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Oparaugo Michael — Sun, 05 Jul 2026 17:37:54 +0000

The Traditional Square of Opposition
First published Fri Aug 8, 1997; substantive revision Mon May 19, 2025
This entry traces the historical development of the Square of Opposition, a collection of logical relationships traditionally embodied in a square diagram. This body of doctrine provided a foundation for work in logic for over two millennia. For most of this history, logicians assumed that negative particular propositions (“Some S is not P”) are vacuously true if their subjects are empty. This validates the logical laws embodied in the diagram, and preserves the doctrine against modern criticisms. Certain additional principles (“contraposition” and “obversion”) were sometimes adopted along with the Square, and they genuinely yielded inconsistency. By the nineteenth century an inconsistent set of doctrines was widely adopted. Strawson’s 1952 attempt to rehabilitate the Square does not apply to the traditional doctrine; it does salvage the nineteenth century version but at the cost of yielding inferences that lead from truth to falsity when strung together.

Introduction 1.1 The Modern Revision of the Square 1.2 The Argument Against the Traditional Square
Origin of the Square of Opposition 2.1 The Diagram 2.2 Aristotle’s Formulation of the O Form 2.3 The Rewording of the O Form
The (Ir)relevance of Syllogistic
The Principles of Contraposition and Obversion
Later Developments 5.1 Empty Terms in Medieval Semantics 5.2 Negative Propositions with Empty Terms 5.3 Affirmative Propositions with Empty Terms 5.4 An Oddity 5.5 Renaissance, Modern, and Nineteenth Centuries
Strawson’s Defense Bibliography Academic Tools Other Internet Resources Related Entries
Introduction 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:

Name Form Title
A Every S is P Universal Affirmative
E No S is P Universal Negative
I Some S is P Particular Affirmative
O Some S is not P Particular Negative
The diagram for the traditional square of opposition is:

A square: upper-left corner has 'A'/'Every S is P', upper-right corner has 'E'/'No S is P', lower-left corner has 'I'/'Some S is P', lower-right corner has 'O'/'Some S is not P'. Lines from A to I and E to O is labelled 'subalterns'; line from A to E is labelled 'contraries'; line from I to O is labelled 'subcontraries'; lines from A to O and E to I are labelled 'contradictories'.
Figure: The Traditional Square of Opposition

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. Whether it was Aristotle’s view is a matter of contention, but the doctrine 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 P ∀x(Sx → Px)
No S is P ∀x(Sx → ¬Px)
Some S is P ∃x(Sx & Px)
Some S is not P ∃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:

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.

Origin of the Square of Opposition The doctrine that I call [SQUARE] can be traced to 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:

A square: upper-left corner has 'A'/'Every S is P', upper-right corner has 'E'/'No S is P', lower-left corner has 'I'/'Some S is P', lower-right corner has 'O'/'Some S is not P'. Lines crossing from A to O and from E to I are labelled 'contradictories'. Line from A to E is labelled 'contraries'.
Figure: 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.

The (Ir)relevance of Syllogistic 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.

The Principles of Contraposition and Obversion 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, in Paul of Venice’s eclectic and widely published Logica Parva dating from the end of the fourteenth century, the traditional, the square occurs with simple conversion;[15] but Paul rejects conversion by contraposition, essentially for Buridan’s reasons.

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]

Later Developments 5.1 Empty Terms in Medieval Semantics Empty terms are traditionally considered problematic in medieval logic, even beyond the issues involved in [SQUARE]. This is due to several factors related to the development of theories regarding the properties of terms in medieval semantics and in particular to the workings of supposition theory.

All empty terms lack a suppositum (roughly, “referent”), but they may do so to varying degrees. Some terms are contingently empty—that is, they might happen to be empty at present but have had or will have supposita at some point. This applies to presently empty natural classes, such as dodos. Medieval logicians often discussed examples like “roses in winter” and “thunder on a clear day” (Bur., Q. Eth. VI, q. 6).

Other terms will never be anything but empty due to the world’s inherent structure. Medieval authors regarded these terms as signifying natural or relative impossibilities. For instance, when we speak of ideal gases, we speak of entities that cannot be found in nature. Under the medieval Aristotelian worldview, “void” falls into this category (since nature abhors a vacuum), as do terms for mathematical entities (e.g., points, lines), instants, atoms, and all minima. These terms are not inherently flawed; they simply denote things that were not, are not, will not, and could not exist in nature—given the period’s understanding—but are not self-contradictory.

But there are also terms that are necessarily empty in a stronger sense because they signify what medieval authors called simple or absolute impossibilities, viz. something that is intrinsically contradictory. The first thing coming to the mind of a modern reader would be “the round square cupola on Berkeley College” (Quine 1948). The standard medieval example is the Chimera, defined as a composite of incompossible essences. The semantics of necessarily empty terms has been a matter of contention since Late Antiquity and throughout the Middle Ages (Ebbesen 1986). Issues with empty terms in general were both recognized and discussed throughout the development of medieval semantics (Maierù 1972; de Libera 2002).

The semantics of necessarily empty terms has been a matter of contention since Late Antiquity and throughout the Middle Ages (Ebbesen 1986). Issues with empty terms in general were recognized and discussed throughout the development of medieval semantics (Maierù 1972; de Libera 2002).

5.2 Negative Propositions with Empty Terms
By the end of the 14th century there is a common understanding that negative propositions do not have existential import.

For example, 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.

By the time of Paul of Venice, 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.

5.3 Affirmative Propositions with Empty Terms
The case of emptiness that medieval authors discussed at length is that of affirmative statements with an empty subject term. Under the assumption of Existential Import, all these statements are false by default, including some sentences that you might want to be able to verify, namely:

Statements with subjects signifying natural classes empty at present. Typical medieval examples are: “rose is a flower” when, in winter, there are no roses; or “thunder is a sound made in the clouds” on a clear day. The same applies to “dodos are birds” or even “all dodos are dodos” since there are no presently existing dodos.
Statements with subjects signifying idealized or abstract items that are not possible in nature, but are not intrinsically contradictory. Typical medieval examples of such statements are: “the void is place with no matter”, which is naturally impossible in Aristotelian physics; “point is space with no extension”, along with all affirmative statements about geometrical entities and mathematical ficta, as well as instants, atoms, and minima. But the same would apply to “ideal gases are infinitely compressible” and other such statements about useful abstractions that are not given in nature.
Trivial identities with necessarily empty subjects signifying intrinsically contradictory items, such as the round square or the chimera. Statements such as “the round square is the round square” or “the chimera is the chimera” are unavoidably false. Most late-medieval logicians subscribe to this claim, including John Buridan (TC 1.5.3) and Albert of Saxony (QcL q. 13)
Note that subjects of (A) statements are merely presently empty but have had and can have referents at some point in time. Instead, there are no possibly actual referents for subjects of (B) and (C) statements, since those subjects signify what, by the mid-14th century, were considered respectively relative and absolute impossibilities. The default falsity of affirmative sentences with an empty subject term raises a plethora of serious problems for late medieval philosophers, who generally operate within an Aristotelian framework and have theological concerns.
For instance, the fact that universal affirmatives with empty subject terms are false by default runs into a problem with the 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 ll.34–45]
However, the broader problem affecting any affirmative statement with an empty subject remains. These affected statements include sentences that are relevant for scientific discourse, such as (A) and (B) statements. While trivial identities about chimeras or round squares might have limited impact on Aristotelian science, their default falsity still seems counterintuitive to ordinary language interpretation.

In the 14th century, logicians address these issues using different strategies. If you want to have your cake and eat it, you could either tweak the proper interpretation of empty affirmative statements or their domain of reference.

For example, one option is to claim that universal affirmatives are understood in scientific theory as 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 it is a human, then it is a rational animal’ “because this is a conditional and not a categorical” [Ockham SL III-2.5 ll. 46–47]. This is a major concern for Ockham and any medieval Aristotelian philosopher, since scientific reasoning is articulated in scientific syllogisms, which are described in the Posterior Analytics. But scientific syllogisms require premises that are universal, affirmative, necessarily true, and categorical.

Another option is to claim that the copula in (some) scientific statements is not quite a proper copula; instead, it stands for something like “meaning” or “signifying”. Rather than normal predications, these sentences explain the use of an empty term like “void” or “point”. Albert of Saxony deploys this strategy, for example, in his Questions on Physics (e.g., IV q. 8).

Buridan’s view is more nuanced. He focuses on the reading of the copula and on the domain of reference. For Buridan, the copula remains a proper coupla and does its proper copulative job of subject and predicate. However, Buridan holds that when engaged in scientific theory, the subject matter is not limited to presently existing things. In other words, in general scientific statements, the copula is not importing the present only, viz. it is not predicating only present predicates of present subjects. In such cases, the sentences have their usual meanings, and the copula is a proper copula. But the tense of the copula in sentences like “dodos are birds” or “Every human is an animal” is not really the present tense. Instead of co-signifying the present, the copula is omnitemporal, signifying all times (De Rijk 1973). In this way, these sentences remain cases of proper categorical predication with an expanded subject matter (Ciola 2020). When the word “human” is used in “Every human is an animal”, one is discussing every human, past and future, and maybe 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.

Another option expanding on the subject matter of affirmative sentences with empty subjects appeals to the doctrine of ampliation. Ampliation is one of the ways that medieval logicians use to treat tensed and modal contexts. In some versions of the doctrine, ampliation “expands” a term’s reference across tensed and modal contexts. These tensed and modal slices of the domain are all grouped as “distinctions of time” (differentiae temporum) and they usually include the present, the past, the future and the possible. In the third quarter of the 14th century, Marsilius of Inghen adds a fifth “distinction of time” – that of the impossible or the imaginable. This approach became widely adopted in the 15th century and in the 16th (Ashworth 1977). With this addition, affirmative sentences with empty subject terms can receive a uniform semantic treatment that avoids default falsity across scientific statements and ordinary language (Ciola 2019; 2020).

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]. Ashworth [1974, 201–02] 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.

5.4 An Oddity
There is one odd view that occurs a handfil of times, which entails the claim that there are no empty terms. In the 13th 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] In the first half of the 14th century, Richard Brinkley made this claim explicit: no term is empty (Logica, Tractatus de Suppositione; see Cesalli 2013). 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, since true nonbeings can be proper referents for terms that would otherwise be necessarily empty.

5.5 Renaissance, Modern, 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.

Strawson’s Defense 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.

Truth, Validity and Form

Oparaugo Michael — Sat, 04 Jul 2026 20:03:32 +0000

Arguments exhibit different combinations of true and false premises and conclusions. This is illustrated very well by the following examples:

All Judges are lawyers

All Chief Justices are Judges

Therefore, all Chief Justices are Lawyers.

All writers are creative

All novelists are writers

Therefore, all novelists are creative

All mathematicians are college graduates

All college graduates are intelligent

Therefore, all mathematician are intelligent

If James was allowed to rule then he will be famous

James was not allowed to rule

Therefore, James was not famous.

Logical reasoning

Oparaugo Michael — Fri, 03 Jul 2026 19:41:55 +0000

Mike

Mike when are you going the us?

This text is italic

This text is important

This text is bold

Emphasized text

Mark text

Smaller text

~~Deleted text~~

Inserted text

Subscript

^Superscript

This is man.

This is Jane.

This is a car.

This is a bus.

This is a train.

This is a cap.

Centered.

This is a Pen. my pentext.

Elementary Set Theory and Relational Propositions.

Background Analysis.

The logicist programme of A.N. Whitehead and Bertrand Russell reached its zenith in Principia Mathematica, a work of immense significance in the domain of symbolic logic and pure mathematics. The book is a thorough and ingenuous expose of pure mathematics with a few primitive propositions of logic. In the work, the authors discussed extensively different kinds of relations which form a common ground of theoritical disquisitions amongst logicians and mathematicians. Additionally, Principia Mathematheca elucidated the theory of classes and the relations that may obtain between them. In order to appreciate fully the ground-breaking achievements of that work, it is necessary to remember that before its publication the theory of numbers and the calculus of relations had floated for quite some time in the thick fog of metaphysics and mysticism. For example, Aristotle's theory of substance and essences obfuscated the distinction between names and predicates and consequently engendered untenable theories about relations. Leibniz, whose work on mathematical logic would have superseded Aristle's logic even made mistakes in his analysis of infinite collections or sets. He had noticed that the number of even numbers must be equal to the number of all whole numbers, and thought it was a contradiction. The mathematician, George Cantor rejected Leibniz's position and posited a better theory of numbers. Again, Hegel's amalgamation of logic with methaphysics was very inimical to the growth of the former. Infact, Hegelians either ignored relations altogether or advanced specious arguments which purport to prove the unreality of relations.

Russel and Whitehead provided a mathematical framework for the detail spelling out of relations and classes amongst others. The theoritical framework necessitated new algorithms which made it possible to treat symbolically many technical questions previously left to the fuzziness and nebulosity of ordinary language. This is particularly true in the field of relations.

In this chapter we shall consider some of the elementary principles and methods employed by logicians and mathematical logicians in the analyses of sets and relational propositions.

Sets: Membership and Operations on Sets

The idea of grouping things into bundles, aggregates, or sets is a very common one. It is presupposed when we collect a bundle of wirewood, or when someone thinks about the collection of books in a university library. Even, a bag of groundnuts or a bunch of plantain constitute a set. Generally speaking, the word "set" is used to refer to any kind of collection of things of any sort. Individual entities that make up a set are called element or members of that set. In most cases, it is possible for us to say with regard to any entity and any set that a particular entity is an element of that set or it is not. Thus, an apple is not a number of the set of all insects, a butterfly is. The square-root of the number 2 is not a member of the set of integers, but the number 2 belongs to that set.

It is customary, in the discussion of sets, to commence with a definite class of objects on which the analysis is focused. This is referred to as the universal set. The universal or basic set E varies from one problem to another. Sometimes, for example, we may focus our attention on the positive integers, or on the set of legislators, etc.

Usually, a set can be defined in two ways. One, we may enumerate all its members and then enclose them with braces. This is the roster method. The set of even numbers less than 14 is {2, 4, 6, 8, 10, 12}. Ellipses can occur within the braces if and only if one can tell precisely what and what are the intervening entities whose spaces are taken up by the ellipses. The expression {1, 2, 3, 4,...50} is the set of positive integers from 1 to 50. But we cannot define an infinite collection or a collection whose numbers are unknown in this way. So we introduce the second method in which a set is defined by stating the property or properties that must be possessed by each and every member of the set. The second definition is more popular amongst scholars. It is based on the principle of abstraction, namely, that every property defines a set. All the alphabets in Igbo language, for instance, constitute a set. So do all the fingers in a hand. The sets we have discussed so far are finite. Of course, there are infinite sets, that is sets that contain an infinite number of elements. The set of natural numbers is a good example. Given a basic set, we can talk about its subset. A subset s of t is a set whose elements are all members of another set. The philosophy lecturers in the University of Boston form a subset of all human beings. Again, the footballs used by FIFA in last year's world cup tournament are a subset of all footballs. Certainly a subset, just like a set, does not have to be tangible; infact sets and subsets are generally intangibles. In mathematical logic, for example, the odd integers are a subset of integers, both of which are intangibles. Assuming ther are two set s and t such that every member of s is a member of t also, we say that s is included in t and that s is a subset of t. Symbolically the relation is written as

s⊂t

If we wish to affirm that s is not a subset of t, we symbolize it thus

s⊆t

For instance, the set of motor cars is not a subset of the set comprising aeroplanes. It may happen that both s⊂t and t⊂s are equal, meaning that the two sets have exactly identical numbers. This implies that s and t are equal. The equality of sets is referred to as the principle of existensionlity. For instance, the sets s = {A, B, C, D} and t = {D, A, C, B} are equal inspite of the fact that the elements are arranged differently in each set.

If it is stated that s⊆t and t⊂s, s is said to be a proper subset of t. It is written sd.

s⊂t

There are some basic operations that can be performed on sets. such operations are generally in agreement with the Boolean interpretation of classes.

The first operation we shall consider is the union of sets. The union-set of two sets s and t is the set of objects which belong either to s or to t. A collection of students may belong to the philosophy department or to mathematics. The union of s and t is designated as:

........ s ∪ t The intersection-set or product of s and t is the set of all those objects that are elements or members of both s and t. The intersection of these sets is symbolically stated in this manner.
........ s ∩ t The complement of a set s is made up of those objects that do not belong to s. It is rendered in symbols as S (read "s bar") In mathemathics and mathematical logic, a set is exemplified usually by an imaginary number, indicating clearly that sets and subsets are intangible. Generally, a subset is a set in its own right, and the number of logically possible sets in infinite. A property that delineates a set does not have to be tangible even when the objects to which it applies are tangible. To illustrate this point, I may be thinking about the set of all the triangles in a concrete pyramid, but triangularity is not a tangle property. Now, we express the relation between any individual x, that is an element of s, as x ∈ s The symbol ∈ is the Greek letter epsilon, and symbolizes the relation of being "a member of or is an element in" a set. AS an illustration, supposing that s represents the set of "fairly-used cars" and t symbolizes the set of "cars" then s is a subset of t and any member of s is also a member of t. This is plain enough, since every fairly-used car is a car. Logically, the relation involved here can be generalized as (x) : (x ∈ A) ⊂ (x ∈ B) Since a set may have any number of elements, it may also have just a single number. A set with only one element is called a unit set. In set theory, we distinguish the element 1 and the set {1}. The symbol {1}, is the unit set of the number 1. Thus, it is the set whose only member is the element 1. in general, given any indidual a, we may define ia, the unit set of a, as: ia= {(x ∈ v) | x= a} This follows from the generalization of the principle of abstraction which is written thus x { ...|x ...} and read as "the set of all x's such that..." There is an important set in mathematical logic which is usually neglected in ordinary discussion. It is the set with no elements at all. We call such a set the null or empty set, and in the notation of Boolean algebra, it is symbolised as Ø. Accordingly, if there is no x we then have x ∈ Ø. A radical difference exists between ∈ on the one hand and ⊆ or ⊂, on the other. To begin with,⊆ and ⊂ (together with ∪,∩ and") apply to sets alone. When we affirm that the set whose numbers are footballers is included in the set of athletes (using s and t to substitute for the set of footballers and the set of athletes respectively), we then write, using the usual notation, s ⊂ t So, the relation of inclusion cannot hold between elements of sets or between elements and sets. It is possible to state that {1,2} ⊂ {1,2,3,4,} but never I ⊂ I nor I ⊂ {1,2} Similarly since the relation ∈ holds between an element and the corresponding set, we cannot in general assert of an element x that x ∈ x but we may say x ∈ s it is important to note that sets can be members of other sets, as in {1,2}3} and {1},2}3} It does not really matter the order in which the elements in a set are written down. The following are different arrangements of four alphabets which constitute a set: {q, r, s, t} = {t, s, r, q} = {r, q, s, t} = {r, t, q, s} ... and so on. The number of elements which a set contains is called the cardinality of that set. In our example, the cardinality of the set is four. By definition, a set s contains each of its element exactly once and no repetitions are permitted. Its subsets, as we said earlier, are those sets whose elements are contained in s. To determine the number of subses in the universal set E, we must include the set itself and the null set as the subsets of E also. Given any set whatsoever, the number of its subsets is derivable from the formula N = 2n, where N represents the number of its subsets in s. As an illustration consider the number of students who distinguished themselves in a logic examination. These students (let us call them S1, S2S3) constitute the universal set. The number of possible subsets we can derive from the situation are {S1,S2,S3,},{S1,S2,},{S1,S3,}, {S2,S3,}, {S2,S3,},{S1,},{S2,},{S3,} and {o}. This follows from N = 2n. Note also that here both the universal set and the null set are included as subsets of the universal set. A set with a single member, as you can see, has 2 possible subsets. There is an important method for forming sets which is different from unions and intersections. This is the method whereby sets are constructed from ordered pairs. An ordered pair is made up of two elements taken in a specified order; for example a and b in that order. Thus, to assert that (a, b) is an ordered pair, we intend to say that a comes before b or, (a, b) ≠ (b, a) When we say that "Abel comes before cane in the alphabetical order", we are saying something totally different from "Cane comes before Abel in the alphabetical order". Now, taking a and b as arbitrary non empty sets, we define the Cartesian product a x b of these two sets as the ensemble of all ordered pairs which can be constructed by taking a first term from a and a second one from b. From the definition, it follows that if {a,b} is an ordered pair, we can affirm, in relation to another ordered pair, that there is a one-to-one correspondence between their elements. Thus, if the second ordered pair b {1,2} then {a,b} = {1,2} if and only if a = 1 and b = 2. But {a,b} ≠ {1,2} Also, a x b is greater than (>) a or b since if a has x elements and b has y elements then a x b has xy elements. We shall try an example to illustrate this point.

Let a = {0, 1} and b = {0, 2} we then have

a x b = {(0, 0),(0,2),(4,0),(1,2)}

b x a = {(0,0),(0,1),(2,0),(2,1)}

Given a universal set E some relations may link pairs of elements of E in a particular sequence. Take the relation of "taller than", it selects precisely those pairs of individuals actually related in such a manner that the first is taller than the second. Assume futher than a set constructed with the relation "taller" than has just four elements. By using the cartesian product in a tabular form, we can construct all possible pairs of elements of the set s. Labelling the four elements 1,2,3 and 4 respectively, the set in question can generate sixteen ordered pairs:

4 (4,1), (4,2), (4,3), (4,4)

3 (3,1), (3,2), (3,3), (3,4)

2 (2,1), (2,2), (2,3), (2,4)

1 (1,1), (1,2), (1,3), (1,4)

1 2 3 4

Our table depicts all possible ordered pairs that can be generated from the relation of "taller man" in a set of four elements. The same principle can be used to generate ordered pairs for finite and infinite sets. In the case of an infinite set is it enough to write down the first few pairs as illustrations, followed by three dotted lines to indicate that the list go on and on.

To instaciate what can be referred to as ordered "4-tuples" the following diagram is quite representative.

Consider a logician who takes gambling as a hobby. In each game he played on a particular occasion he has two different options. Let the two options be a set of elements A and B. If the logician has to play four times, we calculate the possibilities before him as follows:

B,B | A,A,B,B,| A,B,B,B, | B,A,B,B, | B,B,B,B, |

B,A | A,A,B,A,| A,B,B,A, | B,A,B,A, | B,B,B,A, |

A,B | A,A,A,B,| A,B,A,B, | B,A,A,B, | B,B,A,B, |

A,A | A,A,A,A,| A,B,A,A, | B,A,A,A, | B,B,A,A, |

A,A A,B B,A B,B

The table completely exhausts all the possibilities available within the logic of the situation. In otherwords, we have generated sixteen ordered 4-tuples. Applying the expression to the ordered pairs we constructed initially, we refer to them as 2-tuples.

Propositions of predicate logic can be translated into set-theoretical symbols without any loss in meaning. Since any predicate or property determines a set, a monadic formula such as f(x), where the possession of f determines a set s, can be rewritten as x s. Thus since if in f(x1...x2) determines s we can rewrite the whole formula as (x1...x2) s. Logicians such as Russell and whitehead have fruitfully exploited the ramifications of set theory into other domains to set up an expressive theory of relations.

Elementary Postulates of the Algebra of Sets.

Following upon what we stated about the translatability of predicate logic into set-theoretic terms, some basic postulates of set theory clearly resemble the rules of inference in propositional logic. We shall list fourteen of these postulates and illustrate a few of them with venn diagram. But before we go into all that, two other operations on sets have to be introduced. The two operations are difference and exclusive union.

Let s and t be any two subsets of the universal set E. The difference between s and t, written as s-t, is the subset that consists of those elements which are members of s but are not numbers of t. The difference s-t is clearly exhibited in the diagram below.

The exclusive union of s and t, denoted by sUt is the set of elements of the universal set that belong to s or t but not both. The following venn Diagram illustrates sUt.

On closer inspection, it can be observed that s-t and sUt can be derived from the three operations on sets namely union, intersection and complementation.

Thus s-t can be expressed as s n t, while 5 u t can be penned down as (s n t) n (s u t) or as (t n s) u (s n t). Returning now to the postulates we undumbrated earlier, a few of interest to the student of Symbolic logic. They include:

Postulates 10 down to 12, rely on the definition of the empty set Q universal set E and the complement set. They are exhibited clearly in this Venn Diagram:

This diagram illustrates clearly that the intersection of the set s and the universal E (that is, S n E) is precisely covered by the area enclosed within the circle. This This diagram illustrates clearly that the
intersection of the set s and the universal E (that
is, S n E) is precisely covered by the area enclosed
within the circle. This establishes postulate 10.
Further, the union of s and 5 includes the space
within the circle and the space outside the circle
enclosed in the boundaries of ABCD, that is within
the universal set E. But these spaces are all within
E. Therefore, S U S equals E. Finally, S is the
complement set of S. Our diagram does not
contain any space corresponding to the intersection of
f s and its complementary set. This means that s
intersection 5 represents an empty or null set. The
null set, you would recall, has no members at all.
From the diagram then, it follows that nothing
belong to both 5 and s. The use of the set algebra
to illustrate sets is demonstrated further in the
Venn Diagram below:

This ﬁgure represents the
universal set and two subsets within it. Different
spaces in the diagram are demarked by the letters
a, b, c and u. Those areas or spaces consisting of a, b, c, u make up the universal set E. The set 5
consists of a and b whereas b and c make up set t.
All the possible subsets contained in the Venn
Diagram can be constructed using the operations
on sets earlier introduced:

SetSpace
as n t v s - t
bs
nt
csntvt-s
u (sUt)
a, bs
a,c(snt) U (s
nt)n(sUt),v(sUt),v(s-t)U (t-s)
a,ut
b,ct
c,u s
a, b, cs U t
a, c, u (s n t)
a, b, c, u E

Example
1Determine w

which of the following statements are true and
which are false.

(a) D E {A, B, C, E, F}

(b) A E {1, 2, 3, 4}
(c) F E {2, 4, 6, 8,
10
(d) {1}E]

Solution

(a)
False (b) False
(c) True
(d) False

Example
2

(a) {1,2, 3, 4} U {2, 3, 5} =
(b) {1, 2, 3,
4} n {4, 5, 3} =
(c) {5, 6, 7, 8} U {2, 3, 5} =
(d) {2, 4,
6, 8} n {5, 6, 7} =
(e) {1, 2, 5, 6} n {A, B, C}
=

Solution

br>
(a) 1, 2, 3, 4, 5 (b) 3,
4 (c) 2, 3, 5, 6, 7, 8
(d)
6 (e) Q

Example
3

Enumerate the subsets of 1, 2, 3,
4, 5 which have exactly two numbers.

1,
2, 1, 3, 1, 4, 1, 5, 2,3, 2, 4, 2, 5, 3, 4, 3,
5

Example 4

(a) Emumerate the
subsets of 1, 2, 3
(b) Which of them are
proper subsets of 1, 2, 3

So|ution

(a) 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3, Q
(b) 1, 2, 3, 1, 2, 1, 3, 2, 3, Q

Example
5

For the sets 5 = a, b, c, d and t = b, d, e,
determine the membership of s U t, s n t, s U s, s n
Q, s U Q,

Solution

s U t = a, b, c,
d,e;snt=b,d;sUs=s;sns=s;an=Q;sU
Q: s.

Relations

Although
relational propositions could be interpreted in set

theoretical terms, there are well - developed alternatives for articulating and symbolizing such propositions. Relational propositions differ from subject-predicate propositions in that the latter links a subject and predicate together through a copula whereas as the latter does not. Really available illustrations of relational propositions are contained in phrases "to the east of", "on top of", "greater than" etc. In the statement "Obi is older than Ada", or "Aba is to the east of Owerri", Obi or Aba is the referent, and Ada or Owerri is the relatum. The relation between two individuals or objects as in the examples above is called a dyadic relation. In "A PDP Faithful gave one hundred and twenty

million naira to Obasanjo", we have a Relationship

between PDP faithful, one hundred and twenty
million naira and Obasanjo. We call such a relation
between three entities a triadic relation. An
example of tetradic relation is contained in the
proposition "Enwerem insisted that the letter referred to in magazine was rejected by a competent court of law,’I

Relations can be characterized in other
ways. For instance, from the proposition "Unoka is
the father of Okonkwo", it is not possible to infer
that "Okonkwo Is the father of Unoka". Thus the
relation "father of" is asymmetrical. But in "One
hundred kobo are equal to one naira," the
relation of equality asserted is symmetrical, since
a = b is the same as b = 3. Therefore we can infer
from the given proposition the statement "One
naira is equal to one hundred kobo.‘I A
relationship like "loving" is somet sometimes symmetrical and sometimes
asymmetrical. We say that kind of relation is not
symmetrical.
Transitive relation is
illustrated in the following: If A is bigger than B
and B is bigger than C, then A is bigger than C. The
relation "bigger than" is transitive. But even if it is
true that A is the father of B and B is the father of
C, it does not follow that A is the father of C. Here
the relation is intransitive. Again, some relations
may be transitive in some contexts, and
intransitive in o others. They are called nontransltlve relations. A
good example of such a relation is "being a friend
of."
From the foregoing given any
relational proposition whatsoever, it appears that
some individuals or objects stand in some relation
to other individuals or objects.

Symbolizing Relational
Propositions

ln Symbolizing
propositions containing relational words, phrases
and expressions, the use of variables to replace
individuals is complemented with symbols for relations. In the proposition "David was a teacher
of Jane", we replace David with x and Jane with y
to get "x was a teacher of y". Now, representing
"teacher of" with the capital letter T, "x is a
teacher of y'I becomes Txy. The order of
substitution is very important in Symbolizing
relations, for, in this instance, the converse, "y is a
teacher of x" is false. By the same token, "b is
adjacent to c and d" can be written as Abcd. Other
relational proposition and their symbolizations
include:

John i is the father of Chicketaram .....
ch
lshiowerre is between Samuel and
Ruth ....
Bian
Chioma is the sister of
Anthony ..... Sne

ln order to improve the reader's
ability to symbolize relations, it is convenient to
start with a dyadic relation involving two
individual constans, p and q, for instance. We can
then assert a relation which is claimed to hold
between p and q. The proposition "p contains q"
is logically equivalen equivalent to "q is contained in p" and both can
be symbolized as Cpq. Of course its negation can
be symbolized as Cpq. Other simple
symbolizations of relational propositions involving
quantifiers this time around include:

P
contains everything = (x)
Cpx
Everything does not contain p
= (x) Cxp
p contains something
= (x) Cpx
Something
does not contain p = (x)
Cxp
Everything contain p =(x) Cxp
p does not contain everything
: (x) Cpx
Something
contains p : (x) Cxp
p
does not contain something
: (x) Cps

Multiply general
relational propositions as well can be symbolized.
A few of them are listed below:

Everything contains everything = (x) (y)
xy
Something contains something = (Ex) (Ey)
ny
Nothing contains anything = (x) (y)
ny

A table of translation from ordinary language to symbols can facilitate our
understanding of how to symbolize propositions
containing relations:

Everything contains
everything (x) (y)
ny
Everything is contained by everything
(y) (x) ny
Something contains
something (x) (Ey) ny
Something is
contained by something (Ey) (Ex)
ny
Nothing contains (x) (y)
ny
Nothing is contained by anything
(y) (x) ny
Everything contains
something (x) (Ey) ny

Logical opposition diagram

Oparaugo Michael — Fri, 03 Jul 2026 16:16:13 +0000

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CONTENTS
AN INTRODUCTION TO LOGIC
Module 3: Aristotelian Logic

3.2 THE SQUARE OF OPPOSITION
We have now established the boundaries of our domain of logically well-behaved natural language — A, E, I, and O propositions. These four types of categorical propositions are related to one another in systematic ways; we will look into those relationships.

3.2.1 Introducing the Square
The relationships are inferential: for example, we can often infer from the truth of one of the four types, whether the other three are true or false. These inferential relationships among the four categorical propositions are summarized graphically in a diagram known as The Square of Opposition. The diagram looks like this:

Large square, A at upper left, E upper right, I lower left, O lower right and operations depict relationships between prop types in corners
Figure 3.2-1
The four types of categorical propositions are arranged at the four corners of the square, and the relationships that apply between pairs of the proposition types are marked along the sides and diagonals.

First we will look at each of these four relationships:

Contradictories
Contraries
Subcontradictories
Subalterns
After that, we will work with inferences that can be made, if we know the truth status of one proposition type (e.g., an A proposition), about the truth/falsehood of the other three proposition types.

3.2.2 Contradictories
Contradictory pairs of categorical propositions are at opposite corners from one another on the Square of Opposition. A and O propositions are contradictory; E and I propositions are contradictory. What it means for a pair of propositions to be contradictory is this: they have opposite truth-values; when one is true, the other must be false, and vice versa.

This is pretty intuitive. Consider an A proposition—all sailors are pirates. Suppose I make that claim. How do you contradict me, how do you prove I’m wrong? My brother’s in the Navy, you might protest. He’s a sailor, but he’s not a pirate. That would do the trick. The way you contradict a universal affirmative claim—a claim that all S are P—is by showing that there’s at least one S (a sailor in this case, your brother) who’s not a P (not a pirate, as your brother is not). At least one S that’s not a P—that’s just the particular negative, O proposition, that some S are not P. (Remember: ‘some’ means ‘there is at least one’.) A and O propositions make opposite, contradictory claims. If it’s false that all sailors are pirates, then it must be true that some of them aren’t; that’s just how you show it’s false. Likewise, if it’s true that all dogs are animals (it is), then it must be false that some of them are not (you’re not going to find even one dog that’s not an animal). A and O propositions have opposite truth-values.

Likewise for E and I propositions. If we claim that no saints are priests, and someone wants to contradict, what they need to do is come up with a saint who was a priest. It’s not hard: Saint Thomas Aquinas (who was the most prominent medieval interpreter of Aristotle, by the way, and a terrific philosopher in his own right) was a priest. So, to contradict a universal negative claim—that no S are P—you need to show that there’s at least one S (a saint in this case, Thomas Aquinas) who is in fact a P (a priest, as Aquinas was). At least one S that is a P—that’s just the particular affirmative, I proposition, that some S are P. (Again, ‘some’ means ‘there is at least one’.) E and I propositions make opposite, contradictory claims. If it’s false that no saints are priests, it must be true that some of them are; that’s just how you show it’s false. Likewise, it’s true that no cats are dogs (it is), then it must be false that some of them are (you’re not going to find even one cat that’s a dog). E and I propositions have opposite truth-values.

Check Your Understanding

3.2.3 Contraries
The two universal propositions—A and E, along the top of the Square—are a contrary pair. This is a slightly weaker form of opposition than being contradictory. Being contrary means that they can’t both be true, but they could both be false—though they needn’t both be false; one could be true and the other false.

Again, this is intuitive. Suppose I claim the universal affirmative, All dogs go to heaven, and you claim the corresponding universal negative, No dogs go to heaven. (Those sentences aren’t in standard form, but the translation is easy.) Obviously, we can’t both be right; that is, both claims can’t be true. On the other hand, we could both be wrong. Suppose getting into heaven, for dogs, is the way they say it is for people: if you’re good and all that, then you get in; but if you’re bad, oh boy—it’s the Other Place for you. In that case, both of our claims are false: some dogs (the good ones) go to heaven, but some dogs (the bad ones, the ones who bite kids, maybe) don’t. But that picture might be wrong, too. I could be right and you could be wrong: God loves all dogs equally and they get a free pass into heaven. Or, I could be wrong and you could be right: God hates dogs and doesn’t let any of them in; or maybe there is no heaven at all, and so nobody goes there, dogs included.

Check Your Understanding

3.2.4 Subcontraries
Along the bottom of the Square, we have the two particular propositions—I and O—and they are said to be subcontraries. This means they can’t both be false, but they could both be true—though they needn’t be; one could be true and the other false.;

It’s easy to see how both I and O could be true. As a matter of fact, some sailors are pirates. That’s true. Also, as a matter of fact, some of them are not. It’s also easy to see how one of the particular propositions could be true and the other false, provided we keep in mind that ‘some’ just means ‘there is at least one’. It’s true that some dogs are mammals—that is, there is at least one dog that’s a mammal—so that I proposition is true. In fact, all of them are—the A proposition is true as well. This means that since A and O are contradictories, the corresponding O proposition—that some dogs are not mammals—must be false. Likewise, it’s true that some women are not (Catholic) priests (at least one woman isn’t a priest); and it’s false that some women are priests (the Church doesn’t allow it). So O can be true while I is false.

It’s a bit harder to see why both particular propositions can’t be false. Why can’t ‘Some surfers are priests’ and ‘Some surfers are not priests’ both be false? It’s not immediately obvious. But think it through: if the I (some surfers are priests) is false, that means the E (no surfers are priests) must be true since I and E are contradictory; and if the O (some surfers are not priests) is false, that means the A (all surfers are priests) must be true since O and A are contradictory. That is to say, if I and O were both false, then the corresponding A and E propositions would both have to be true. But, as we’ve seen already, this is (obviously) impossible: if I claim that all surfers are priests and you claim that none of them are, we can’t both be right.

Check Your Understanding

3.2.5 Subalterns
The particular propositions at the bottom of the table—I and O—are subalterns of the universal propositions directly above them—A and E, respectively. (The universal propositions are called superalterns.) This means that the pairs have the following relationship: if the universal proposition is true, then the particular proposition (it’s subaltern) must also be true. That is, if an A propositions is true, it’s corresponding I proposition must also be true; if an E proposition is true, its corresponding O proposition must also be true.

This is intuitive if we keep in mind, as always, that ‘some’ means ‘there is at least one’. Suppose the A proposition that all whales are mammals is true (it is); then the corresponding I proposition, that some whales are mammals, must also be true. Again, ‘some whales are mammals’ just means ‘at least one whale is a mammal’; if all of them are, then at least one of them is! Similarly, on the negative side of the square, if it’s true that no priests are women (universal negative, E), then it’s got to be true that some priests are not women (particular negative, O)—that at least one priest is not a woman. If none of them are, then at least one isn’t!

Notice that these relationships are depicted in a slightly different way from the others on the Square of Opposition — there’s an arrow pointing toward the bottom. This is because the relationship is not symmetrical. If the proposition on top is true, then the one on the bottom must also be true; but the reverse is not the case. If an I proposition is true—some sailors are pirates—it doesn’t follow that the corresponding A proposition—that all sailors are pirates—is true. Likewise, the truth of an O proposition—some surfers are not priests—does not guarantee the truth of the corresponding E proposition—that no surfers are priests. (This could be true, but there is no guarantee. that there isn’t a surfing priest out there.)

Truth, as it were, travels down the side of the Square. Falsehood does not: if the universal proposition is false, that doesn’t tell us anything about the truth or falsehood of the corresponding particular. You could have a false A proposition—all men are priests—with a true corresponding I—some men are priests. But you could also have a false A proposition—all cats are dogs—whose corresponding I—some cats are dogs—is also false. Likewise, you could have a false E proposition—no men are priests—with a true corresponding O—some men are not priests. But you could also have a false E proposition—no whales are mammals—whose corresponding O— some whales are not mammals—is also false.

Falsehood doesn’t travel down the side of the Square, but it does travel up. That is, if a particular proposition—I or O—is false, then its corresponding universal proposition—A or E, respectively—must also be false. Think about it in the abstract: if it’s false that some S are P, that means that there’s not even one S that’s also a P; well, in that case, there’s no way all the Ss are Ps! False I, false A. Likewise on the negative side: if it’s false that some S are not P, that means you won’t find even one S that’s not a P, which is to say all the Ss are Ps; in that case, it’s false that no S is P (A and E are contraries). False O, false E.

Check Your Understanding

3.2.6 Making Inferences
Given information about the truth or falsehood of a particular categorical proposition, we can use the relationships in the Square of Opposition to make inferences about the truth values of the other three types of categorical propositions.

Suppose a universal affirmative proposition—an A proposition—is true. What are the truth-values of the corresponding E, I, and O propositions? (“Corresponding” means propositions with the same subject and predicate classes.) The Square can help us answer these questions.

First of all, A is in the opposite corner from O—they’re contradictory. That means A and O have to have opposite truth-values. Well, if A is true, as we’re supposing, then the corresponding O proposition has to be false.
Also, A and E are contraries. That means that they can’t both be true. Well, we’re supposing that the A is true, so then the corresponding E must be false.
What about the I proposition? Three ways to attack this one, and they all agree that the I must be true:
I is the subaltern of A, so if A is true, then I must be true as well.
I is the contradictory of E, and we’ve already determined that E must be false, so I must be true.
I and O are subcontraries, meaning they can’t both be false, and since we’ve already determined that O is false, it follows that I must be true.
In summary: If an A proposition is true, the corresponding O is false, E is false, and I is true.

Next, for example, let’s suppose a universal negative – E proposition – is true. What about the corresponding A, I, and O propositions? Well, again, A and E are contraries—can’t both be true— so A must be false. I is the contradictory of E, so it must be false—the opposite of I’s truth-value. And since O is subaltern to E, it must be true because E is.

In summary: If an E proposition is true, the corresponding A is false, I is false, and O is true.

And one more: Suppose a particular affirmative – I proposition – is true. What about the other three? Well, E is its contradictory, so it must be false. And if some S are P, that means some of them aren’t— so the O is also true. And since A is the contradictory of O… WAIT JUST A MINUTE! Go back and read that again. Do you see what happened? “And if some S are P, that means some of them aren’t….” No it doesn’t! Remember, ‘some’ means ‘there is at least one’. If some S are P, that just means at least one S is a P—and for all we know, all of them might be; and then again, maybe not. I and O are subcontraries: they can’t both be false, they could both be true, and one could be true and the other false. Knowing that I is true tells us nothing about the truth-value of the corresponding O or the corresponding A. That some are, meaning at least one is, leaves open the possibility that all of them are; but then again, maybe not. The fact is, based on the supposition that an I is true, we can only know the truth-value of the corresponding E for sure.

In summary: If an I proposition is true, then the corresponding E is false, and A and O have an undetermined (unknown) truth-value.

Skills Practice

Suppose that a particular negative – O proposition – is true. What inferences can you make about the truth values of the other three corresponding proposition types?

Pictures

Oparaugo Michael — Thu, 02 Jul 2026 13:30:59 +0000

Forever living products.

Oparaugo Michael — Mon, 29 Jun 2026 17:07:03 +0000

Click this link for more information.
https://mikeflpforme.wordpress.com

Philosophy lecture.

Oparaugo Michael — Mon, 29 Jun 2026 15:52:53 +0000

Logic Table

Oparaugo Michael — Fri, 26 Jun 2026 15:10:27 +0000

https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html

https://amateurlogician.com/categorical-syllogisms-venn-diagrams/

Logic Diagram

Oparaugo Michael — Fri, 26 Jun 2026 15:06:13 +0000

Categorical Syllogisms
The Structure of Syllogism
Now, on to the next level, at which we combine more than one categorical proposition to fashion logical arguments. A categorical syllogism is an argument consisting of exactly three categorical propositions (two premises and a conclusion) in which there appear a total of exactly three categorical terms, each of which is used exactly twice.

One of those terms must be used as the subject term of the conclusion of the syllogism, and we call it the minor term of the syllogism as a whole. The major term of the syllogism is whatever is employed as the predicate term of its conclusion. The third term in the syllogism doesn't occur in the conclusion at all, but must be employed in somewhere in each of its premises; hence, we call it the middle term.

Since one of the premises of the syllogism must be a categorical proposition that affirms some relation between its middle and major terms, we call that the major premise of the syllogism. The other premise, which links the middle and minor terms, we call the minor premise.

Consider, for example, the categorical syllogism:

            No geese are felines.
            Some birds are geese.
 Therefore, Some birds are not felines.

Clearly, "Some birds are not felines" is the conclusion of this syllogism. The major term of the syllogism is "felines" (the predicate term of its conclusion), so "No geese are felines" (the premise in which "felines" appears) is its major premise. Simlarly, the minor term of the syllogism is "birds," and "Some birds are geese" is its minor premise. "geese" is the middle term of the syllogism.

Standard Form
In order to make obvious the similarities of structure shared by different syllogisms, we will always present each of them in the same fashion. A categorical syllogism in standard form always begins with the premises, major first and then minor, and then finishes with the conclusion. Thus, the example above is already in standard form. Although arguments in ordinary language may be offered in a different arrangement, it is never difficult to restate them in standard form. Once we've identified the conclusion which is to be placed in the final position, whichever premise contains its predicate term must be the major premise that should be stated first.

Medieval logicians devised a simple way of labelling the various forms in which a categorical syllogism may occur by stating its mood and figure. The mood of a syllogism is simply a statement of which categorical propositions (A, E, I, or O) it comprises, listed in the order in which they appear in standard form. Thus, a syllogism with a mood of OAO has an O proposition as its major premise, an A proposition as its minor premise, and another O proposition as its conclusion; and EIO syllogism has an E major premise, and I minor premise, and an O conclusion; etc.

Since there are four distinct versions of each syllogistic mood, however, we need to supplement this labelling system with a statement of the figure of each, which is solely determined by the position in which its middle term appears in the two premises: in a first-figure syllogism, the middle term is the subject term of the major premise and the predicate term of the minor premise; in second figure, the middle term is the predicate term of both premises; in third, the subject term of both premises; and in fourth figure, the middle term appears as the predicate term of the major premise and the subject term of the minor premise. (The four figures may be easier to remember as a simple chart showing the position of the terms in each of the premises:

    M   P        P   M        M   P        P   M
 1    \       2      |     3  |         4    /
    S   M        S   M        M   S        M   S

All told, there are exactly 256 distinct forms of categorical syllogism: four kinds of major premise multiplied by four kinds of minor premise multiplied by four kinds of conclusion multiplied by four relative positions of the middle term. Used together, mood and figure provide a unique way of describing the logical structure of each of them. Thus, for example, the argument "Some merchants are pirates, and All merchants are swimmers, so Some swimmers are pirates" is an IAI-3 syllogism, and any AEE-4 syllogism must exhibit the form "All P are M, and No M are S, so No S are P."

Form and Validity
This method of differentiating syllogisms is significant because the validity of a categorical syllogism depends solely upon its logical form. Remember our earlier definition: an argument is valid when, if its premises were true, then its conclusion would also have to be true. The application of this definition in no way depends upon the content of a specific categorical syllogism; it makes no difference whether the categorical terms it employs are "mammals," "terriers," and "dogs" or "sheep," "commuters," and "sandwiches." If a syllogism is valid, it is impossible for its premises to be true while its conclusion is false, and that can be the case only if there is something faulty in its general form.

Thus, the specific syllogisms that share any one of the 256 distinct syllogistic forms must either all be valid or all be invalid, no matter what their content happens to be. Every syllogism of the form AAA-1 is valid, for example, while all syllogisms of the form OEE-3 are invalid.

This suggests a fairly straightforward method of demonstrating the invalidity of any syllogism by "logical analogy." If we can think of another syllogism which has the same mood and figure but whose terms obviously make both premises true and the conclusion false, then it is evident that all syllogisms of this form, including the one with which we began, must be invalid.

Thus, for example, it may be difficult at first glance to assess the validity of the argument:

            All philosophers are professors.
            All philosophers are logicians.
 Therefore, All logicians are professors.

But since this is a categorical syllogism whose mood and figure are AAA-3, and since all syllogisms of the same form are equally valid or invalid, its reliability must be the same as that of the AAA-3 syllogism:
All terriers are dogs.
All terriers are mammals.
Therefore, All mammals are dogs.

Both premises of this syllogism are true, while its conclusion is false, so it is clearly invalid. But then all syllogisms of the AAA-3 form, including the one about logicians and professors, must also be invalid.
This method of demonstrating the invalidity of categorical syllogisms is useful in many contexts; even those who have not had the benefit of specialized training in formal logic will often acknowledge the force of a logical analogy. The only problem is that the success of the method depends upon our ability to invent appropriate cases, syllogisms of the same form that obviously have true premises and a false conclusion. If I have tried for an hour to discover such a case, then either there can be no such case because the syllogism is valid or I simply haven't looked hard enough yet.

Diagramming Syllogisms
The modern interpretation offers a more efficient method of evaluating the validity of categorical syllogisms. By combining the drawings of individual propositions, we can use Venn diagrams to assess the validity of categorical syllogisms by following a simple three-step procedure:

First draw three overlapping circles and label them to represent the major, minor, and middle terms of the syllogism.

Next, on this framework, draw the diagrams of both of the syllogism's premises.
Always begin with a universal proposition, no matter whether it is the major or the minor premise.
Remember that in each case you will be using only two of the circles in each case; ignore the third circle by making sure that your drawing (shading or × ) straddles it.

Finally, without drawing anything else, look for the drawing of the conclusion. If the syllogism is valid, then that drawing will already be done.
Since it perfectly models the relationships between classes that are at work in categorical logic, this procedure always provides a demonstration of the validity or invalidity of any categorical syllogism.

Consider, for example, how it could be applied, step by step, to an evaluation of a syllogism of the EIO-3 mood and figure,

            No M are P.
            Some M are S.
 Therefore, Some S are not P.

three circles
First, we draw and label the three overlapping circles needed to represent all three terms included in the categorical syllogism:

Second, we diagram each of the premises:
major premise

Since the major premise is a universal proposition, we may begin with it. The diagram for "No M are P" must shade in the entire area in which the M and P circles overlap. (Notice that we ignore the S circle by shading on both sides of it.)
minor premise

Now we add the minor premise to our drawing. The diagram for "Some M are S" puts an × inside the area where the M and S circles overlap. But part of that area (the portion also inside the P circle) has already been shaded, so our × must be placed in the remaining portion.

Third, we stop drawing and merely look at our result. Ignoring the M circle entirely, we need only ask whether the drawing of the conclusion "Some S are not P" has already been drawn.conclusion

Remember, that drawing would be like the one at left, in which there is an × in the area inside the S circle but outside the P circle. Does that already appear in the diagram on the right above? Yes, if the premises have been drawn, then the conclusion is already drawn.

But this models a significant logical feature of the syllogism itself: if its premises are true, then its conclusion must also be true. Any categorical syllogism of this form is valid.

Here are the diagrams of several other syllogistic forms. In each case, both of the premises have already been drawn in the appropriate way, so if the drawing of the conclusion is already drawn, the syllogism must be valid, and if it is not, the syllogism must be invalid.AAA-1

AAA-1 (valid)
All M are P.
All S are M.
Therefore, All S are P.

AAA-3
AAA-3 (invalid)
All M are P.
All M are S.
Therefore, All S are P.

OAO-3
OAO-3 (valid)
Some M are not P.
All M are S.
Therefore, Some S are not P.

EOO-2
EOO-2 (invalid)
No P are M.
Some S are not M.
Therefore, Some S are not P.

IOO-1
IOO-1 (invalid)
Some M are P.
Some S are not M.
Therefore, Some S are not P.

Practice your skills in using Venn Diagrams to test the validity of Categorical Syllogisms by using Ron Blatt's excellent Syllogism Evaluator.

Categorical Propositions

Oparaugo Michael — Fri, 26 Jun 2026 15:02:07 +0000

Categorical Propositions
Now that we've taken notice of many of the difficulties that can be caused by sloppy use of ordinary language in argumentation, we're ready to begin the more precise study of deductive reasoning. Here we'll achieve the greater precision by eliminating ambiguous words and phrases from ordinary language and carefully defining those that remain. The basic strategy is to create a narrowly restricted formal system—an artificial, rigidly structured logical language within which the validity of deductive arguments can be discerned with ease. Only after we've become familiar with this limited range of cases will we consider to what extent our ordinary-language argumentation can be made to conform to its structure.

Our initial effort to pursue this strategy is the ancient but worthy method of categorical logic. This approach was originally developed by Aristotle, codified in greater detail by medieval logicians, and then interpreted mathematically by George Boole and John Venn in the nineteenth century. Respected by many generations of philosophers as the the chief embodiment of deductive reasoning, this logical system continues to be useful in a broad range of ordinary circumstances.

Terms and Propositions
We'll start very simply, then work our way toward a higher level. The basic unit of meaning or content in our new deductive system is the categorical term. Usually expressed grammatically as a noun or noun phrase, each categorical term designates a class of things. Notice that these are (deliberately) very broad notions: a categorical term may designate any class—whether it's a natural species or merely an arbitrary collection—of things of any variety, real or imaginary. Thus, "cows," "unicorns," "square circles," "philosophical concepts," "things weighing more than fifty kilograms," and "times when the earth is nearer than 75 million miles from the sun," are all categorical terms.

Notice also that each categorical term cleaves the world into exactly two mutually exclusive and jointly exhaustive parts: those things to which the term applies and those things to which it does not apply. For every class designated by a categorical term, there is another class, its complement, that includes everything excluded from the original class, and this complementary class can of course be designated by its own categorical term. Thus, "cows" and "non-cows" are complementary classes, as are "things weighing more than fifty kilograms" and "things weighing fifty kilograms or less." Everything in the world (in fact, everything we can talk or think about) belongs either to the class designated by a categorical term or to its complement; nothing is omitted.

Now let's use these simple building blocks to assemble something more interesting. A categorical proposition joins together exactly two categorical terms and asserts that some relationship holds between the classes they designate. (For our own convenience, we'll call the term that occurs first in each categorical proposition its subject term and other its predicate term.) Thus, for example, "All cows are mammals" and "Some philosophy teachers are young mothers" are categorical propositions whose subject terms are "cows" and "philosophy teachers" and whose predicate terms are "mammals" and "young mothers" respectively.

Each categorical proposition states that there is some logical relationship that holds between its two terms. In this context, a categorical term is said to be distributed if that proposition provides some information about every member of the class designated by that term. Thus, in our first example above, "cows" is distributed because the proposition in which it occurs affirms that each and every cow is also a mammal, but "mammals" is undistributed because the proposition does not state anything about each and every member of that class. In the second example, neither of the terms is distributed, since this proposition tells us only that the two classes overlap to some (unstated) extent.

Quality and Quantity
Since we can always invent new categorical terms and consider the possible relationship of the classes they designate, there are indefinitely many different individual categorical propositions. But if we disregard the content of these propositions, what classes of things they're about, and concentrate on their form, the general manner in which they conjoin their subject and predicate terms, then we need only four distinct kinds of categorical proposition, distinguished from each other only by their quality and quantity, in order to assert anything we like about the relationship between two classes.

The quality of a categorical proposition indicates the nature of the relationship it affirms between its subject and predicate terms: it is an affirmative proposition if it states that the class designated by its subject term is included, either as a whole or only in part, within the class designated by its predicate term, and it is a negative proposition if it wholly or partially excludes members of the subject class from the predicate class. Notice that the predicate term is distributed in every negative proposition but undistributed in all affirmative propositions.

The quantity of a categorical proposition, on the other hand, is a measure of the degree to which the relationship between its subject and predicate terms holds: it is a universal proposition if the asserted inclusion or exclusion holds for every member of the class designated by its subject term, and it is a particular proposition if it merely asserts that the relationship holds for one or more members of the subject class. Thus, you'll see that the subject term is distributed in all universal propositions but undistributed in every particular proposition.

Combining these two distinctions and representing the subject and predicate terms respectively by the letters "S" and "P," we can uniquely identify the four possible forms of categorical proposition:

A universal affirmative proposition (to which, following the practice of medieval logicians, we will refer by the letter "A") is of the form
All S are P.
Such a proposition asserts that every member of the class designated by the subject term is also included in the class designated by the predicate term. Thus, it distributes its subject term but not its predicate term.

A universal negative proposition (or "E") is of the form
No S are P.
This proposition asserts that nothing is a member both of the class designated by the subject term and of the class designated by the predicate terms. Since it reports that every member of each class is excluded from the other, this proposition distributes both its subject term and its predicate term.

A particular affirmative proposition ("I") is of the form
Some S are P.
A proposition of this form asserts that there is at least one thing which is a member both of the class designated by the subject term and of the class designated by the predicate term. Both terms are undistributed in propositions of this form.

Finally, a particular negative proposition ("O") is of the form
Some S are not P.
Such a proposition asserts that there is at least one thing which is a member of the class designated by the subject term but not a member of the class designated by the predicate term. Since it affirms that the one or more crucial things that they are distinct from each and every member of the predicate class, a proposition of this form distributes its predicate term but not its subject term.
Although the specific content of any actual categorical proposition depends upon the categorical terms which occur as its subject and predicate, the logical form of the categorical proposition must always be one of these four types.

The Square of Opposition
When two categorical propositions are of different forms but share exactly the same subject and predicate terms, their truth is logically interdependent in a variety of interesting ways, all of which are conveniently represented in the traditional "square of opposition."

"All S are P."  (A)- - - - - - -(E)  "No S are P."
                 | *           * |
                     *       *
                 |     *   *     |
                         *
                 |     *   *     |
                     *       *
                 | *           * |

"Some S are P." (I)--- --- ---(O) "Some S are not P."
Propositions that appear diagonally across from each other in this diagram (A and O on the one hand and E and I on the other) are contradictories. No matter what their subject and predicate terms happen to be (so long as they are the same in both) and no matter how the classes they designate happen to be related to each other in fact, one of the propositions in each contradictory pair must be true and the other false. Thus, for example, "No squirrels are predators" and "Some squirrels are predators" are contradictories because either the classes designated by the terms "squirrel" and "predator" have at least one common member (in which case the I proposition is true and the E proposition is false) or they do not (in which case the E is true and the I is false). In exactly the same sense, the A and O propositions, "All senators are politicians" and "Some senators are not politicians" are also contradictories.

The universal propositions that appear across from each other at the top of the square (A and E) are contraries. Assuming that there is at least one member of the class designated by their shared subject term, it is impossible for both of these propositions to be true, although both could be false. Thus, for example, "All flowers are colorful objects" and "No flowers are colorful objects" are contraries: if there are any flowers, then either all of them are colorful (making the A true and the E false) or none of them are (making the E true and the A false) or some of them are colorful and some are not (making both the A and the E false).

Particular propositions across from each other at the bottom of the square (I and O), on the other hand, are the subcontraries. Again assuming that the class designated by their subject term has at least one member, it is impossible for both of these propositions to be false, but possible for both to be true. "Some logicians are professors" and "Some logicians are not professors" are subcontraries, for example, since if there any logicians, then either at least one of them is a professor (making the I proposition true) or at least one is not a professor (making the O true) or some are and some are not professors (making both the I and the O true).

Finally, the universal and particular propositions on either side of the square of opposition (A and I on the one left and E and O on the right) exhibit a relationship known as subalternation. Provided that there is at least one member of the class designated by the subject term they have in common, it is impossible for the universal proposition of either quality to be true while the particular proposition of the same quality is false. Thus, for example, if it is universally true that "All sheep are ruminants", then it must also hold for each particular case, so that "Some sheep are ruminants" is true, and if "Some sheep are ruminants" is false, then "All sheep are ruminants" must also be false, always on the assumption that there is at least one sheep. The same relationships hold for corresponding E and O propositions.