Kinds of Simple and Compound Propositions
Chapter Thirty-Four
Syllabus topic 2.6, "Kinds of simple and compound propositions"
Pages 163 to 167 of 334
In one line
Simple propositions are divided by what they assert of what, and compound propositions by the connective that joins their components.
In the wording a student can write in an examination: a simple proposition contains no component proposition, and is either subject-predicate or relational in form. A compound proposition contains at least one component, and is classified by its connective as negative, conjunctive, disjunctive, implicative or equivalent.
The notation, once and for all
| Name | Symbol | Read as | Example |
|---|---|---|---|
| Negation | ~p | not p | ~p |
| Conjunction | p • q | p and q | p • q |
| Disjunction | p ∨ q | p or q | p ∨ q |
| Implication | p ⊃ q | if p then q | p ⊃ q |
| Equivalence | p ≡ q | p if and only if q | p ≡ q |
Simple propositions are written as single letters, p, q, r. Brackets group, exactly as in arithmetic: ~(p • q) is the denial of the conjunction, while ~p • q is the conjunction of a denial with q, and the two are different propositions.
Kinds of simple proposition
Subject-predicate propositions, also called attributive. A quality is asserted of a thing. "Ravi is a minor." "The document is registered." This is the only form traditional logic recognised.
Relational propositions. A relation is asserted to hold between two or more things. "Ravi owes Meena two lakh rupees." "The mortgagee is entitled to redeem from the mortgagor." "A paid B on behalf of C." Modern logic treats the relation as the predicate and the things related as its arguments, of which there may be two, three or more. This is the answer to failure one of sequence 320.
Class-membership and class-inclusion propositions. "Ravi is a lawyer" puts an individual into a class. "All lawyers are graduates" puts one class inside another. Traditional logic ran the two together, because both look like "S is P", and the difference matters: the first has an individual as its subject and the second a class.
The two terms of a relation have names, and MU asks for them. The term from which the relation proceeds is the referent; the term to which it proceeds is the relatum. In "Ravi owes Meena two lakh rupees", Ravi is the referent and Meena the relatum. In "the mortgagor redeemed from the mortgagee", the mortgagor is the referent. Inference by converse relation, at sequence 530, is the operation that exchanges the two.
A caution about relational propositions. The order of the terms is part of the proposition, which is to say that referent and relatum cannot be swapped without changing what is asserted. "Ravi owes Meena" and "Meena owes Ravi" are different propositions, and no rearrangement of one gives the other. This is obvious and it is exactly what the traditional form could not record, since both would reduce to a subject with a predicate attached.
Kinds of Simple and Compound Propositions
Kinds of compound proposition
Negative propositions, ~p. The denial of a proposition is itself a compound in the modern scheme, because it has a component. "It is not the case that the notice was served." ~p is true exactly when p is false.
Conjunctive propositions, p • q. Two or more components asserted together, each of them asserted. "The notice was served and the rent was unpaid." Both are claimed, and the whole is true only if both are true.
Disjunctive propositions, p ∨ q. At least one component is asserted, without saying which. "Either the notice was defective or the tenant waived the defect." Read inclusively, as sequence 290 established.
Implicative propositions, p ⊃ q. One component is asserted to follow from another, and neither is asserted by itself. "If the notice was served, the tenancy ended."
Equivalent propositions, p ≡ q. Each component is asserted to follow from the other. "A person is a major if and only if he has completed eighteen years." Statutory definitions are frequently equivalences, and reading them as mere implications is a common error: "means" in a definition clause is usually an equivalence, while "includes" is not.
Which components are asserted
This is the point of the classification and it is worth a table of its own, because it decides what an opponent has to disprove.
| Compound | Components asserted | To make the whole false, disprove |
|---|---|---|
| ~p | The denial of p | Establish p |
| p • q | Both | Either one |
| p ∨ q | Neither individually | Both |
| p ⊃ q | Neither | Establish p and disprove q |
| p ≡ q | Neither | Show the two differ in truth value |
Read that table as a litigator. A pleading that asserts a conjunction has given the other side two targets and needs both to survive. A pleading that asserts a disjunction has given two targets and needs only one to survive. That asymmetry is the whole reason alternative pleading exists, and it is a consequence of the truth conditions of the connectives rather than of any rule of procedure.
A worked example: symbolising a section
Section 14 of the Indian Contract Act 1872 provides that consent is said to be free when it is not caused by coercion as defined in section 15, or undue influence as defined in section 16, or fraud as defined in section 17, or misrepresentation as defined in section 18, or mistake, subject to the provisions of sections 20, 21 and 22.
Assign letters.
f: the consent is free
c: the consent is caused by coercion
u: the consent is caused by undue influence
d: the consent is caused by fraud
m: the consent is caused by misrepresentation
k: the consent is caused by mistake within sections 20, 21 and 22
Kinds of Simple and Compound Propositions
Symbolise. The section states an equivalence, because "is said to be free when" in a definitional provision fixes both directions:
f ≡ ~(c ∨ u ∨ d ∨ m ∨ k)
Read what the symbolism shows. Consent is free exactly when none of the five vitiating factors is present. The section is a negated disjunction, and a negated disjunction is true only when every alternative is false. So a party alleging that consent was not free needs to establish only one of the five, while a party asserting that it was free must be able to meet all of them.
And a second thing, which the words hide. Because the five are joined by "or" and then denied, the section is equivalent to a conjunction of five denials:
f ≡ (~c • ~u • ~d • ~m • ~k)
That transformation is one of De Morgan's rules and it is worked at sequence 350. The practical point is that the section can be read either as "not any of these" or as "none of these, and none of these, and none of these", and the second reading is the one that tells a drafter what has to be pleaded.
Distinctions that carry marks
| Kind of simple proposition | Form | Example |
|---|---|---|
| Subject-predicate | A quality of a thing | The document is registered |
| Relational | A relation among two or more things | Ravi owes Meena two lakh rupees |
| Class-membership | An individual in a class | Ravi is a lawyer |
| Class-inclusion | One class inside another | All lawyers are graduates |
| Kind of compound | Symbol | True when |
|---|---|---|
| Negative | ~p | p is false |
| Conjunctive | p • q | both are true |
| Disjunctive | p ∨ q | at least one is true |
| Implicative | p ⊃ q | not the case that p is true and q false |
| Equivalent | p ≡ q | both have the same truth value |
What this does not mean
A negative proposition is not the same as a negative categorical. "No S is P" is an E proposition and is simple in the modern scheme, because it contains no component; "it is not the case that all S is P" is compound, because it denies a whole proposition. The English is close and the analysis is not.
"Or" in a statute is not automatically exclusive. Sequence 290 dealt with this, and it is worth repeating because it changes what a party has to prove.
Brackets are not decoration. ~(p • q) and ~p • q are different propositions with different truth conditions, and dropping the brackets changes the meaning rather than the appearance.
Kinds of Simple and Compound Propositions
Quick revision
Notation: ~ negation, • conjunction, ∨ disjunction, ⊃ implication, ≡ equivalence. Simple propositions are single letters. Brackets group.
Simple propositions: subject-predicate, relational, class-membership, class-inclusion. In a relational proposition the term from which the relation proceeds is the referent and the term to which it proceeds is the relatum, and the order is part of what is asserted.
Compound propositions: negative, conjunctive, disjunctive, implicative, equivalent.
Which components are asserted: both in a conjunction, neither in a disjunction, an implication or an equivalence.
Litigation consequence: a conjunction gives the opponent two targets and needs both; a disjunction gives two targets and needs one.
Definitions using "means" are usually equivalences; those using "includes" are not.
Test yourself
1. Name the kinds of simple proposition with an example of each.
Subject-predicate propositions, asserting a quality of a thing, as in "the document is registered". Relational propositions, asserting a relation among two or more things, as in "Ravi owes Meena two lakh rupees". Class-membership propositions, placing an individual in a class, as in "Ravi is a lawyer". And class-inclusion propositions, placing one class within another, as in "all lawyers are graduates".
2. Name the kinds of compound proposition with their symbols.
Negative, ~p; conjunctive, p • q; disjunctive, p ∨ q; implicative, p ⊃ q; and equivalent, p ≡ q. Each is classified by its connective, and the connective is what determines the truth of the whole from the truth of the components.
3. Which components of a compound proposition are asserted?
Both components of a conjunction are asserted, so both must be true for the whole to be true. Neither component of a disjunction is asserted individually; what is asserted is that at least one holds. Neither component of an implication is asserted; only the connection is. Neither component of an equivalence is asserted; only that they stand or fall together.
4. Why does the difference between a conjunction and a disjunction matter to a pleader?
Because it decides how many of the assertions must survive. A case pleaded as a conjunction fails if the other side disproves any one of its parts, so every part is a target and all must hold. A case pleaded in the alternative is a disjunction and survives if any one part holds, so the other side must defeat every alternative. Alternative pleading is a direct application of the truth conditions of the connectives.
5. Symbolise "Consent is free when it is not caused by coercion, undue influence, fraud, misrepresentation or mistake", and say what a party must prove.
Writing f for free consent and c, u, d, m and k for the five factors, the section is f ≡ ~(c ∨ u ∨ d ∨ m ∨ k). A party alleging that consent was not free needs to establish any one of the five, since a disjunction is true if any alternative is. A party asserting that consent was free must be in a position to meet all five, since a negated disjunction is true only when every alternative is false.
Kinds of Simple and Compound Propositions
7. In a relational proposition, what are the referent and the relatum?
The referent is the term from which the relation proceeds and the relatum is the term to which it proceeds. In "Ravi owes Meena two lakh rupees", Ravi is the referent and Meena the relatum; in "the tenant holds under the landlord", the tenant is the referent. The two cannot be exchanged without changing the proposition, and inference by converse relation is the operation that exchanges them while replacing the relation by its converse.
6. What is the difference between ~(p • q) and ~p • q?
The first denies the conjunction, and is true whenever at least one of p and q is false. The second asserts the conjunction of ~p with q, and is true only when p is false and q is true. They differ in truth value in two of the four possible cases, so the brackets are not a matter of style: removing them states a different proposition.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself, or the past papers, for the same subject.