The Traditional Square of Opposition (Stanford Encyclopedia of Philosophy)

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:

NAMEFORMTITLE
AEvery S is PUniversal Affirmative
ENo S is PUniversal Negative
ISome S is PParticular Affirmative
OSome S is not PParticular Negative

The diagram for the traditional square of opposition is:

traditional square

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 + “the E and I forms convert
simply”

where

A proposition converts simply iff 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 S is Px(Sx
Px)
No S is Px(Sx
¬Px)
Some S is Px(Sx &
Px)
Some S is not Px(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:

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 Ss, 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
Ss. 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(Sx
Px)’ 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
Ss, 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(Sx
Px)
&
xSx.

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 the I form: ‘Some
S is P’ is false. But then its contradictory
E form: ‘No S is P
must be true. But then the subaltern O form:
‘Some S is not P’ must be true. But that
is wrong, since there aren’t any Ss.

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:

square fragment

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 C is B
Every C is A
So, some A is B

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 C is B
Every A is C
So, some A is not B

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 S is P=No S is non-P
No S is P=Every S is non-P
Some S is P=Some S is not non-P
Some S is not P=Some S is 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 S is P=Every not-P is 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 A and O forms
entail each other’s negations, as do the E and
I forms. The negation of the A form
entails the (unnegated) O form, and vice versa;
likewise for the E and I forms.
Contraries:The A and E forms
entail each other’s negations
Subcontraries:The negation of the I form entails the
(unnegated) O form, and vice versa.
Subalternation:The A form entails the I
form, and the E form entails the O form.
Converses:The E and I forms each
entail their own converses.
Contraposition:The A and O forms 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 As but no
Bs, 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.

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