of PIS).

Obversion. . .

Obversion is obversion! The traditional and contemporary definition is the same.

In an obversion, the original subject is retained but the original predicate is complemented (i.e., contradicted or negated). The new proposition’s quality is changed.

SAP is obverted to SEP’.
“All good men are virtuous” obverts to “no good men are non-virtuous.”

SIP is obverted to SOP’.
“Some objects are conductors” obverts to “some objects are not non-conductors.”

SEP is obverted to SAP’.
“No angels are material” obverts to “all angels are non-material.”

SOP is obverted to SIP’.
“Some craters are not large” obverts to “some craters are non-large.”

The Laws of Thought are sufficient to prove obversion. Insofar as “All S is P” is true, it follows that “No S is non-P” by the Law of Contradiction (since “All S is P,” the Law of Contradiction tells us, it is impossible that “All S is non-P”). Similarly, insofar as “No S is P” is true, it follows that “All S is non-P” by the Law of Excluded Middle (since by that law S is either P or non-P, and it is given that No S is P, it thereby follows that All S is non-P).

Obverted Converse. . .

Performing a conversion and then an obversion gets an “obverted converse.”

(Traditional) Contraposition. . .

Traditional contraposition is defined differently than the contemporary version!

This is unfortunate, but it is what it is.
(Though sometimes traditional contraposition has been called “partial contraposition.”)

To get to the point, in traditional contraposition, the derived proposition’s subject will be the complemented (i.e., contradicted or negated) original predicate. This can be obtained by converting the obverse of an originally given proposition.

Question: Can SAP be contraposed? Yes!
Step 1 (obvert): SEP’
Step 2 (convert): P’ES
“All men are mortal” contraposes to “No non-mortals are men.”

Note that this is valid since distribution is preserved. S is distributed in the original and remains distributed in the derived proposition; moreover, though P is not distributed in the original, this implies P’ is distributed in the original, and thus, as P’ is distributed in the derived proposition, it is a valid inference.

Question: Can SIP be contraposed? No!
Step 1 (obvert): SOP’
Step 2 (convert): N/A

Since it’s impossible to convert SOP’, we cannot contrapose the SIP proposition.

Question: Can SEP be contraposed? Yes!
Step 1 (obvert): SAP’
Step 2 (convert): P’IS
“No metals are cheap” contraposes to “Some non-cheap things are metals.”

Question: Can SOP be contraposed? Yes!
Step 1 (obvert): SIP’
Step 2 (convert): P’IS
“Some bookshelves are not organized” contraposes to “Some non-organized things are bookshelves.”

That both SEP and SOP validly contrapose to P’IS should remind us that traditional immediate inference is not about equivalences.

A More “Contemporary” Approach Can Reach These Conclusions. . .

We can reach the same results by using the Square of Opposition (“oppositional inference”) and an Equivalency Circle.

Consider how the truth of SEP implies the truth of P’IS by noting the following: on the Square of Opposition, SEP allows us to derive SOP; and with an Equivalency Circle, SOP can lead us to SIP’ and then to P’IS.

Consider how the truth of SOP implies the truth of P’IS by noting the following: with an Equivalency Circle, SOP can lead us to SIP’ and then to P’IS.

Given this, by the way, SOP is equivalent to P’IS (because that derivation was only on the Equivalency Circle); but SEP is not equivalent to P’IS (because that derivation, in part, required the Square of Opposition to go from SEP to SOP).

Obverted Contrapositive. . .

Performing a contraposition and then an obversion gets an “obverted contrapositive.”

Inversion. . .

Any inversion has it that the derived proposition’s subject will be the complemented (i.e., contradicted or negated) original subject while retaining the original predicate.

A trial-and-error process can help us find inverses.

Using only obversion and (traditional) conversion, we can get all of our traditional immediate inferences by alternating between the two.

First, we’ll start with obversion.
Original: SAP
Obvert: SEP’
Convert: P’ES
Obvert: P’AS’
Convert: S’IP’
Obvert: S’OP

Second, we’ll start with conversion.
Original: SAP
Convert: PIS
Obvert: POS’

Thus, the inverse of SAP is S’OP.
“All priests are male” has the inverse of “some non-priests are not male.”

A More “Contemporary” Approach Can Reach This Conclusion. . .

Reaching the conclusion S’OP from the premise SAP takes a bit of work here, though some intuition may help provide a short-cut or two.

Here’s one method…
This will make sure we have a valid inference!

To be exhaustive, we can create an Equivalency Circle for SAP and an Equivalency Circle for S’OP. The goal then is to find an instance where both have the same subject and predicate. They do with P’ (subject) and S’ (predicate).

From SAP we can infer P’AS’ on the Equivalency Circle; and from S’OP we can infer P’IS’ on the Equivalency Circle. Next, we can relate P’AS’ and P’IS’ on the Square of Opposition. Thus, given P’A’S as true, P’I’S’ follows as true.

Question: Can SIP be inversed? No!
Question: Can SOP be inversed No!

Various methods can reveal that neither SIP nor SOP can be validly inversed.

For example. . .

Using only obversion and (traditional) conversion, we can get all of our traditional immediate inferences by alternating between the two.

First, we’ll start with obversion.
Original: SIP
Obvert: SOP’

Second, we’ll start with conversion.
Original: SIP
Convert: PIS
Obvert: POS’

No inverse shows up. There is no valid way to get to the inverse of SIP.
(Don’t believe this result? Try a more “contemporary” approach. Be exhaustive!)

Question: Can SEP be inversed? Yes!
Convert: PES
Obvert: PAS’
Convert: S’IP
“No brute is rational” is inversed to “some non-brute is rational.”
(Caution: doesn’t this assume there exists some non-brutes?)^[1]

Obverted Inverse. . .

Performing an inversion and then an obversion gets an “obverted inverse.”

A Summary. . .

Traditional immediate inference is more expansive than the more “contemporary” approach. It becomes clear that for any categorical proposition, at least three more categorical propositions can be immediately inferred. The universality of A and E propositions (versus the particularity of I and O propositions) is what provides us with more possibilities.

***
Existential Import?

https://archive.org/details/thescienceoflogi01coffuoft/page/248/mode/2up

^[1] “No brute is rational” is inversed to “some non-brute is rational.”

The example above with inversion might get us worried.

We inferred from the proposition “no brute is rational” that “some non-brute is rational.” It might be argued that this is formally invalid. Why? Because it assumes (or seems to assume) that there exists “non-brutes.” Likewise, other such inferences seem to assume that S, P, S’, and P’ represent real things that exist.

To be careful, we can modify our inferences. Instead of inferring that “some non-brute is rational,” we rather could infer that “if some non-brute exists, then it is rational.” This yields a conditional.

Still, it might be asked, must the original proposition itself imply the existence of anything (or the derived proposition imply the existence of anything)? For a detailed discussion, see chapter seven of The Science of Logic by Peter Coffey (volume one).