The doctrine of the square of opposition originated with Aristotle in

the fourth century BC and has occurred in logic texts ever since.

Although severely criticized in recent decades, it is still regularly

referred to. The point of this entry is to trace its history from the

vantage point of the early twenty-first century, along with closely

related doctrines bearing on empty terms.

The square of opposition is a group of theses embodied in a diagram.

The diagram is not essential to the theses; it is just a useful way to

keep them straight. The theses concern logical relations among four

logical forms:

NAMEFORMTITLEAEvery SisPUniversal Affirmative ENo SisPUniversal Negative ISome SisPParticular Affirmative OSome Sis notPParticular Negative

The diagram for the traditional square of opposition is:

The theses embodied in this diagram I call ‘SQUARE’.

They are:

**SQUARE**

- ‘Every
*S*is*P*’ and ‘Some

*S*is not*P*’ are contradictories.

- ‘No
*S*is*P*’ and ‘Some

*S*is*P*’ are contradictories.

- ‘Every
*S*is*P*’ and ‘No

*S*is*P*’ are contraries.

- ‘Some
*S*is*P*’ and ‘Some

*S*is not*P*’ are subcontraries.

- ‘Some
*S*is*P*’ is a subaltern of

‘Every*S*is*P*’.

- ‘Some
*S*is not*P*’ is a

subaltern of ‘No*S*is*P*’.

These theses were supplemented with the following explanations:

- Two propositions are contradictory iff they cannot both be true and

they cannot both be false.

- Two propositions are contraries iff they cannot both be true but

can both be false.

- Two propositions are subcontraries iff they cannot both be false

but can both be true.

- A proposition is a subaltern of another iff it must be true if its

superaltern is true, and the superaltern must be false if the subaltern

is false.

Probably nobody before the twentieth century ever held exactly these

views without holding certain closely linked ones as well. The most

common closely linked view that is associated with the traditional

diagram is that the **E** and **I**

propositions *convert simply*; that is, ‘No *S* is

*P*’ is equivalent in truth value to ‘No *P*

is *S*’, and ‘Some *S* is *P*’

is equivalent in truth value to ‘Some *P* is

*S*’. The traditional doctrine supplemented with simple

conversion is a very natural view to discuss. It is Aristotle’s view,

and it was widely endorsed (or at least not challenged) before the late 19th century. I call this

total body of doctrine ‘[SQUARE]’:

[SQUARE] =_{df}SQUARE + “theEandIforms convert

simply”

where

A proposition

converts simplyiff it is

necessarily equivalent in truth value to the proposition you get by

interchanging its terms.

So [SQUARE] includes the relations illustrated in the diagram plus the

view that ‘No *S* is *P*’ is equivalent to

‘No *P* is *S*’, and the view that

‘Some *S* is *P*’ is equivalent to

‘Some *P* is *S*’.

### 1.1 The Modern Revision of the Square

Most contemporary logic texts symbolize the traditional forms as

follows:

Every SisP∀ x(Sx→

Px)No SisP∀ x(Sx→

¬Px)Some SisP∃ x(Sx&

Px)Some Sis notP∃ x(Sx&

¬Px)

If this symbolization is adopted along with standard views about the

logic of connectives and quantifiers, the relations embodied in the

traditional square mostly disappear. The modern diagram looks like

this:

THE MODERN REVISED SQUARE:

This has too little structure to be particularly useful, and so it

is not commonly used. According to Alonzo Church, this modern view

probably originated sometime in the late nineteenth

century.^{[1]}

This representation of the four forms is now generally accepted,

except for qualms about the loss of subalternation in the left-hand

column. Most English speakers tend to understand ‘Every

*S* is *P*’ as requiring for its truth that there

be some *S*s, and if that requirement is imposed, then

subalternation holds for affirmative propositions. Every modern logic

text must address the apparent implausibility of letting ‘Every

*S* is *P*’ be true when there are no

*S*s. The common defense of this is usually that this is a

logical notation devised for purposes of logic, and it does not claim

to capture every nuance of the natural language forms that the symbols

resemble. So perhaps

‘∀*x*(*S**x* →

*P**x*)’ does fail to do complete justice to

ordinary usage of ‘Every *S* is *P*’, but

this is not a problem with the logic. If you think that ‘Every

*S* is *P*’ requires for its truth that there be

*S*s, then you can have that result simply and easily: just

represent the recalcitrant uses of ‘Every *S* is

*P*’ in symbolic notation by adding an extra conjunct to

the symbolization, like this:

∀*x*(*S**x*

→ *P**x*)

&

∃*x**S**x*.

This defense leaves logic intact and also meets the objection, which

is not a logical objection, but merely a reservation about the

representation of natural language.

Authors typically go on to explain that we often wish to make

generalizations in science when we are unsure of whether or not they

have instances, and sometimes even when we know they do not, and they

sometimes use this as a defense of symbolizing the **A**

form so as to allow it to be vacuously true. This is an

argument from convenience of notation, and does not bear on logical

coherence.

### 1.2 The Argument Against the Traditional Square

Why does the traditional square need revising at all? The argument

is a simple

one:^{[2]}

Suppose that ‘

S’ is an empty term; it

is true of nothing. Then theIform: ‘Some

SisP’ is false. But then its contradictory

Eform: ‘NoSisP’

must be true. But then the subalternOform:

‘SomeSis notP’ must be true. But that

is wrong, since there aren’t anySs.

The puzzle about this argument is why the doctrine of the

traditional square was maintained for well over 20 centuries in the

face of this consideration. Were 20 centuries of logicians so obtuse as

not to have noticed this apparently fatal flaw? Or is there some other

explanation?

One possibility is that logicians previous to the 20th century must

have thought that no terms are empty. You see this view referred to

frequently as one that others

held.^{[3]}

But with a few very

special exceptions (discussed below) I have been unable to find anyone

who held such a view before the nineteenth century. Many authors do not

discuss empty terms, but those who do typically take their presence for

granted. Explicitly rejecting empty terms was never a mainstream

option, even in the nineteenth century.

Another possibility is that the particular **I** form

might be true when its subject is empty. This was a common view

concerning *indefinite propositions* when they are read

generically, such as ‘A dodo is a bird’, which (arguably)

can be true now without there being any dodos now, because being a

bird is part of the essence of being a dodo. But the truth of such

indefinite propositions with empty subjects does not bear on the forms

of propositions that occur in the square. For although the indefinite

‘A dodo ate my lunch’ might be held to be equivalent to

the particular proposition ‘Some dodo ate my lunch’,

generic indefinites like ‘A dodo is a bird’, are quite

different, and their semantics does not bear on the quantified

sentences in the square of opposition.

In fact, the traditional doctrine of [SQUARE] is completely coherent

in the presence of empty terms. This is because on the traditional

interpretation, the **O** form lacks existential

import. The **O** form is (vacuously) true if its subject

term is empty, not false, and thus the logical interrelations of

[SQUARE] are unobjectionable. In what follows, I trace the development

of this view.

The *doctrine* that I call [SQUARE], occurs in Aristotle. It

begins in *De Interpretatione* 6–7, which contains three

claims: that **A** and **O** are

contradictories, that **E** and **I** are

contradictories, and that **A** and **E**

are contraries (17b.17–26):

I call an affirmation and a negation contradictory

opposites when what one signifies universally the other signifies not

universally, e.g. every man is white—not every man is white, no man

is white—some man is white. But I call the universal affirmation and

the universal negation contrary opposites, e.g. every man is just—no

man is just. So these cannot be true together, but their opposites may

both be true with respect to the same thing, e.g. not every man is

white—some man is white.

This gives us the following fragment of the square:

But the rest is there by implication. For example, there is enough to

show that **I** and **O** are subcontraries:

they cannot both be false. For suppose that **I** is

false. Then its contradictory, **E**, is true. So

**E**’s contrary, **A**, is false. So

**A**’s contradictory, **O**, is true. This

refutes the possibility that **I** and **O**

are both false, and thus fills in the bottom relation of

subcontraries. Subalternation also follows. Suppose that the

**A** form is true. Then its contrary **E**

form must be false. But then the **E** form’s

contradictory, **I**, must be true. Thus if the

**A** form is true, so must be the **I**

form. A parallel argument establishes subalternation from

**E** to **O** as well. The result is

SQUARE.

In *Prior Analytics* I.2, 25a.1–25 we get the additional

claims that the **E** and **I** propositions

convert simply. Putting this together with the doctrine of *De Interpretatione* we have the full

[SQUARE].

^{[4]}

### 2.1 The Diagram

The diagram accompanying and illustrating the doctrine shows up

already in the second century CE; Boethius incorporated it into his

writing, and it passed down through the dark ages to the high medieval

period, and from thence to today. Diagrams of this sort were popular

among late classical and medieval authors, who used them for a variety

of purposes. (Similar diagrams for modal propositions were especially

popular.)

### 2.2 Aristotle’s Formulation of the O Form

Ackrill’s translation contains something a bit unexpected: Aristotle’s

articulation of the **O** form is *not* the

familiar ‘Some *S* is not *P*’ or one of its

variants; it is rather ‘Not every *S* is

*P*’. With this wording, Aristotle’s doctrine

automatically escapes the modern criticism. (This holds for his views

throughout *De Interpretatione*.

^{[5]})

For assume again that ‘

*S*’ is an empty term, and

suppose that this makes the

**I**form ‘Some

*S*is

*P*’ false. Its contradictory, the

**E**form: ‘No

*S*is

*P*’, is

thus true, and this entails the

**O**form in Aristotle’s

formulation: ‘Not every

*S*is

*P*’, which

must therefore be true. When the

**O**form was worded

‘Some

*S*is not

*P*’ this bothered us, but

with it worded ‘Not every

*S*is

*P*’ it

seems plainly right. Recall that we are granting that ‘Every

*S*is

*P*’ has existential import, and so if

‘

*S*’ is empty the

**A**form must be

false. But then ‘Not every

*S*is

*P*’

*should*be true, as Aristotle’s square requires.

On this view *affirmatives* have existential import, and

*negatives* do not—a point that became elevated to a general

principle in late medieval

times.^{[6]}

The ancients thus did

not see the incoherence of the square as formulated by Aristotle

because there was no incoherence to see.

### 2.3 The Rewording of the O Form

Aristotle’s work was made available to the Latin west principally via

Boethius’s translations and commentaries, written a bit after 500

CE. In his translation of *De interpretatione*, Boethius

preserves Aristotle’s wording of the **O** form as “Not

every man is white.” But when Boethius comments on this text he

illustrates Aristotle’s doctrine with the now-famous diagram, and he

uses the wording ‘Some man is not

just’.^{[7]}

So this must have seemed

to him to be a natural equivalent in Latin. It looks odd to us in

English, but he wasn’t bothered by it.

Early in the twelfth century Abelard objected to Boethius’s

wording of the **O**

form,^{[8]}

but Abelard’s writing was not widely influential, and except for him

and some of his followers people regularly used ‘Some *S*

is not *P*’ for the **O** form in the

diagram that represents the square. Did they allow the

**O** form to be vacuously true? Perhaps we can get some

clues to how medieval writers interpreted these forms by looking at

other doctrines they endorsed. These are the theory of the syllogism

and the doctrines of contraposition and obversion.

One central concern of the Aristotelian tradition in logic is the

theory of the categorical syllogism. This is the theory of

two-premised arguments in which the premises and conclusion share

three terms among them, with each proposition containing two of

them. It is distinctive of this enterprise that everybody agrees on

which syllogisms are valid. The theory of the syllogism partly

constrains the interpretation of the forms. For example, it determines

that the **A** form has existential import, at least if

the **I** form does. For one of the valid patterns (Darapti) is:

Every

CisB

EveryCisA

So, someAisB

This is invalid if the **A** form lacks existential

import, and valid if it has existential import. It is held to be

valid, and so we know how the **A** form is to be

interpreted. One then naturally asks about the **O**

form; what do the syllogisms tell us about it? The answer is that they

tell us nothing. This is because Aristotle did not discuss weakened

forms of syllogisms, in which one concludes a particular proposition

when one could already conclude the coresponding universal. For

example, he does not mention the form:

No

CisB

EveryAisC

So, someAis notB

If people had thoughtfully taken sides for or against the validity of

this form, that would clearly be relevant to the understanding of

the **O** form. But the weakened forms were typically

ignored.

One other piece of subject-matter bears on the interpretation of the **O**

form. People were interested in Aristotle’s discussion of “infinite”

negation,^{[9]}

which is the use of negation to form a term from a term instead of a

proposition from a proposition. In modern English we use “non” for

this; we make “non-horse,” which is true of exactly those things that

are not horses. In medieval Latin “non” and “not” are the same word,

and so the distinction required special discussion. It became common

to use infinite negation, and logicians pondered its logic. Some

writers in the twelfth and thirteenth centuries adopted a principle

called “conversion by contraposition.” It states that

- ‘Every
*S*is*P*’ is equivalent to

‘Every non-*P*is non-*S*’

- ‘Some
*S*is not*P*’ is equivalent to

‘Some non-*P*is not non-*S*’

Unfortunately, this principle (which is not endorsed by

Aristotle^{[10]})

conflicts with the idea that there may be empty or universal

terms. For in the universal case it leads directly from the truth:

Every man is a being

to the falsehood:

Every non-being is a non-man

(which is false because the universal affirmative has existential

import, and there are no non-beings). And in the particular case it

leads from the truth (remember that the **O** form has no

existential import):

A chimera is not a man

to the falsehood:

A non-man is not a non-chimera

These are Buridan’s examples, used in the fourteenth century to show

the invalidity of contraposition. Unfortunately, by Buridan’s time the

principle of contraposition had been advocated by a number of authors.

The doctrine is already present in several twelfth century

tracts,^{[11]}

and it is endorsed in the thirteenth

century by Peter of

Spain,^{[12]}

whose work was republished for centuries,

by William

Sherwood,^{[13]}

and by Roger Bacon.^{[14]}

By the fourteenth century, problems

associated with contraposition seem to be well-known, and authors

generally cite the principle and note that it is not valid, but that it

becomes valid with an additional assumption of existence of things

falling under the subject term. For example, Paul of Venice in his

eclectic and widely published *Logica Parva* from the end of the

fourteenth century gives the traditional square with simple

conversion^{[15]}

but rejects conversion by contraposition,

essentially for Buridan’s reason.

A similar thing happened with the principle of obversion. This is

the principle that states that you can change a proposition from

affirmative to negative, or vice versa, if you change the predicate

term from finite to infinite (or infinite to finite). Some examples

are:

Every SisP= No Sis non-PNo SisP= Every Sis non-PSome SisP= Some Sis not non-PSome Sis notP= Some Sis non-P

Aristotle discussed some instances of obversion in *De Interpretatione*. It is apparent, given the truth conditions for

the forms, that these inferences are valid when moving from affirmative

to negative, but not in the reverse direction when the terms may be

empty, as Buridan makes

clear.

^{[16]}

Some medieval writers

before Buridan accepted the fallacious versions, and some did

not.

^{[17]}

### 5.1 Negative Propositions with Empty Terms

In Paul of Venice’s other major work, the *Logica Magna*

(*circa* 1400), he gives some pertinent examples of particular

negative propositions that follow from true universal negatives. His

examples of true particular negatives with patently empty subject terms

are

these:^{[18]}

Some man who is a donkey is not a donkey.

What is different from being is not.

Some thing willed against by a chimera is not willed against by

a chimera.A chimera does not exist.

Some man whom a donkey has begotten is not his son.

So by the end of the 14th century the issue of empty terms was

clearly recognized. They were permitted in the theory, the

**O** form definitely did not have existential import,

and the logical theory, stripped of the incorrect special cases of

contraposition and obversion, was coherent and immune to 20th century

criticism.

### 5.2 Affirmative Propositions with Empty Terms

The fact that universal affirmatives with empty subject terms are

false runs into a problem with Aristotelian scientific theory.

Aristotle held that ‘Every human is an animal’ is a

necessary truth. If so, it is true at every time. So at every time

its subject is non-empty. And so there are humans at every time. But

the dominant theology held that before the last day of creation there

were no humans. So there is a contradiction.

Ockham avoids this problem by abandoning parts of Aristotle’s theory:

Although it conflicts with the texts of Aristotle, yet

according to the truth no proposition among those which concern

precisely corruptible things [which is] entirely affirmative and

entirely about the present is able to be a principle or a conclusion

of a demonstration because any such is contingent. For if some such

were necessary this would seem to be so especially for this one

“A human is a rational animal”. But this is contingent

because it follows “A human is a rational animal, therefore a

human is an animal” and further “therefore a human is

composed of a body and a sensitive soul”. But this is contingent

because if there was no human that would be false because of the false

[thing] implied because it would imply that something is composed from

a body and soul which would then be false. [Ockham SL

III.2.5]

The contradiction might also vanish if propositions in scientific

theory have unusual meanings. One option is that universal

affirmatives are understood in scientific theory as universalized

conditionals, as they are understood today. This would not interfere

with the fact that they are not conditionals in uses outside of

scientific theory. Although De Rijk (1973, 52) states that Ockham

holds such a view, he seems to explicitly reject it, stating that

‘A human is a rational animal’ is not equivalent to

‘If a human is then a human is a rational animal “because

this is a conditional and not a categorical”. [Ockham SL

II.11]

Buridan’s view is neater. He holds that when engaged in

scientific theory, the subject matter is not limited to presently

existing things. Instead, the propositions have their usual meanings,

but an expanded subject matter. When the word ‘human’ is

used, one is discussing every human, past and future, and even

possible humans. [Buridan SdD 4.3.4] With such an understanding, the

subject of ‘Every human is an animal’ is not empty at

all.

Work on logic continued for the next couple of centuries, though

most of it was lost and had little influence. But the topic of empty

terms was squarely faced, and solutions that were given within the

Medieval tradition were consistent with [SQUARE]. I rely here on

Ashworth 1974, 201–02, who reports the most common themes in the

context of post-medieval discussions of contraposition. One theme is

that contraposition is invalid when applied to universal or empty

terms, for the sorts of reasons given by Buridan. The **O** form is

explicitly held to lack existential import. A second theme, which

Ashworth says was the most usual thing to say, is also found in

Buridan: additional inferences, such as contraposition, become valid

when supplemented by an additional premise asserting that the terms in

question are non-empty.

### 5.3 An Oddity

There is one odd view that occurs at least twice, which may have as a

consequence that there are no empty terms. In the thirteenth century,

Lambert of Lagny (sometimes identified as Lambert of Auxerre) proposed

that a term such as ‘chimera’ which stands for no existing

thing must “revert to nonexistent things.” So if we

suppose that no roses exist, then the term ‘rose’ stands

for nonexistent

things.^{[19]}

A related view also occurs much later;

Ashworth reports that Menghus Blanchellus Faventinus held that

negative terms such as ‘nonman’ are true of non-beings,

and he concluded from this that ‘A nonman is a chimera’ is

true (apparently assuming that ‘chimera’ is also true of

nonbeings).^{[20]}

However, neither of these views seems to have been clearly developed,

and neither was widely

adopted.^{[21]}

Nor is it clear that either of them is

supposed to have the consequence that there are no empty terms.

### 5.4 Modern, Renaissance, and Nineteenth Centuries

According to

Ashworth,^{[22]}

serious and sophisticated investigation of logic ended at about the

third decade of the sixteenth century. The *Port Royal Logic*

of the following (seventeenth) century seems typical in its approach:

its authors frequently suggest that logic is trivial and

unimportant. Its doctrine includes that of the square of opposition,

but the discussion of the **O** form is so vague that

nobody could pin down its exact truth conditions, and there is

certainly no awareness indicated of problems of existential

import, in spite of the fact that the authors state that

the **E** form entails the **O** form (4th

corollary of chapter 3 of part 3). This seems to typify popular texts

for the next while. In the nineteenth century, the apparently most

widely used textbook in Britain and America was Whately’s *Elements of Logic*. Whately gives the traditional doctrine of the square,

without any discussion of issues of existential import or of empty

terms. He includes the problematic principles of contraposition (which

he calls “conversion by negation”):

Every SisP= Every not- Pis not-S

He also endorses

obversion:^{[23]}

- Some
*A*is not*B*is equivalent to Some*A*is

not-*B*, and thus it converts to Some not-*B*is*A*.

He says that this principle is “not found in Aldrich,” but that it is

“in frequent

use.”^{[24]}

This “frequent use” continued; later

nineteenth and early twentieth century text books in England and

America continued to endorse obversion (also called “infinitation” or

“permutation”), and contraposition (also called “illative

conversion”).^{[25]}

This full nineteenth century tradition is consistent only on the

assumption that empty (and universal) terms are prohibited, but

authors seem unaware of this; Keynes 1928, 126, says generously “This

assumption appears to have been made implicitly in the traditional

treatment of logic.” De Morgan is atypical in making the assumption

explicit: in his 1847 text (p. 64) he forbids universal terms (empty

terms disappear by implication because if *A* is empty,

non-*A* will be universal), but later in the same text (p. 111) he

justifies ignoring empty terms by treating this as an idealization, adopted

because not all of his readers are

mathmeticians.^{[26]}

In the twentieth century Łukasiewicz also developed a version

of syllogistic that depends explicitly on the absence of empty terms;

he attributed the system to Aristotle, thus helping to foster the

tradition according to which the ancients were unaware of empty

terms.

Today, logic texts divide between those based on contemporary logic

and those from the Aristotelian tradition or the nineteenth century

tradition, but even many texts that teach syllogistic teach it with the

forms interpreted in the modern way, so that e.g. subalternation is

lost. So the traditional square, as traditionally interpreted, is now

mostly abandoned.

In the twentieth century there were many creative uses of logical

tools and techniques in reassessing past doctrines. One might

naturally wonder if there is some ingenious interpretation of the

square that attributes existential import to the **O**

form *and* makes sense of it all without forbidding empty or

universal terms, thus reconciling traditional doctrine with modern

views. Peter Geach, 1970, 62–64, shows that this can be done

using an unnatural interpretation. Peter Strawson, 1952, 176–78,

had a more ambitious goal. Strawson’s idea was to justify the square

by adopting a nonclassical view of truth of statements, and by

redefining the logical relation of validity. First, he suggested, we

need to suppose that a proposition whose subject term is empty is

neither true nor false, but lacks truth value altogether. Then we say

that *Q* entails *R* just in case there are no instances

of *Q* and *R* such that the instance of *Q* is

true and the instance of *R* is false. For example, the

**A** form ‘Every *S* is *P*’

entails the **I** form ‘Some *S* is

*P*’ because there is no instance of the

**A** form that is true when the corresponding instance

of the **I** form is false. The troublesome cases

involving empty terms turn out to be instances in which one or both

forms lack truth value, and these are irrelevant so far as entailment

is concerned. With this revised account of entailment, all of the

“traditional” logical relations result, if they are worded as follows:

Contradictories: The AandOforms

entail each other’s negations, as do theEand

Iforms. The negation of theAform

entails the (unnegated)Oform, andvice versa;

likewise for theEandIforms.Contraries: The AandEforms

entail each other’s negationsSubcontraries: The negation of the Iform entails the

(unnegated)Oform, andvice versa.Subalternation: The Aform entails theI

form, and theEform entails theOform.Converses: The EandIforms each

entail their own converses.Contraposition: The AandOforms each

entail their own contrapositives.Obverses: Each form entails its own obverse.

These doctrines are not, however, the doctrines of [SQUARE]. The

doctrines of [SQUARE] are worded entirely in terms of the possibilities

of truth values, not in terms of entailment. So “entailment” is

irrelevant to [SQUARE]. It turns out that Strawson’s revision of truth

conditions *does* preserve the principles of SQUARE (these can

easily be checked by

cases),^{[27]}

but not the additional conversion principles of [SQUARE], and also

not the traditional principles of contraposition or obversion. For

example, Strawson’s reinterpreted version of conversion holds for the

**I** form because any **I** form

proposition entails its own converse: if ‘Some *A* is

*B*’ and ‘Some *B* is *A*’ both

have truth value, then neither has an empty subject term, and so if

neither lack truth value and if either is true the other will be true

as well. But the original doctrine of conversion says that an

*I* form and its converse always have the same truth value, and

that is false on Strawson’s account; if there are *A*s but no

*B*s, then ‘Some *A* is *B*’ is false

and ‘Some *B* is *A*’ has no truth value at

all. Similar results follow for contraposition and obversion.

The “traditional logic” that Strawson discusses is much closer to

that of nineteenth century logic texts than it is to the version that

held sway for two millennia before

that.^{[28]}

But even though he

literally salvages a version of nineteenth century logic, the view he

saves is unable to serve the purposes for which logical principles are

formulated, as was pointed out by Timothy Smiley in a short note in

*Mind* in

1967.^{[29]}

People have always taken the square to

embody principles by which one can reason, and by which one can

construct extended chains of reasoning. But if you string together

Strawson’s entailments you can infer falsehoods from truths, something

that nobody in any tradition would consider legitimate. For example,

begin with this truth (the subject term is non-empty):

No man is a chimera.

By conversion, we get:

No chimera is a man.

By obversion:

Every chimera is a non-man.

By subalternation:

Some chimera is a non-man.

By conversion:

Some non-man is a chimera.

Since there are non-men, the conclusion is not truth-valueless, and

since there are no chimeras it is false. Thus we have passed from a

true claim to a false one. (The example does not even involve the

problematic **O** form.) All steps are validated by

Strawson’s doctrine. So Strawson reaches his goal of preserving

certain patterns commonly identified as constituting traditional

logic, but at the cost of sacrificing the application of logic to

extended reasoning.