# Abductive Logic

**Origin.** Peirce introduced abduction (late 19th c.). Formalized in AI by Pople, Poole (1970s-80s). Inference to best explanation. Given observation and theory, find hypothesis explaining it. Foundation for diagnosis, planning, theory formation.

**Models.** Reasoning from effects to causes. Deduction: theory + cause → effect. Induction: observations → theory. Abduction: theory + effect → cause. "If H explains E, and E is observed, maybe H." Hypothesis selection problem.

**Formalism.**

*Abduction problem:*
Given:
- Background theory T
- Observation O
- Hypothesis space H

Find H ⊆ H such that:
1. T ∪ H ⊨ O (H explains O)
2. T ∪ H ⊬ ⊥ (H is consistent with T)
3. H is "best" (minimal, most probable, etc.)

*Logic programming abduction:*
T: logic program (rules)
H: abducible predicates (assumable)
O: goal to explain

Find: set of ground abducibles making O true.

*Preference criteria:*
- Subset minimality: no smaller H works
- Cardinality minimality: fewest hypotheses
- Probability: most probable H
- Simplicity: Occam's razor

*Integrity constraints:*
IC: rules that must not be violated.
Abductive solution must satisfy ICs.

*Example:*
T: wet_grass ← rain. wet_grass ← sprinkler.
O: wet_grass.
H = {rain} or H = {sprinkler} — both explain O.

**Symbols.**

| Symbol | Unicode | Name | Meaning |
|--------|---------|------|---------|
| ⊨ | U+22A8 | Entails | Logical consequence |
| ← | U+2190 | Rule | If body then head |
| H | — | Hypothesis | Explanation |
| O | — | Observation | Given data |
| T | — | Theory | Background knowledge |
| ⊥ | U+22A5 | Inconsistent | Contradiction |

**Metatheory.** Abduction is NP-complete in general. Multiple solutions typical (hypothesis selection). Non-monotonic: new observations may invalidate explanations. Probabilistic abduction: Bayesian reasoning. Complexity depends on hypothesis space structure.

**Applies to.** Medical diagnosis. Fault diagnosis. Scientific discovery. Natural language understanding. Plan recognition. Legal reasoning (evidence). Debugging programs.

**Limitations.** Multiple explanations problem. Computational hardness. Preference criteria not always clear. Ignoring unlikely hypotheses. Integration with learning. Iterated abduction (revising hypotheses). Open-world assumption issues.

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