Inference and Implication

The four kinds of implication

MU asks for implication as a basic concept, and the standard treatment distinguishes four senses in which one proposition is said to imply another. Only the last is the one modern logic works with, and knowing why is worth marks.

Logical implication. "This is a square" implies "this has four sides". The connection holds because of the meanings and the logic alone, and denying the second while asserting the first is a contradiction.

Definitional implication. "He is a bachelor" implies "he is unmarried". The connection comes from the definition of a word rather than from the structure of the propositions.

Causal implication. "The glass was heated to a thousand degrees" implies "the glass melted". The connection is a matter of physical fact discovered by observation, not of meaning at all.

Decisional implication. "This is a public nuisance" implies "an ordinary member of the public cannot sue for it without leave". The connection is neither logical nor definitional nor causal; somebody has decided that it shall hold. Legal implications are overwhelmingly of this kind, which is why they can be altered by legislation and the others cannot.

Material implication

Modern logic uses a fifth and deliberately thin notion, the one written with the horseshoe. p ⊃ q is defined to be false in exactly one case, where p is true and q is false, and true in the other three. Nothing else is required: p and q need have no connection of meaning, cause or decision whatsoever.

This produces results that look wrong on first meeting, and they are called the paradoxes of material implication. A false proposition materially implies anything: "the Contract Act was passed in 1900 ⊃ the moon is made of cheese" is true, because its antecedent is false. A true proposition is materially implied by anything.

The answer is that material implication is not an analysis of the English "if". It is a truth function, adopted because it is the weakest relation strong enough to make deduction work, and because a truth-functional connective is one a truth table can handle. Ordinary conditionals carry more than material implication does, and Module II's chapter on truth tables returns to this.

The legal sense of "implication", which is different

A law student meets the word "implication" every day and almost never in the logician's sense, so the two must be separated deliberately.

An implied term in a contract is a term the parties did not write and which the law reads in. An implied repeal is a repeal Parliament did not enact in words and which a court finds because two Acts cannot stand together. A necessary implication in a statute is a meaning not stated but forced by the words used.

In all three the word describes something that has been read in by a court because it was not expressed. That is close to the logician's decisional implication and quite unlike logical implication, and a student who explains the legal doctrine of implied terms in an answer about the basic concepts of logic has answered a different question.

A worked example

An inspector finds a shop closed at ten in the morning on a working day, the shutters locked, and the licence displayed in the name of the accused. He records: "the accused was absent from the shop, therefore the shop was being run in contravention of the licence condition requiring the licensee to be present during business hours."

Find the inference. The act performed by the inspector: from the closed shutters and the displayed licence he passed to the conclusion that the licensee was absent, and from that to a contravention.

Find the implication, if any. Does "the shutters were down at ten" imply "the licensee was absent"? No. It is consistent with the licensee being inside. The inspector performed an inference for which no implication existed, which is precisely what an invalid argument is.

Now the second step. Does "the licensee was absent during business hours" imply "the licence condition was contravened"? Yes, and it is a decisional implication: it holds because the licensing authority laid down the condition, and it could be repealed tomorrow. A student who called it a logical implication would have said something false about the source of the connection.

What the analysis gives the defence. Two distinct answers, and they are answers of different kinds. Against the first step, evidence: the licensee was inside doing stock. Against the second, argument: the condition on its true construction requires presence only when the shop is open for business. Only separating inference from implication makes it obvious that these are two answers and not one.

Distinctions that carry marks

What this does not mean

To imply is not to hint. In ordinary English "he implied that the witness was lying" means he suggested it without saying so. In logic implication is a relation between propositions and nobody performs it. The word for hinting is insinuating.

Inferring is not implying. The listener infers; the speaker implies, in the loose sense. In the strict sense the speaker does neither, because implication holds between propositions.

Material implication is not the English "if". It is a deliberately weakened substitute that a truth table can handle, and its oddities are a known price rather than a discovery about conditionals.

Quick revision

Inference: the act of passing from premises to a conclusion. Personal, dated, may be performed badly.

Implication: the relation in virtue of which the conclusion follows. Impersonal, timeless, either holds or does not.

The link: an inference is correct when there is a corresponding implication; validity is the name of an argument that exhibits one.

Four kinds: logical, definitional, causal, decisional. Legal implications are decisional, which is why statute can change them.

Material implication: p ⊃ q, false only when p is true and q false. Its oddities are the paradoxes of material implication.

The legal word "implication", as in implied terms and implied repeal, means something read in because it was not expressed. Do not answer with it.

Test yourself

1. Distinguish inference from implication.

Inference is an act performed by a person at a particular time, by which they pass from one or more propositions to another; it can be done well or badly and it is studied by psychology as well as by logic. Implication is a relation between propositions, holding independently of whether anybody has ever thought of them, and it is timeless and impersonal. An inference is correct exactly when there is an implication for it to follow.

2. How are implication and validity connected?

Validity is the property an argument has when its premises imply its conclusion. So implication is the relation, validity is the name for an argument that exhibits it, and inference is the act of drawing the conclusion by relying on it. To say that an argument is valid and to say that its premises imply its conclusion are two ways of saying one thing.

3. Name the four kinds of implication with an example of each.

Logical: "this is a square" implies "this has four sides", the connection coming from logic and meaning. Definitional: "he is a bachelor" implies "he is unmarried", the connection coming from the definition of the word. Causal: "the water was cooled below zero" implies "the water froze", the connection being a discovered fact about the world. Decisional: "this agreement was made by a minor" implies "this agreement is void", the connection existing because the law has so decided.

4. What is material implication and what are its paradoxes?

Material implication is the truth-functional connective written p ⊃ q, defined to be false only when p is true and q false and true in every other case, so that no connection of meaning between p and q is needed. Its paradoxes are that a false proposition materially implies every proposition, and that a true proposition is materially implied by every proposition, both of which are absurd if the horseshoe is read as the English "if".

5. Why does modern logic use material implication despite the paradoxes?

Because it is truth functional, so the truth value of the compound is fixed entirely by the truth values of its parts, and only such a connective can be handled by a truth table. It is also the weakest relation that still supports valid deduction: any stronger reading would import claims about meaning or cause that logic has no way to test. The paradoxes are accepted as the price of a workable calculus, not defended as an account of ordinary conditionals.

6. A statute is said to repeal an earlier one by necessary implication. Is this implication in the logician's sense?

Not in the sense of logical implication. It is closest to decisional implication: a court decides that the later Act cannot stand together with the earlier one and reads the repeal in although Parliament did not enact it in words. The connection exists because an authority has held that it does, and it could be displaced by an express saving in a later statute, which no logical implication could be.