Why FOPL; rules of inference

Why do we need FOPL?

Propositional logic can only state whole facts as true or false; it cannot talk about objects, their properties, relations or about "all" and "some". First-order predicate logic (FOPL) is needed because:

1. It represents objects, properties and relations with predicates, e.g. Teacher(Ram), Likes(Ram, Sita).
2. It uses quantifiers (∀, ∃) to make general statements, e.g. ∀x Man(x) → Mortal(x), which would need thousands of propositions otherwise.
3. It uses functions and variables, so it is far more expressive and compact.
4. It has a precise syntax and semantics, with sound and complete inference (resolution), which allows automated theorem proving and is the basis of Prolog and expert systems.

Three rules of inference

1. Modus Ponens: from P → Q and P, infer Q.
   • Example: "If it rains, the ground is wet. It rains. So the ground is wet."
2. Modus Tollens: from P → Q and ¬Q, infer ¬P.
   • Example: "If it rains, the ground is wet. The ground is not wet. So it did not rain."
3. Universal Instantiation: from ∀x P(x), infer P(a) for any constant a.
   • Example: ∀x Man(x) → Mortal(x) gives Man(Socrates) → Mortal(Socrates).

(Others: And-elimination, And-introduction, Existential instantiation, Resolution, Hypothetical syllogism.)

Making the machine learn

Learning means improving performance from experience. To give a machine learning capacity:

1. Collect experience data (examples, sensor readings, feedback from the environment or a teacher).
2. Choose a representation for what is learned (rules, decision tree, neural network, probabilities).
3. Use a learning algorithm that adjusts the representation to fit the data:
   • Supervised learning: learn from labelled examples (classification, regression; decision trees, neural networks).
   • Unsupervised learning: find structure in unlabelled data (clustering).
   • Reinforcement learning: learn actions from rewards and penalties.
   • Other forms: rote learning, learning by instruction, induction, analogy, explanation-based learning.
4. Test and refine the learned model on new data, and keep updating it as new experience arrives.

A learning agent therefore has a learning element (improves the system), a performance element (acts), a critic (feedback) and a problem generator (suggests new experiments).