22  Understanding the structure of arguments: argument forms, structure of categorical propositions, Mood and Figure, Formal and Informal fallacies, Uses of language, Connotations and denotations of terms, Classical square of opposition

22.1 What the Syllabus Covers

This syllabus head bundles seven examined heads:

  1. Argument forms — what an argument is, its components.
  2. Categorical propositions — A, E, I, O; subject-predicate structure; distribution.
  3. Mood and Figure of categorical syllogism.
  4. Formal vs Informal fallacies.
  5. Uses of language — informative, expressive, directive.
  6. Connotations and denotations of terms.
  7. Classical square of opposition.

PYQs reliably test: (a) identifying the A/E/I/O type of a proposition, (b) naming the figure (1st, 2nd, 3rd, 4th) of a given syllogism, (c) naming a fallacy from a worked example, and (d) the distribution of terms.

22.2 What an Argument Is

An argument in logic is a set of statements in which one statement (the conclusion) is claimed to follow from one or more other statements (the premises). An argument is not a quarrel; it is a structured claim with reasons.

22.2.1 Components

TipComponents of an Argument
  • Premise(s) — the supporting statements offered as evidence.
  • Conclusion — the statement that the premises are supposed to establish.
  • Inference / Linkage — the logical relation between premises and conclusion.

22.2.2 Indicator Words

TipIndicator Words
  • Premise indicators: since · because · for · given that · as · the reason is · in view of.
  • Conclusion indicators: therefore · thus · hence · so · it follows that · consequently · we may infer.

22.3 Argument Forms — Two Basic Kinds

TipTwo Kinds of Argument
Form Claim Strength
Deductive Conclusion must follow from premises Valid / Invalid; Sound / Unsound
Inductive Conclusion probably follows Strong / Weak; Cogent / Uncogent
TipValidity, Truth, Soundness
  • Validity — formal property; the premises if true guarantee the conclusion.
  • Truth — actual correspondence with reality.
  • Soundness — valid argument with all true premises.

A valid argument can have false premises; a sound argument cannot.

22.4 Categorical Propositions — A, E, I, O

A categorical proposition affirms or denies a relation between two classes (subject and predicate).

22.4.1 The Four Forms

TipThe Four Categorical Propositions (A, E, I, O)
Code Form Quantity Quality Latin source
A All S are P Universal Affirmative Affirmo
E No S are P Universal Negative Nego
I Some S are P Particular Affirmative Affirmo
O Some S are not P Particular Negative Nego

The vowels in AffIrmo (I affirm) give the affirmative codes (A, I); the vowels in nEgO (I deny) give the negative codes (E, O).

22.4.2 Distribution of Terms

A term is distributed when the proposition refers to all members of its class.

TipDistribution by Proposition Type
Type Subject (S) Predicate (P)
A — All S are P Distributed Undistributed
E — No S are P Distributed Distributed
I — Some S are P Undistributed Undistributed
O — Some S are not P Undistributed Distributed

Memory aid: A distributes S; E distributes both; I distributes neither; O distributes P.

22.5 Classical Square of Opposition

The square of opposition shows the logical relations among A, E, I, O propositions with the same subject and predicate.

TipFour Relations on the Square
Relation Pair Truth-value link
Contradictories A ↔︎ O; E ↔︎ I Cannot both be true; cannot both be false
Contraries A ↔︎ E Cannot both be true; can both be false
Subcontraries I ↔︎ O Cannot both be false; can both be true
Subalterns A → I; E → O Truth of universal entails truth of particular 

flowchart TB
  A["A: All S are P<br/>(Universal Affirmative)"] -- "Contraries" --- E["E: No S are P<br/>(Universal Negative)"]
  I["I: Some S are P<br/>(Particular Affirmative)"] -- "Subcontraries" --- O["O: Some S are not P<br/>(Particular Negative)"]
  A -- "Subalterns" --- I
  E -- "Subalterns" --- O
  A -. "Contradictories" .- O
  E -. "Contradictories" .- I
    classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

TipQuick Inferences
  • If A is true, then O is false (contradictories), E is false (contraries), I is true (subalterns).
  • If I is false, then A is false (subalterns), E is true (contradictories with I).

22.6 Categorical Syllogism — Mood and Figure

A categorical syllogism is a deductive argument with two premises and one conclusion, all categorical propositions. It contains three terms: Major (P), Minor (S), and Middle (M).

TipTerm Placement
  • Major term (P) — the predicate of the conclusion.
  • Minor term (S) — the subject of the conclusion.
  • Middle term (M) — appears in both premises but not in the conclusion.
  • Major premise — contains the major term.
  • Minor premise — contains the minor term.

22.6.1 Mood

The mood of a syllogism is the ordered sequence of A/E/I/O types of its three propositions. 64 possible moods (4³).

22.6.2 Figure

The figure is the position of the middle term in the two premises. Four figures:

TipThe Four Figures
Figure Major premise Minor premise
1st M – P S – M
2nd P – M S – M
3rd M – P M – S
4th P – M M – S

Memory aid (medieval mnemonic): “sub-prae / prae-prae / sub-sub / prae-sub” for the position of M.

22.6.3 Valid Moods (Medieval Names)

Medieval logicians named the 24 valid moods using mnemonic words whose vowels gave the mood. Famous examples:

TipSome Famous Valid Moods
  • Figure 1: Barbara (AAA), Celarent (EAE), Darii (AII), Ferio (EIO).
  • Figure 2: Cesare (EAE), Camestres (AEE), Festino (EIO), Baroco (AOO).
  • Figure 3: Darapti (AAI), Disamis (IAI), Datisi (AII), Felapton (EAO).
  • Figure 4: Bramantip (AAI), Camenes (AEE), Dimaris (IAI), Fesapo (EAO).

22.6.4 Six Rules for Valid Syllogism

TipSix Rules of Validity
  1. Three terms only, each used in the same sense.
  2. Middle term distributed at least once.
  3. No term distributed in conclusion that is not distributed in its premise.
  4. No conclusion from two negative premises.
  5. If one premise is negative, conclusion must be negative.
  6. No conclusion from two particular premises (must have at least one universal).

22.7 Formal Fallacies in Syllogisms

Violating any of the six rules produces a formal fallacy:

TipSix Formal Fallacies
  1. Fallacy of Four Terms (Quaternio Terminorum) — Rule 1.
  2. Undistributed Middle — Rule 2 violated.
  3. Illicit Major / Illicit Minor — Rule 3 violated.
  4. Exclusive Premises — Rule 4 violated.
  5. Drawing affirmative conclusion from negative premise — Rule 5.
  6. Existential Fallacy — drawing a particular conclusion from two universal premises (Boolean reading).

22.7.1 Three Propositional Fallacies

Already covered in Topic 18:

TipFormal Propositional Fallacies
  • Affirming the Consequent.
  • Denying the Antecedent.
  • Improper Disjunctive Inference.

22.8 Informal Fallacies

Informal fallacies are errors in content or relevance, not in form. Aristotle named 13 in Sophistical Refutations; modern lists exceed 100.

22.8.1 Fallacies of Relevance

TipFallacies of Relevance
  • Ad hominem — attack the person.
  • Ad populum — appeal to popularity.
  • Ad verecundiam — appeal to authority.
  • Ad baculum — appeal to force/threat.
  • Ad misericordiam — appeal to pity.
  • Ad ignorantiam — appeal to ignorance (“not disproved, therefore true”).
  • Ignoratio elenchi / Red herring — irrelevant conclusion.
  • Tu quoque — “you too” — turning the criticism back.
  • Genetic fallacy — judging a claim by its origin.
  • Straw man — misrepresenting the opponent’s view.

22.8.2 Fallacies of Presumption

TipFallacies of Presumption
  • Begging the question (petitio principii) — assumes the conclusion.
  • Complex question — loaded question (“Have you stopped beating your wife?”).
  • False cause (post hoc, ergo propter hoc) — temporal sequence as causation.
  • False analogy — comparing across irrelevant differences.
  • Hasty generalisation.
  • Slippery slope.
  • False dilemma.

22.8.3 Fallacies of Ambiguity

TipFallacies of Ambiguity
  • Equivocation — word used in two senses.
  • Amphiboly — sentence structurally ambiguous.
  • Composition — what holds of parts holds of whole.
  • Division — what holds of whole holds of parts.
  • Accent — meaning depends on stress.

22.9 Uses of Language

The most-asked classification (Copi & Cohen): three primary uses.

TipThree Uses of Language
  1. Informative — to assert facts (used in argument).
  2. Expressive — to express emotion (poetry, exclamations).
  3. Directive — to influence behaviour (commands, requests).

Mixed uses are common (a sermon may be informative, expressive, and directive).

22.9.1 Other Linguistic Functions

TipAdditional Functions
  • Ceremonial — greetings, condolences, oaths.
  • Performative — utterance does the act (“I promise”, “I now pronounce you…”).
  • Phatic — channel-maintenance (“Hello?”).

These align loosely with Jakobson’s six (Topic 14).

22.10 Connotation and Denotation

J.S. Mill’s A System of Logic (1843) distinguishes two aspects of a term’s meaning:

TipConnotation vs Denotation
Aspect Definition Example for “human”
Connotation (intension) The set of essential attributes rational, biped, social, mortal
Denotation (extension) The set of things referred to Socrates, Riya, every human
TipThree Inverse Rules (Approximate)
  • The greater a term’s connotation, the smaller its denotation (e.g., “Indian engineer” denotes fewer things than “engineer”).
  • A proper name has denotation but no fixed connotation (debated).
  • A purely abstract idea may have connotation but no denotation.

22.11 Worked Examples

22.11.1 Identify the Proposition Type

“Some politicians are not honest.” Quantity: particular (“Some”). Quality: negative (“not”). → O.

22.11.2 Identify the Figure

All mammals (M) are warm-blooded (P). All whales (S) are mammals (M). Therefore all whales (S) are warm-blooded (P).

Middle term M is subject of major and predicate of minor → Figure 1. Mood AAA → Barbara.

22.11.3 Spot the Formal Fallacy

All cats (M) are mammals (P). Some pets (S) are mammals (M, undistributed). Therefore some pets (S) are cats (P).

Undistributed Middle — neither premise distributes M. Invalid.

22.11.4 Spot the Informal Fallacy

“Smoking is fine. My grandfather smoked all his life and lived to 95.” Hasty generalisation (one case → universal claim).

22.11.5 Use of Language

“Don’t touch the wet paint.” — Directive. “It is raining heavily.” — Informative. “What a tragic loss!” — Expressive.

22.12 Theory Anchors

TipPersons, Years and Key Ideas
Person Year Contribution
Aristotle 4th c. BCE Categorical syllogism; A, E, I, O; Sophistical Refutations (informal fallacies)
Theophrastus 4th c. BCE Refined syllogistic figures
Medieval logicians 12th–14th c. Barbara-Celarent mnemonic; 24 valid moods
William of Ockham 14th c. Razor of parsimony
John Stuart Mill 1843 Connotation/Denotation distinction
George Boole 1847 Boolean algebra; modern reading of syllogism
Gottlob Frege 1879 Predicate logic
John Venn 1881 Venn diagrams (Topic 24)
Irving Copi & Carl Cohen mid-20th c. Standard introduction; 3 uses of language
Stephen Toulmin 1958 The Uses of Argument; Claim-Data-Warrant model

22.13 Practice Questions

Q 01 Components Easy

An argument in logic consists of:

  • APremises and a conclusion
  • BSpeaker and audience
  • CEncoder, channel, decoder
  • DQuarrelling and rebuttal
View solution
Correct Option: A
Argument = premises + conclusion + inference.
Q 02 Validity Medium

A SOUND argument is one that is:

  • AValid AND has all true premises
  • BValid but with at least one false premise
  • CPersuasive even if invalid
  • DInductive with strong evidence
View solution
Correct Option: A
Sound = valid + all true premises.
Q 03 Proposition Easy

"No politicians are honest" is which type of categorical proposition?

  • AA
  • BE
  • CI
  • DO
View solution
Correct Option: B
"No S are P" = Universal Negative = E.
Q 04 Proposition Medium

"Some children are dancers" is which type?

  • AA
  • BE
  • CI
  • DO
View solution
Correct Option: C
"Some S are P" = Particular Affirmative = I.
Q 05 Distribution Hard

In an "A" proposition (All S are P), which term(s) is/are distributed?

  • AOnly S
  • BOnly P
  • CBoth S and P
  • DNeither
View solution
Correct Option: A
A distributes S only. (E distributes both; O distributes P only; I distributes neither.)
Q 06 Square Medium

In the classical square of opposition, A and O are:

  • AContraries
  • BSubcontraries
  • CContradictories
  • DSubalterns
View solution
Correct Option: C
Contradictories — A↔O, E↔I. Cannot both be true, cannot both be false.
Q 07 Square Hard

If "All S are P" (A) is TRUE, then which of the following must be TRUE?

  • ANo S are P (E)
  • BSome S are P (I)
  • CSome S are not P (O)
  • DBoth A and B
View solution
Correct Option: B
By subalternation A → I. If "all" is true, "some" must be true. (E is false by contraries; O is false by contradictories.)
Q 08 Syllogism Terms Medium

In a categorical syllogism, the MIDDLE term:

  • AAppears only in the conclusion
  • BAppears in both premises but not in the conclusion
  • CAppears in only one premise
  • DAppears in all three statements
View solution
Correct Option: B
Middle term M bridges premises but is absent from the conclusion.
Q 09 Figure Hard

When the middle term is the SUBJECT of the major premise AND the PREDICATE of the minor premise, the syllogism is in:

  • A1st Figure
  • B2nd Figure
  • C3rd Figure
  • D4th Figure
View solution
Correct Option: A
M–P (subject of major) and S–M (predicate of minor) = Figure 1.
Q 10 Mood Hard

The valid Figure-1 mood AAA is known by the medieval name:

  • ABarbara
  • BCelarent
  • CDarii
  • DFerio
View solution
Correct Option: A
Barbara (AAA-1). Celarent = EAE-1; Darii = AII-1; Ferio = EIO-1.
Q 11 Rule Medium

"No valid conclusion can be drawn from two negative premises." This is:

  • ARule of distribution of middle term
  • BRule of exclusive premises
  • CRule of existential import
  • DRule of mood
View solution
Correct Option: B
Rule 4 = Exclusive Premises. Violating it = fallacy of exclusive premises.
Q 12 Fallacy Medium

"All artists are creative. Some students are creative. Therefore some students are artists." This commits the fallacy of:

  • AFour Terms
  • BUndistributed Middle
  • CIllicit Major
  • DExclusive Premises
View solution
Correct Option: B
Middle term "creative" is the predicate of both A (undistributed) and I (undistributed). Undistributed Middle.
Q 13 Language Use Medium

"Please close the door behind you." This sentence is an example of which use of language?

  • AInformative
  • BExpressive
  • CDirective
  • DCeremonial
View solution
Correct Option: C
A command/request aimed at action = Directive.
Q 14 Connot-Denot Hard

"The greater the connotation of a term, the smaller its denotation." This relation is associated with:

  • AAristotle
  • BJohn Stuart Mill
  • CGeorge Boole
  • DKarl Popper
View solution
Correct Option: B
J.S. Mill, A System of Logic (1843) — formalised the connotation/denotation distinction.
Q 15 Fallacy Hard

"Have you stopped cheating in exams?" — this is the fallacy of:

  • AComplex Question
  • BFalse Cause
  • CBegging the Question
  • DHasty Generalisation
View solution
Correct Option: A
Loaded question that presupposes the answerer cheats = Complex Question.
Q 16 Fallacy Hard

"Each member of this team is excellent. Therefore the team is excellent." This is the fallacy of:

  • ADivision
  • BComposition
  • CEquivocation
  • DAmphiboly
View solution
Correct Option: B
Composition — what holds of parts assumed to hold of the whole.
Q 17 Latin Fallacy Hard

"Argumentum ad populum" is the appeal to:

  • AAuthority
  • BPopularity
  • CPity
  • DForce
View solution
Correct Option: B
Ad populum = popularity ("everyone believes it"). Ad verecundiam = authority; ad misericordiam = pity; ad baculum = force.
Q 18 Square Medium

A and E propositions stand in the relation of:

  • AContradictories
  • BContraries
  • CSubcontraries
  • DSubalterns
View solution
Correct Option: B
A and E: cannot both be true, but CAN both be false = Contraries.
Q 19 Mnemonic Hard

In the medieval mnemonic "BARBARA, CELARENT, DARII, FERIO" for Figure-1 valid moods, the letters AEIO encoded are:

  • AThe names of the four philosophers
  • BThe three vowels of each word encoding the mood AEI/O
  • CThe four classical elements
  • DThe four Aristotelian causes
View solution
Correct Option: B
BArbArA = AAA; CElArEnt = EAE; DArII = AII; FErIO = EIO.
Q 20 Match Hard

Match each fallacy with its type:

(i) Equivocation (a) Relevance
(ii) Ad hominem (b) Presumption
(iii) Post hoc (c) Ambiguity
(iv) Undistributed Middle (d) Formal
  • A(i)-c, (ii)-a, (iii)-b, (iv)-d
  • B(i)-a, (ii)-b, (iii)-c, (iv)-d
  • C(i)-d, (ii)-c, (iii)-a, (iv)-b
  • D(i)-b, (ii)-d, (iii)-c, (iv)-a
View solution
Correct Option: A
Equivocation → ambiguity; Ad hominem → relevance; Post hoc → presumption (false cause); Undistributed Middle → formal (syllogistic).

22.14 Quick Recall

ImportantQuick recall
  • Argument = premises + conclusion + inference.
  • Premise indicators: since, because, for, given that. Conclusion indicators: therefore, hence, so, thus.
  • Deductive (valid/invalid; sound/unsound) vs Inductive (strong/weak; cogent/uncogent).
  • Sound = valid + true premises.
  • 4 categorical propositions: A (All S are P, U+) · E (No S are P, U−) · I (Some S are P, P+) · O (Some S are not P, P−). Codes from Latin affirmo/nego.
  • Distribution: A → S only · E → both · I → neither · O → P only.
  • Square of opposition: Contradictories (A↔︎O, E↔︎I) · Contraries (A↔︎E) · Subcontraries (I↔︎O) · Subalterns (A→I, E→O).
  • Syllogism = 2 premises + 1 conclusion; 3 terms — Major (P), Minor (S), Middle (M). Middle appears in both premises, never in conclusion.
  • Mood = sequence of A/E/I/O types (64 possibilities); Figure = position of middle term (4 figures).
  • 4 figures: F1 M-P / S-M · F2 P-M / S-M · F3 M-P / M-S · F4 P-M / M-S.
  • 24 valid moods. F1 mnemonic: Barbara (AAA), Celarent (EAE), Darii (AII), Ferio (EIO).
  • 6 rules of validity: 3 terms only · M distributed at least once · no illicit distribution · no two negatives · negative premise → negative conclusion · no two particulars.
  • 6 formal fallacies: Four Terms · Undistributed Middle · Illicit Major/Minor · Exclusive Premises · Affirmative-from-Negative · Existential.
  • Informal fallacies — 3 groups:
    • Relevance: ad hominem, ad populum, ad verecundiam (authority), ad baculum (force), ad misericordiam (pity), ad ignorantiam, ignoratio elenchi / red herring, tu quoque, genetic, straw man.
    • Presumption: begging the question (petitio principii), complex question, false cause (post hoc), false analogy, hasty generalisation, slippery slope, false dilemma.
    • Ambiguity: equivocation, amphiboly, composition, division, accent.
  • 3 uses of language (Copi & Cohen): Informative · Expressive · Directive. (+ ceremonial, performative, phatic.)
  • Connotation (intension) vs Denotation (extension) — J.S. Mill, 1843. Inverse rule: ↑ connotation → ↓ denotation.
  • Toulmin (1958): Claim · Data · Warrant model.