On the traditional square of opposition, if "Some S is not P" (O) is false, then:

A is true, E is false and I is true

A is undetermined, E is false and I is true

A is false, E is true and I is false

A is true, E is undetermined and I is true

A is true, E is false and I is true

Assertion (A): Increasing the connotation of a term generally decreases, or leaves unchanged, its denotation. Reason (R): The connotation of a term is the set of objects to which the term applies. Choose the correct option.

Both A and R are true, and R is the correct explanation of A

Both A and R are true, but R is not the correct explanation of A

Assertion (A): On the traditional square, subcontraries can both be false. Reason (R): I and O propositions with the same subject and predicate are subcontraries. Choose the correct option.

Both A and R are true, and R is the correct explanation of A

Both A and R are true, but R is not the correct explanation of A

Logic I: Truth and Validity, Terms and Propositions, the Laws of Thought, the Square of Opposition and the Categorical Syllogism

Unit 9 is formal logic, and it is examined by working problems, not by naming doctrines. This chapter takes traditional (Aristotelian) logic: truth and validity, denotation and connotation, the nature and classification of propositions, the laws of thought, the square of opposition in both its traditional and Boolean readings, and the categorical syllogism with mood, figure, the rules and their fallacies, and the Venn diagram test. Every inference in the chapter is worked out and can be checked step by step. The Paper-1 reasoning chapters introduce the same A, E, I, O forms at an aptitude level; here they are developed to the depth a philosophy paper sets, including the count of valid forms under each interpretation. The next two chapters take propositional logic and quantification.

1. Truth and validity; denotation and connotation

Truth and falsity belong to propositions; validity and invalidity belong to deductive arguments. A deductive argument is valid when it is impossible for its premises to be true and its conclusion false, whatever the actual truth of its parts; validity depends on form alone. A valid argument with all premises true is sound, and the conclusion of a sound argument must be true. Every combination of truth and validity is possible except one: a valid argument cannot have true premises and a false conclusion.

Truth and validity: which combinations can occur
Premises	Conclusion	Valid argument?	Invalid argument?
True	True	Possible (sound)	Possible
True	False	Impossible	Possible
At least one false	True	Possible	Possible
At least one false	False	Possible	Possible

Denotation and connotation, in J. S. Mill's terms, are the two aspects of a general term's meaning. The denotation (extension) of "planet" is the set of objects it applies to, Mercury, Venus, Earth and the rest; its connotation (intension) is the set of attributes an object must have to fall under it. The law of inverse variation says that as connotation increases, denotation decreases or stays the same, and conversely: "animal", "rational animal", "rational animal born in India" name ever-narrower classes. The qualification "or stays the same" matters, because adding an attribute that every member already has (for example "animal that has a body") leaves the denotation unchanged.

2. Propositions, distribution and the laws of thought

Propositions are classified by relation (categorical, hypothetical "if p then q", disjunctive "either p or q"), by modality (assertoric, problematic, apodeictic) and, for categorical propositions, by quality (affirmative or negative) and quantity (universal or particular). The four standard-form categorical propositions are named by the vowels of affirmo (I affirm) and nego (I deny). A term is distributed in a proposition when the proposition refers to every member of the class it names.

The four categorical propositions and the distribution of their terms
Name	Form	Quantity and quality	Subject distributed?	Predicate distributed?
A	All S is P	Universal affirmative	Yes	No
E	No S is P	Universal negative	Yes	Yes
I	Some S is P	Particular affirmative	No	No
O	Some S is not P	Particular negative	No	Yes

🧠 Universals distribute subjects, negatives distribute predicates

Read the table as two rules. Universal propositions (A, E) distribute their subject; negative propositions (E, O) distribute their predicate. So E distributes both terms, I neither, A only its subject and O only its predicate. Almost every syllogism fallacy is a failure of distribution, so this pair of rules does most of the work in Unit 9.

Law of identity: whatever is, is; A is A. In propositional form, p ⊃ p.
Law of non-contradiction: nothing can both be and not be (in the same respect at the same time); ~(p · ~p).
Law of excluded middle: everything either is or is not; p ∨ ~p. Leibniz's principle of sufficient reason is sometimes added as a fourth law of thought, but it is not a law of formal logic.

3. The square of opposition: traditional and Boolean

The four relations of the traditional square
Relation	Pairs	Rule
Contradictories	A and O; E and I	Exactly one is true: they cannot both be true and cannot both be false
Contraries	A and E	Cannot both be true; can both be false
Subcontraries	I and O	Cannot both be false; can both be true
Subalternation	A to I; E to O	Truth passes down from universal to particular; falsity passes up from particular to universal

Worked inferences on the traditional square
Given	A	E	I	O
A true	T	F (contrary)	T (subaltern)	F (contradictory)
E true	F	T	F	T
I true	Undetermined	F	T	Undetermined
O true	F	Undetermined	Undetermined	T
A false	F	Undetermined	Undetermined	T
E false	Undetermined	F	T	Undetermined
I false	F	T	F	T
O false	T	F	T	F

How the table is worked, for "O false": O's contradictory A must be true; A true makes its contrary E false; A true passes truth down to its subaltern I. Every other row is worked the same way, starting from the contradictory, which is always determined. The traditional square assumes that the subject class has members (existential import).

The Boolean (modern) interpretation drops that assumption for universal propositions: "All S is P" means only "there is nothing that is S and not P", which is true if there are no S at all ("all unicorns have one horn" is vacuously true). Particular propositions still assert existence ("some S is P" means there is at least one S that is P). On this reading only the contradictory relations survive. Contrariety, subcontrariety and subalternation all fail, because when S is empty A and E are both true and I and O are both false. An inference that relies on them, such as A to I, commits the existential fallacy. Of the immediate inferences, conversion of E and I, obversion of all four and contraposition of A and O remain valid; conversion by limitation (A to I) and contraposition by limitation (E to O) are valid only traditionally.

4. The categorical syllogism: mood, figure, rules and fallacies

A standard-form categorical syllogism has three categorical propositions and exactly three terms, each used twice: the major term (P, the predicate of the conclusion), the minor term (S, the subject of the conclusion) and the middle term (M, which appears in both premises but not in the conclusion). The major premise, containing P, is written first. The mood is the three letters of the major premise, minor premise and conclusion in that order; the figure is fixed by the position of the middle term.

The four figures
Figure	Major premise	Minor premise	Position of M
1	M – P	S – M	Subject, then predicate
2	P – M	S – M	Predicate in both
3	M – P	M – S	Subject in both
4	P – M	M – S	Predicate, then subject

The rules of the syllogism and the fallacies they catch (Boolean standard)
Rule	Fallacy when broken
1. Exactly three terms, used in the same sense	Four terms (quaternio terminorum)
2. The middle term must be distributed at least once	Undistributed middle
3. A term distributed in the conclusion must be distributed in its premise	Illicit major (P), illicit minor (S)
4. No syllogism with two negative premises	Exclusive premises
5. A negative premise requires a negative conclusion, and a negative conclusion a negative premise	Affirmative conclusion from a negative premise, or negative conclusion from affirmative premises
6. No particular conclusion from two universal premises (Boolean only)	Existential fallacy

Worked, undistributed middle: "All dogs are mammals; all cats are mammals; so all cats are dogs" is AAA-2 (P–M, S–M). The middle term "mammals" is the predicate of two A propositions and is distributed in neither, so rule 2 fails.
Worked, illicit major: "All M are P; no S are M; so no S are P" is AEE-1. The conclusion (E) distributes P, but in the major premise (A) P is the predicate and undistributed, so rule 3 fails for the major term.
Worked, illicit minor: "All M are P; all M are S; so all S are P" is AAA-3. The conclusion (A) distributes S, but in the minor premise (A) S is the predicate and undistributed, so rule 3 fails for the minor term. (Its conclusion weakened to "some S are P" gives Darapti, AAI-3, valid only traditionally.)

The fifteen unconditionally valid forms (Boolean) and the nine conditionally valid (traditional only)
Figure	Unconditionally valid	Valid only with existential import
1	AAA Barbara, EAE Celarent, AII Darii, EIO Ferio	AAI Barbari, EAO Celaront
2	EAE Cesare, AEE Camestres, EIO Festino, AOO Baroco	EAO Cesaro, AEO Camestros
3	IAI Disamis, AII Datisi, OAO Bocardo, EIO Ferison	AAI Darapti, EAO Felapton
4	AEE Camenes, IAI Dimaris, EIO Fresison	AAI Bamalip, AEO Calemos, EAO Fesapo

Counting: each premise and the conclusion can be A, E, I or O, so there are 4 × 4 × 4 = 64 moods, and with four figures 64 × 4 = 256 standard forms. Of these, 15 are valid on the Boolean interpretation (4 + 4 + 4 + 3 by figure) and 24 on the traditional interpretation, the extra 9 being the "weakened" and "strengthened" forms that draw a particular conclusion from universal premises and are valid only if the relevant class is non-empty.

ℹ️ The Venn diagram test

Draw three overlapping circles for S, P and M. Diagram the universal premise first: shade the region it says is empty (for "all M are P", shade the part of M outside P). Then diagram the particular premise: put an X in the region it says is occupied, and if the premises leave that region split by a line, put the X on the line. The syllogism is valid if and only if diagramming the premises has already diagrammed the conclusion. For Barbara, shading M-outside-P and S-outside-M leaves S-outside-P entirely shaded, which is exactly "all S are P".

Key takeaways

Validity is a matter of form: a valid argument cannot have true premises and a false conclusion; a sound argument is valid with true premises. Connotation and denotation vary inversely (denotation falls or stays the same as connotation rises).
A distributes S; E distributes S and P; I neither; O distributes P. Laws of thought: identity (p ⊃ p), non-contradiction ~(p · ~p), excluded middle (p ∨ ~p).
Traditional square: contradictories (A–O, E–I), contraries (A–E), subcontraries (I–O), subalternation (A→I, E→O). Boolean square: only contradictories survive; A to I commits the existential fallacy.
Mood is the three letters in order; figure is the position of M (1: M–P, S–M; 2: P–M, S–M; 3: M–P, M–S; 4: P–M, M–S). Rules: three terms, middle distributed, no illicit process, not two negatives, negative in means negative out, no existential fallacy.
256 forms (64 moods × 4 figures); 15 valid on the Boolean interpretation, 24 traditionally; Barbara is AAA-1, Celarent EAE-1, Darii AII-1, Ferio EIO-1; valid if the Venn diagram of the premises already shows the conclusion.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

Which categorical proposition distributes both its subject and its predicate?
1. O (Some S is not P)
2. I (Some S is P)
3. A (All S is P)
4. E (No S is P)
Show answer
Answer: D — E (No S is P)
E is universal, so it distributes its subject, and negative, so it distributes its predicate. A distributes only the subject, O only the predicate, and I neither.
"No M is P. Some S is M. Therefore, some S is not P." What are the mood and figure?
1. EIO-1 (Ferio)
2. EIO-4 (Fresison)
3. EIO-2 (Festino)
4. EIO-3 (Ferison)
Show answer
Answer: A — EIO-1 (Ferio)
The premises are E, I and the conclusion O, so the mood is EIO. M is the subject of the major premise (M–P) and the predicate of the minor (S–M), which is the first figure: Ferio. EIO is valid in all four figures, so the figure must be read from the position of M.
"All M are P. No S are M. Therefore, no S are P." This syllogism commits the fallacy of:
1. Exclusive premises
2. Undistributed middle
3. Illicit major
4. Illicit minor
Show answer
Answer: C — Illicit major
This is AEE-1. The conclusion, an E proposition, distributes P; in the major premise, an A proposition, P is the predicate and so undistributed. A term distributed in the conclusion but not in its premise is an illicit process, here of the major term. M is distributed in the major premise, and only one premise is negative.
On the traditional square of opposition, if "Some S is not P" (O) is false, then:
1. A is undetermined, E is false and I is true
2. A is false, E is true and I is false
3. A is true, E is undetermined and I is true
4. A is true, E is false and I is true
Show answer
Answer: D — A is true, E is false and I is true
O false makes its contradictory A true. A true makes its contrary E false (contraries cannot both be true) and makes its subaltern I true (truth passes down). So A true, E false, I true: everything is determined.
Assertion (A): Increasing the connotation of a term generally decreases, or leaves unchanged, its denotation. Reason (R): The connotation of a term is the set of objects to which the term applies. Choose the correct option.
1. Both A and R are true, and R is the correct explanation of A
2. Both A and R are true, but R is not the correct explanation of A
3. A is true, but R is false
4. A is false, but R is true
Show answer
Answer: C — A is true, but R is false
A is true: this is the law of inverse variation, with the qualification that denotation may stay the same. R is false: the set of objects a term applies to is its denotation (extension); its connotation (intension) is the set of attributes that determine membership.
Assertion (A): On the traditional square, subcontraries can both be false. Reason (R): I and O propositions with the same subject and predicate are subcontraries. Choose the correct option.
1. Both A and R are true, and R is the correct explanation of A
2. Both A and R are true, but R is not the correct explanation of A
3. A is true, but R is false
4. A is false, but R is true
Show answer
Answer: D — A is false, but R is true
A is false: subcontraries cannot both be false, though they can both be true. If I were false, its contradictory E would be true and E's subaltern O true, so O cannot also be false. R is true: I and O are the subcontrary pair.
Which of the following syllogistic forms are valid on the Boolean interpretation? Select all that apply.
1. AII-2
2. EAO-3
3. AAA-1
4. OAO-3
Show answer
Answer: C — AAA-1; D — OAO-3
AAA-1 (Barbara) and OAO-3 (Bocardo) are unconditionally valid. AII-2 (all P are M; some S are M; so some S are P) leaves M undistributed, since M is the predicate of an A and an I proposition. EAO-3 (Felapton) draws a particular conclusion from universal premises and is valid only traditionally, so on the Boolean reading it commits the existential fallacy.
In which of the following categorical propositions is the predicate term distributed? Select all that apply.
1. A
2. E
3. I
4. O
Show answer
Answer: B — E; D — O
Negative propositions distribute their predicates: E ("no S is P" excludes S from the whole of P) and O ("some S is not P" excludes some S from the whole of P). A and I, being affirmative, leave the predicate undistributed.
How many standard-form categorical syllogism forms are there, counting every combination of mood and figure? Type the number.
Numerical answer — type the value.
Show answer
Answer: 256
Each of the three propositions can be A, E, I or O, giving 4 × 4 × 4 = 64 moods; each mood can occur in 4 figures, giving 64 × 4 = 256 forms.
How many standard-form syllogisms are unconditionally valid on the Boolean interpretation? Type the number.
Numerical answer — type the value.
Show answer
Answer: 15
Fifteen: four in the first figure (Barbara, Celarent, Darii, Ferio), four in the second (Cesare, Camestres, Festino, Baroco), four in the third (Disamis, Datisi, Bocardo, Ferison) and three in the fourth (Camenes, Dimaris, Fresison). The traditional interpretation adds nine conditionally valid forms, making 24.