Equivalent "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 Contemporary Approach to Immediate Inference. . .

With the contemporary approach, there are three kinds of eductions:
(1) Simple Conversion,
(2) Contraposition,
and (3) Obversion.

A valid eduction is such that a term that is not distributed in the original proposition is (likewise) not distribtued in the new proposition.

(Review the page on distribution, if necessary.)

Otherwise, it is invalid.

In a simple conversion, the subject and predicate switch places. This only works on I and E propositions, not with A or O propositions, because the distribution of their terms is preserved.

For example, “no apple is a vegetable” (SEP) can be validly converted to “no vegetable is an apple” (PES). Both the subject and predicate are distributed in the original proposition and they remain distributed in the derived proposition.

“Some difficult books are worth reading” (SIP) can be validly converted to “some things worth reading are difficult books” (PIS). No term is distributed in the original and no term is distributed in the derived proposition. Hence, it is valid.

But consider “all men are mortal” (SAP). That cannot be equivalent to “all mortal beings are men” (PAS). The original proposition’s predicate is not distributed; but the new proposition has it as a distributed term as the subject. Hence, it is invalid.

Contraposition has it that both the subject and predicate flip places and each are complemented (i.e., contradicted or negated) in the derived proposition. This will validly work with O and A propositions only. Contraposition appears convoluted, to be sure; yet it is a good immediate inference!

For example, “some politicians are not liars” (SOP) can be validly contraposed to “some non-liars are not non-politicians” (P’OS’). Or take the classic example: “all men are mortal” (SAP). This can be validly contraposed to “all non-mortal beings are non-men” (P’AS’). When we think about that latter proposition, that has to be true given the original. Non-mortal beings cannot be men! Also, the distribution of the terms hold.

Note there is a change in distribution when complementing a term. Thus (for example) while S is distributed and P is not in SAP, this is consistent with P’AS’ having S’ as not being distributed and P’ as being distributed.

However, something like SEP cannot be equivalent to P’ES’. Since P is distributed in the original, P’ cannot be distributed. But since P’ is distributed in P’ES’, it cannot be validly derived from SEP.

The immediate inference of obversion validly works for all categorical propositions (A, I, E, and O). The derived proposition’s predicate will contain the complemented (i.e., contradicted or negated) predicate of the original proposition. This inference is valid as long as the quality of the derived proposition is changed from the original’s.

For example, “all men are mortal” (SAP) is equivalent to “no men are non-mortal” (SEP’). (The latter can be better expressed as “no men are immortal (SEP’).) This example implicitly makes clear why there must be a change in quality once the predicate is negated. There is a move from an affirmative proposition to a negative proposition.

“Some objects are material” (SIP) is equivalent to “some objects are not non-material [i.e., immaterial]” (SOP’). Also remember — as with all of contemporary forms of immediate inference — that because these two propositions are equivalent, not only does SIP imply SOP’, but SOP’ implies SIP.

“Equivalency Circles” with the More Contemporary Approach. . .

A handy visual approach to contemporary immediate inference is through a circle. With a given categorical proposition, three equivalent propositions can be derived. (As mentioned here, I adopt this approach from Dr. Casey.)

In this example, (1) we started with SAP as given, (2) derived SEP’ from SAP via obversion, (3) derived P’ES from SEP’ via simple conversion. (4) derived P’AS’ from P’ES via obversion, and (5) can get back to SAP from P’AS’ via contraposition.

So, let’s say we have “all men are mortal” (SAP). Since being “non-mortal” is the same as being “immortal,” we can clean up the language to read more easily. (1) “All men are mortal” is equivalent to (2) “no men are not immortal,” to (3) “no immortal beings are men,” and to (4) “all immortal beings are non-men.” Pretty nice, right?

This general approach works with any categorical proposition.

In this example, (1) we started with B’IX as given, (2) derived B’OX’ from B’IX via obversion, (3) derived XOB from B’OX’ via contraposition, (4) derived XIB’ from XOB via obversion, and (5) can get back to B’IX from XIB’ via simple conversion.

Example 3: Given that the proposition Q’AN’ is false, what is the truth, falsehood, or indeterminacy of the proposition QON’?

Notice this required use of “oppositional inference.”

Example 4: Given that the proposition UAB’ is true, what is the truth, falsehood, or indeterminacy of the proposition UIB?

This example also required used of “oppositional inference.”

***

As noted here, my presentation and understanding of eduction was greatly shaped by Professor Gerard Casey at Liberty Classroom. He taught logic for over 30 years, including at the University College Dublin.

Equivalent “Immediate Inference”