Formal Logic in AI

Logics are formal languages with well defined rules for manipulation of representations.

A knowledge base (KB) is a set of sentences in a given logic language.

The family of logics vary on expressivity.

More expressivity comes at the cost of increasing computational complexity.

Logics

Reasoning :

The manipulation of symbols in a meaningful manner.

Maths is replete with algorithms we use -

• Addition and multiplication of multi-digit numbers 
• Long division
• Solving systems of linear equations
• Fourier transformation, convolution...

Soundness and completeness

Given a knowledge base and a reasoning algorithm -

• Entailment : which other sentences in the language are necessarily true ?
• Proof : which other sentences in the language can one produce by the reasoning algorithm ?

Soundness (of the reasoning algorithm) :

• A logic is sound if only true statement in the language can be proved.

Completeness (of the reasoning algorithm) :

• A logic is complete if all true statements in the language can be proved.