Intelligent Systems by Universal Teacher Publications

Formalized Symbolic Logics

Inference Rules

The inference rules of predicate logic provide the means to perform logical proofs or deductions. The problem is, given a set of sentences S = {s1,…sn}, prove the truth of s (the conclusion); that is, show that

Q. Explain the following inference rules:

Modus Ponens (Dec 02, June 03): From P and P à Q infer Q. Symbolically:

P à Q

For example,

Given: (A is a father)
and: (A is a father) à (A has a child)
conclude: (A has a child)

Chain Rule: From P à Q, and Q à R, infer P à R. Symbolically:

P à Q

Q à R

P à R

For example,

Given: (programmer likes LISP) à (programmer hates COBOL)
and: (programmer hates COBOL) à (programmer likes recursion)
conclude: (programmer likes LISP) à (programmer likes recursion)

Substitution: If s is a valid sentence, s’ derived from s by consistent substitution of propositions in s, is also valid. For example, the sentence A Ú ^~A is valid; therefore, B Ú ^~B is also valid.

Simplification (Dec 02, June 03): From P & Q infer P.

Conjunction: From P and from Q, infer P & Q.

Transposition: From P à Q, infer ^~Q à ^~P

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