Conditional Propositions: Hypothetical and Disjunctive

When would the two fallacies become valid? Only if the conditional were replaced by a biconditional, "p if and only if q", which asserts that each is necessary for the other. English "if" is very often used where "if and only if" is meant, which is why the fallacies feel right.

Disjunctive propositions

"Either the notice was defective or the tenant waived the defect."

Each part is an alternative. The proposition asserts that at least one of them is true, and it does not say which.

The inclusive and exclusive senses. In its inclusive sense, "or" allows both alternatives to be true; in its exclusive sense, it means one but not both. Logic takes "or" as inclusive unless something signals otherwise, and modern notation writes the inclusive sense as p ∨ q.

The same pair under MU's own names: weak and strong disjuncts. A weak disjunction is the inclusive one, "p or q or both", and it is weak because it claims less: it is satisfied by any of three situations. A strong disjunction is the exclusive one, "p or q but not both", and it is strong because it claims more, excluding the case where both hold. "The applicant shall be a citizen of India or a person of Indian origin" is strong, the two being mutually exclusive; "a person who obstructs or resists" is weak, since doing both is certainly not outside the section. Weak is the default in logic and in the reading of a penal provision, because it is the wider.

The two senses give different inferences and the difference is practical.

From an inclusive disjunction, denying one alternative establishes the other. Either p or q; not p; therefore q. This is valid on both readings.

From an inclusive disjunction, affirming one alternative establishes nothing about the other. Either p or q; p; therefore not q, is invalid, because both may be true.

On the exclusive reading that last inference is valid, which is why the reading matters and why careful drafting says "or both" or "but not both" instead of leaving it to be guessed.

In statutes. "A person who obstructs or resists" is inclusive: doing both is worse and certainly not outside the section. "The applicant shall be either a citizen of India or a person of Indian origin" is exclusive in effect, since the categories do not overlap. The safest reading of a penal provision is the inclusive one, because it is the wider.

A worked example

Section 91 of the Code of Civil Procedure 1908 provides that in the case of a public nuisance or other wrongful act affecting, or likely to affect, the public, a suit for a declaration and injunction or for such other relief as may be appropriate may be instituted by the Advocate-General, or with the leave of the Court by two or more persons even though no special damage has been caused to them.

Find the conditional. The provision is a hypothetical: if there is a public nuisance or other wrongful act affecting or likely to affect the public, then such a suit may be instituted in the stated ways. The antecedent is the state of affairs and the consequent is the permission.

Find the disjunctions, and there are three. "A public nuisance or other wrongful act": inclusive, since an act may be both. "Affecting or likely to affect": inclusive, since an act already affecting the public is certainly likely to. "By the Advocate-General, or with the leave of the Court by two or more persons": inclusive, since nothing prevents both routes being taken in different proceedings.

Now test an inference. "The Advocate-General has not sued. Therefore two or more persons may sue with leave." Is this valid? It is not, and the reason is instructive: the disjunction is about who may sue, not about who has sued. Denying that one permitted person has acted says nothing at all, because the proposition never asserted that anybody had. Confusing a permissive disjunction with an assertive one is a real error in reading statutes.

And test a fallacy. "Two or more persons have obtained leave to sue. Therefore there is a public nuisance." That is affirming the consequent: leave may have been granted wrongly, or for an "other wrongful act" which is not a nuisance. The section supplies no such inference.

Distinctions that carry marks

What this does not mean

A conditional does not assert its antecedent. A section beginning "where a person fails to appear" does not say that anybody has failed to appear.

"If" in English often means "if and only if". Logic does not read it that way, and the difference is what makes denying the antecedent feel valid when it is not.

A disjunction is not a choice offered. "Either the notice was defective or the tenant waived the defect" does not invite anyone to pick; it asserts that at least one is so.

Quick revision

Hypothetical: if p then q. Antecedent, consequent. Neither asserted.

Valid: affirming the antecedent, and denying the consequent.

Fallacies: denying the antecedent, and affirming the consequent. Both would be valid only for a biconditional.

Affirming the consequent is the standard error in circumstantial reasoning, and Sarda's fourth condition exists to prevent it.

Disjunctive: either p or q. Alternatives. Asserts at least one.

Inclusive by default, written p ∨ q. Denying one alternative establishes the other; affirming one establishes nothing. MU calls the two WEAK and STRONG disjuncts: weak is the inclusive one, claiming less, and strong is the exclusive one, claiming more.

In statutes: "where" and "provided that" introduce antecedents; "shall" often marks a consequent; "or" is normally read inclusively.

Test yourself

1. Define a hypothetical proposition and name its parts.

A hypothetical proposition asserts that one proposition follows from another, taking the form "if p then q". The part introduced by "if" is the antecedent and the part that follows from it is the consequent. It asserts neither part on its own: it does not say that p is true and does not say that q is true, but only that q would hold if p did.

2. State the two valid inferences from a hypothetical proposition and the two fallacies.

Valid: affirming the antecedent, from "if p then q" and p to q; and denying the consequent, from "if p then q" and not q to not p. Fallacious: denying the antecedent, from "if p then q" and not p to not q; and affirming the consequent, from "if p then q" and q to p. Both fallacies assume that p is necessary for q when the conditional says only that it is sufficient.

3. Demonstrate the fallacy of affirming the consequent with a legal example.

"If the accused was at the scene, his fingerprints would be on the door. His fingerprints are on the door. Therefore he was at the scene." The conclusion does not follow, because the prints may have been left at another time or in another way. This is the characteristic error in reasoning from circumstantial evidence, and it is what the fourth of the five golden principles guards against by requiring every other hypothesis to be excluded.

4. Distinguish the inclusive from the exclusive sense of "or", and give the inference each supports.

The inclusive sense allows both alternatives to be true and is written p ∨ q; the exclusive sense means one but not the other. On both readings, denying one alternative establishes the other. On the inclusive reading, affirming one alternative establishes nothing about the other, since both may hold; on the exclusive reading, affirming one establishes that the other is false. Logic takes "or" as inclusive unless something signals otherwise.

5. What words introduce an antecedent in legislative drafting?

"Where" and "provided that" far more often than "if", together with "in case", "on condition that" and "in the event that". The consequent is frequently marked by "shall". Recognising these is necessary because a section written as "where a person does X, he shall be liable to Y" is a hypothetical proposition, and reading it as an assertion that somebody has done X is a plain misreading of the statute.

7. What is meant by weak and strong disjuncts?

They are the two senses of "or". A weak disjunct is the inclusive one, asserting p or q or both, and it is called weak because it claims less, being satisfied in three of the four possible situations. A strong disjunct is the exclusive one, asserting p or q but not both, and it is called strong because it excludes the case in which both hold. Logic takes a disjunction as weak unless something signals otherwise, and a penal provision is read the same way because the weak sense is the wider.

6. Why can a hypothetical proposition not by itself be an argument?

Because an argument asserts its premises and asserts its conclusion, while a hypothetical asserts neither of its parts. It claims only a connection, so nobody has committed to anything being true and nothing has been established. It can be a premise in an argument, and in legal reasoning it usually is: the section supplies the conditional and the evidence supplies the affirmation of the antecedent.