Can a rule apply to compound statements?

Yes, when the pattern matches consistently.

Does one sound rule make a complete proof system?

Not necessarily; completeness concerns the system's overall ability to derive consequences.

AI Glossary / Graphs and symbolic AI / Advanced

Inference rules

Definition

Inference rules are formal patterns for deriving conclusions from premises. In a sound deductive system, applying a rule correctly preserves truth: whenever its premises are true under the chosen semantics, its conclusion is true as well. Repeated rule applications connect initial assumptions to a conclusion through a checkable proof.

What it means

A rule describes a general reasoning pattern, not a particular business fact. Modus ponens, for example, allows Q to be derived from P and “P implies Q.” The letters can stand for many different statements, provided their substitution is consistent.
An implication stored in a knowledge base is a statement within the logic. Modus ponens is a rule for using that statement together with another premise.
Sound reasoning does not establish that the original premises are accurate descriptions of reality. It establishes what follows if those premises hold.

How it works

Identify available premises whose forms match a rule. Substitute formulas consistently into its placeholders, check any side conditions, and record the resulting conclusion together with its supporting steps.
For classical propositional logic, useful patterns include conjunction introduction, which derives P AND Q from P and Q; conjunction elimination, which extracts either part of a conjunction; and modus ponens, which applies an implication to its established antecedent.
Resolution combines clauses containing complementary literals. From P OR Q and NOT P OR R, it derives Q OR R. It removes a selected complementary pair while retaining the other alternatives; it does not generally prove either remaining alternative separately.
A proof procedure chooses how to search for useful applications. The rules define allowable steps, while the search strategy determines which steps are attempted. Soundness of individual rules does not imply that a particular search strategy will find every available proof or finish quickly.

Three practical examples

A verification tool checks that each step in a proof follows an accepted rule.
A knowledge-based assistant derives an eligibility statement from explicit conditions. A trace records both the domain statements and the inference patterns used to combine them.
A satisfiability procedure uses resolution-based reasoning to derive consequences of clauses. The application must preserve the exact literals and logical operators rather than treating the clauses as approximate natural-language suggestions.

Think of it this way

Think of inference rules as permitted moves in a proof. A move is allowed because of its formal pattern, not because the destination looks plausible. Reaching a believable answer through an invalid move still leaves the reasoning unsupported.

Worked example

Consider four invented propositions about a document:
P means its pages are complete. Q means its references are checked. R means it is approved. S means it is ready for publication.
Assume classical propositional logic and these four premises: P; Q; (P AND Q) implies R; and R implies S.
Step 1 uses conjunction introduction on the first two premises to derive P AND Q. Both components are established; the rule does not permit creating the conjunction from just one of them.
Step 2 uses modus ponens on P AND Q and the third premise to derive R. Here the entire formula P AND Q fills the antecedent position of the general rule.
Step 3 uses modus ponens on R and the fourth premise to derive S. The conclusion has a three-step derivation from the stated premises. Each new line identifies a rule and earlier support, making the chain independently checkable.
Now consider a separate argument with only R implies S and S as premises. It tries to conclude R. This is the invalid pattern called affirming the consequent. Set R false and S true: both premises hold, but the proposed conclusion is false. That single counterexample disproves the pattern's general soundness.
For a resolution example, consider the clauses NOT P OR R and P OR Q. Resolving on P produces R OR Q. To understand why this works, consider the two possible values of P. If P is true, the first clause requires R. If P is false, the second clause requires Q. Either way, R OR Q holds when both clauses hold.
However, R alone does not follow. Assign P false, Q true, and R false. Both original clauses and their resolvent R OR Q are true, while R is false. Dropping Q would therefore turn a sound resolution step into an invalid one.
A further trap is canceling every opposite-looking literal at once. From P OR Q and NOT P OR NOT Q, resolving on P yields Q OR NOT Q, a tautology. It does not yield a contradiction. The original clauses can both hold, for example when P is true and Q is false.

Illustrative case study

Fictional Cedar Publishing reviews a rule-based readiness explanation. One version correctly derives readiness from completed checks and approval. Another explanation works backward from readiness to approval without a rule authorizing that direction.
The team adds a counterexample test and requires each explanation step to cite its premises.
Cedar keeps policy review separate from proof checking. A logically correct derivation can still implement an outdated policy if the supplied domain statements are wrong.

Common misconceptions and mistakes

“An implication works in both directions.” Its converse requires separate justification.
“A true conclusion proves the argument valid.” It may have been reached through an invalid step.
“Resolution proves whichever remaining literal I prefer.” The remaining disjunction must be preserved.
“A rule can change meaning mid-proof.” Placeholder substitutions must remain consistent within an application.
“Sound rules guarantee true input facts.” They preserve truth conditionally on the premises.

Business use

Require reasoning systems to distinguish supplied facts, domain rules, and derived conclusions. Review both formal correctness and the accuracy of the represented policy. Use small counterexamples to test suspicious reasoning patterns, and retain enough proof detail to locate the first unsupported step instead of reviewing only the final answer.

Check your understanding

Why must both P and Q be established for Step 1?

Answer: Conjunction introduction requires both premises.

Why does S not establish R?

Answer: R implies S does not state the converse.

Why retain Q in the resolvent?

Answer: The premises allow Q true while R is false.

Related terms

Learn next

Check modus tollens: from P implies Q and NOT Q, derive NOT P. Test every truth assignment and look for any case with true premises and a false conclusion.

Sources

AIMA glossary, inference rules: https://github.com/aimacode/aima-glossary/blob/master/glossary.md Cornell CS3110, logic and inference rules: https://www.cs.cornell.edu/courses/cs3110/2008fa/lectures/logic.html Bart Selman, Cornell CS4700, representation and inference: https://www.cs.cornell.edu/courses/cs4700/2017fa/lectures/pdf/CS4700-09-Knowl_Repr_Reasoning_representation_and_inference_v2.pdf

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