II · Inferential character
Modes
modus · ‘the manner in which a thing is done’
Inference comes in three modes — distinguished not by topic or by formality, but by the character of the move from premises to conclusion. Each preserves something different, each fails in its own way.
At a glance
The three side by side
I
Deductive
From general to particular, with necessity.
- Preserves
- Truth
- Fails when
- Formal invalidity (e.g. affirming the consequent), or a false premise carried as if it were certain.
II
Inductive
From particulars to a general claim, ampliatively.
- Preserves
- Probability
- Fails when
- Hasty generalisation; ignoring base rates; applying a regularity to a domain where it does not project.
III
Abductive
From a phenomenon to its best explanation.
- Preserves
- Explanatory adequacy (provisionally).
- Fails when
- Settling on a single explanation prematurely; treating 'best so far' as 'true'; cherry-picking the explanation that fits a prior preference.
Mode I
Deductive
From general to particular, with necessity.
Deductive inference is the inferential mode in which the truth of the premises guarantees the truth of the conclusion. If the premises are true and the inference is valid, the conclusion cannot be false. This is what is meant by saying that deduction is truth-preserving.
Deduction is non-ampliative: the conclusion contains no information that was not already implicit in the premises. The work of a deductive inference is to make explicit what the premises already commit one to. This is its strength — it is reliable — and its limit: deduction cannot, by itself, produce new content from observation of the world.
All of formal logic — propositional, predicate, modal — is in the first instance an apparatus for codifying deductive inference.
Example · A modus ponens
- P₁.
- If it is raining, the pavement is wet.
- P₂.
- It is raining.
- ∴
- The pavement is wet.
Origin · Aristotle, Prior Analytics, c. 350 BC.
Mode II
Inductive
From particulars to a general claim, ampliatively.
Inductive inference moves from observed cases to claims about unobserved cases or to a general rule. The premises support the conclusion but do not entail it: the conclusion is rendered probable, not necessary. Induction is ampliative — the conclusion goes beyond the content of the premises.
The standard form: P₁ has property Q; P₂ has property Q; …; Pₙ has property Q; therefore the next P will have property Q, or all P have property Q. The inferential warrant is some principle of uniformity or projectibility — a premise about the regularity of nature that is rarely stated.
The formal articulation of induction is the probability calculus. A Bayesian update treats the premises as evidence and quantifies its impact on a hypothesis through Bayes' rule: P(H | E) = P(E | H) · P(H) / P(E). This is induction made numerical; the inductive move and the Bayesian update are not two things but the same one, formalised.
Hume's celebrated problem — that the principle of uniformity is itself only knowable inductively, so induction cannot be non-circularly justified — applies to the formal and informal cases alike. The accompanying paper Induction Reduced argues that the inferential structure of induction is in fact deductive, once the projectibility premise is made explicit; on that view the deductive/inductive distinction is one of presentation, not of mechanism.
Example · The sunrise inference
- P.
- The sun has risen on each of the past ten thousand mornings.
- ∴
- The sun will rise tomorrow. (Probably; with high confidence.)
Origin · Aristotle's epagōgē (c. 350 BC); systematic treatments from Bacon's Novum Organum (1620) onward.
For a longer argument that inductive inference is, structurally, deductive inference with a suppressed premise, see Induction Reduced.
Mode III
Abductive
From a phenomenon to its best explanation.
Abductive inference works backward from a phenomenon to a hypothesis that, if true, would explain it. The hypothesis is not entailed by the phenomenon (so abduction is not deductive); nor is it a simple generalisation of cases (so abduction is not inductive in the textbook sense). It is a selection: among the hypotheses that would, if true, account for what is observed, abduction picks the one with the most explanatory virtue.
Abductive inference is the mode by which medical diagnoses, scientific hypotheses, criminal investigations, and most of everyday explanation actually proceed. The detective who concludes that the butler did it is not deducing this from the evidence (the evidence is consistent with other suspects); he is choosing the hypothesis that best accounts for what he has seen.
Because abduction selects from a space of hypotheses, its quality is bounded by the richness of that space. The signature failure is to settle on the best-explanation-so-far without having canvassed the relevant alternatives — Peirce's warning against premature closure.
Example · A simple diagnosis
- P₁.
- The patient has a fever and a sore throat.
- P₂.
- Streptococcal infection would account for both symptoms; few other common causes do.
- ∴
- The patient probably has a streptococcal infection.
Origin · Charles Sanders Peirce, late nineteenth century; refined as 'inference to the best explanation' by Gilbert Harman (1965).
How they relate
Three modes, one mechanism?
The pedagogical convenience of three modes obscures a deeper question: are these really three mechanisms of inference, or one mechanism presented three ways? The reducibility thesis — defended in the paper Induction Reduced — argues for the latter. Inductive inference, on that view, is enthymematic deduction: make the suppressed premise of projectibility explicit and the inference becomes deductively valid. Abductive inference admits a parallel treatment: make the comparative-explanatory premise explicit and the inference, too, is a deduction.
Whether to accept this collapse, or to maintain the three modes as genuinely distinct, is a substantive question. The modes do at least describe three recognisably different shapes that ordinary reasoning takes; whether they describe three different inferential acts is what the unity-of-inference thesis denies.