Judgments, negation, and square of opposition

A word by itself is neither true nor false. Truth arises when we assert or deny something about something.

Scene: “everyone is late”

After the meeting, someone says: “Everyone was late.” You remember that two people arrived on time. The phrase turned out to be not an approximate description, but a general statement refuted by one counterexample.

In logic, a thought in which something is affirmed or denied about something is called judgment. To understand its strength, ask:

about how many objects it is: all, some, or one;
what is done: affirmed or denied;
in what time and under what conditions.

Four Useful Forms

Traditional logic denotes them by the letters A, E, I, O, but first it is enough to read them in words:

Form	Example	How to refute
All S are P	All tickets are named	find one unnamed ticket
No S is P	No window is open	find one open
Some S are P	Some books are illustrated	show at least one
Some S are not P	Some seats are not occupied	show at least one

Second example

Not all guests responded to the invitation.

This does not mean “no one answered.” The phrase means: there exists at least one guest who did not answer. The difference between not all and none often changes the solution completely.

Repair: uncover the denial: who exactly did not do what and under what conditions.

Square of Opposition without Memorization

Traditional square of opposition: relationships between “all”, “none”, “some”, and “some not”

The main practical benefit of the square is not in the names of relations, but in the discipline of negation:

negation of “everyone paid” is “someone did not pay”;
negation of “nobody came” is “someone came”;
from “everyone came” it follows “nobody was absent”, if it is about the same group, time and meaning of the word “came”.

The traditional square assumes that the discussed class is not empty. Modern logic is more careful: the phrase “all unicorns are white” does not prove that unicorns exist.

Denial Area

Compare:

I did not promise to come today.
I promised not to come today.
I promised to come not today.

The words are almost the same, but different parts are negated. In public debates, a quote is often distorted precisely by shifting the negation.

Repair: rewrite the phrase so that the negation is next to what it refers to, and add a time or condition.

Short practice

Deny the phrase «All participants agreed with the outcome» in order to get:

strict logical negation;
stronger statement;
a checkable question.

Answers:

“At least one participant did not agree with the outcome”.
“No participant agreed” — stronger and does not follow automatically.
“Who exactly did not agree and with which item?”

Chapter Rule: deny exactly what you want to deny. “Not all,” “none” and “some not” describe different situations.

Academic and Engineering Walkthrough

In Chepanov judgment is a thought in which something is affirmed or denied; expressed in words, it becomes a sentence. One judgment can be expressed by different sentences:

Ни один запрос без токена не авторизован.
Все авторизованные запросы имеют токен.
Если запрос авторизован, у него есть токен.

In the selected model, these phrases may convey a similar relationship, although the grammar differs.

Classic form:

S есть P
S не есть P

S — subject, P — predicate, the copula connects or separates them.

2. Quantity and quality

Categorical judgments are classified by:

quantity: general or specific;
quality: affirmative or negative.

From here four forms:

Name	Form	Example	Modern notation
A	All S are P	All administrators have MFA	`∀x(Sx → Px)`
E	No S is P	No plaintext password is secure	`∀x(Sx → ¬Px)`
I	Some S are P	Some tasks are blocked	`∃x(Sx ∧ Px)`
O	Some S are not P	Some events are not delivered	`∃x(Sx ∧ ¬Px)`

Letters originate from Latin affirmo and nego.

3. “Some” in logic

In classical logic некоторые S суть P means at least one. The statement does not exclude that all S are P.

In colloquial speech:

Some tests have failed.

often hints at “not all.” This is a pragmatic implicature, not a logical content. It is better to write explicitly in the specification:

Как минимум один тест упал.
Не все тесты упали.
Ровно 17 тестов упали.

4. Term Distribution

A term is distributed, if the judgment speaks of its entire extension.

Form	Subject S	Predicate P	Why
A: all S are P	distributed	not distributed	said about all S, but not about all P
E: no S is P	distributed	distributed	classes are completely separated
I: some S are P	not distributed	not distributed	only about the intersection
O: some S are not P	not distributed	distributed	selected S are excluded from all P

This is a table from which syllogism rules are later derived.

Example A:

Все SRE — инженеры.

We talk about all SREs, but we do not claim that all engineers are SREs. Therefore, SRE is distributed, инженеры is not.

5. Square of Opposition

Contradiction

A and O: “all S are P” / “some S are not P”;
E and I: “no S is P” / “some S are P”.

Contradictory judgments cannot be simultaneously true and cannot be simultaneously false.

Antonymy

A and E cannot be simultaneously true, but can be simultaneously false:

Все задачи автоматизированы.      — ложно
Ни одна задача не автоматизирована. — тоже ложно

Part of the tasks is automated, part is not.

Subcontrariety

I and O in the traditional square cannot be simultaneously false if the S class is non-empty.

Subordination

Traditionally from A to I, from E to O. This requires the existence of S.

6. Modern Caveat About Empty Classes

Let there be no unicorns:

Все единороги имеют MFA.        ∀x(Ux → Mx)
Ни один единорог не имеет MFA.  ∀x(Ux → ¬Mx)

Both universal formulas are vacuously true: there is no object violating either one. Therefore, in modern predicate logic, traditional relations of opposition and subordination do not work without an existential presupposition.

Reliable recording:

∃x Ux                 класс U непуст
∀x(Ux → Mx)           все U есть M
──────────────────
∃x(Ux ∧ Mx)           некоторые U есть M

Do not throw away the historical schema; remember its model.

7. Converse, Inversion and Contraposition

7.1. Converse

Changing quality and replacing the predicate with a complement:

Все S есть P.
Ни одно S не есть не-P.

The converse is valid for A, E, I and O.

7.2. Addressing

Swap the subject and predicate.

Ни одно S не P  →  Ни одно P не S.          E, корректно
Некоторые S есть P → Некоторые P есть S.    I, корректно

Simple invocation of A is incorrect:

Все кошки — животные.
Следовательно, все животные — кошки.

Traditional access A “with restriction” to Некоторые P есть S requires the existence of S.

7.3. Contradiction to the Predicate

For A:

Все S есть P.
Все не-P не есть S.

In modern notation this is a counterposition:

∀x(Sx → Px)  ⟺  ∀x(¬Px → ¬Sx)

But the converse statement P → S does not appear.

8. Conditional Judgments

Если P, то Q.

Material implication is false only when P is true and Q is false.

P	Q	P → Q
0	0	1
0	1	1
1	0	0
1	1	1

Sufficient and Necessary

In P → Q:

P is sufficient for Q;
Q is necessary for P.

Если запрос успешно авторизован, его токен валиден.

A valid token is required in this model, but may be insufficient: account is blocked, scope is invalid.

“Only If” and “If”

P, только если Q   = P → Q
P, если Q          = Q → P
P тогда и только тогда, когда Q = P ↔ Q

The word “only” changes the direction. It is a source of a huge number of errors in requirements.

9. Delimiting Judgments

Modern disjunction is usually not exclusive:

P ∨ Q

True when both parts are true. “Either tea or coffee” in a menu often excludes joint choice, but logical “or” is not required to do so.

исключающее или: P ⊕ Q

Before using the disjunctive syllogism clarify:

Are the options exhaustive?
Are the options incompatible?
Is it inclusive or exclusive “or”?

10. Modality

Chelpanov distinguishes:

problematic: possibly P;
assertoric: factually P;
apodictic: necessarily P.

Modern notation:

◇P  возможно P
□P  необходимо P

Common fallacy:

P возможно.
Следовательно, P фактически.

And the reverse:

P фактически.
Следовательно, P необходимо.

The fact that the system has withstood the load now does not prove that it must withstand it in all permissible states.

11. Negation Scope

Compare:

Не все запросы успешны.
Некоторые запросы не успешны.
Ни один запрос не успешен.

The first two formulas are equivalent in a non-empty domain:

¬∀x Success(x)  ⟺  ∃x ¬Success(x)

The third is much stronger:

∀x ¬Success(x)

Quantifier Order

∀x ∃y Reviewer(y, x)

For each PR there is some reviewer.

∃y ∀x Reviewer(y, x)

There is one person reviewing each PR.

Quantifier permutation turns distributed responsibility into a single point of failure.

12. Laws of Thinking

In the traditional formulation:

Law of identity

A есть A

The term should retain its meaning within the reasoning.

Law of non-contradiction

It is impossible to assert P and ¬P simultaneously in one respect, time, and sense.

¬(P ∧ ¬P)

The statuses “task is ready for development” and “task is not ready for production” do not contradict each other: the relationships are different.

Law of excluded middle

P ∨ ¬P

In classical logic, a precisely defined statement has no third value. This does not mean we know which side is true, and does not turn any everyday pair into an exhaustive dilemma.

Principle of Sufficient Reason

It is often added to the three formal laws: a statement must have a basis. This is a methodological principle, not a tautology of classical calculus.

13. Common Judgment Level Errors

Fallacy	Example	Fix
Condition inversion	«without MFA you cannot enter» → «with MFA you can always enter»	separate necessary and sufficient
Contradiction substitution for opposite	«not fast» → «slow»	allow intermediate values
Quantifier shift	«everyone has a reviewer» → «there is a common reviewer»	write out the order `∀/∃`
Incorrect negation	`¬∀` read as `∀¬`	push negation through de Morgan
Existential import	from «all S are P» to «some S are P» immediately	prove `∃S`
Ambiguous disjunction	«either A or B»	clarify completeness and exclusivity
Modal leap	«happened» → «inevitable»	separate `P`, `◇P`, `□P`

14. Engineering example: access policy

Requirement:

Only administrators with MFA can export data.

Possible readings:

Export(x) → Admin(x) ∧ MFA(x)

Administrator and MFA are required for export.

But incorrect:

Admin(x) ∧ MFA(x) → Export(x)

For legal matters, a role in a specific tenant, a work device, no blocking, and separate agreement may be required.

Tests should check the original direction, not accidentally strengthened equivalence.

15. Checklist

Does the statement have a truth value at all?
Is the quantity specified: all, some, exactly, most?
Does “some” mean at least one or “not all”?
Does the subject class exist?
Is a necessary condition not mistaken for a sufficient one?
Is the direction of “if” and “only if” preserved?
Is the disjunction complete and correctly understands option compatibility?
Does the negation have a clear scope?
Has the order of quantifiers not been changed?
Are the actual, possible, and necessary not mixed?
Are the terms distributed correctly?

16. Tasks

Formalize “every task has an owner” and “there is an owner of all tasks”.
Deny without ambiguity: “all tests passed”.
Determine what is necessary and sufficient in the phrase “access is granted only to employees after training”.
Can “all critical incidents require postmortem” be reversed?
Check the traditional square for the class “perpetual engines”.

Sources

Aristotle. On Interpretation.
Aristotle. First Analytics.
Georgy Chelpanov. Logic Textbook, chapters VII–XII.
The Traditional Square of Opposition.
forall x: Calgary, sections on TFL and FOL.

What’s next

Now four forms of A/E/I/O can be combined in deduction. The next chapter analyzes syllogism terms, figures, modes, distribution, and the method of checking through diagrams.

Categorical syllogisms

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Judgments, negation, and the square of opposition

Judgments, negation, and square of opposition

Scene: “everyone is late”

Four Useful Forms

Second example

Square of Opposition without Memorization

Denial Area

Short practice

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