Propositional Logic
Introduction
- Propositional logic (PL) is a simple but powerful way to represent knowledge using symbols and logical connectives.
- It is the foundation of many AI systems, including expert systems, question-answering systems, and automated planners.
Syntax of Propositional Logic
- Atomic sentences: These consist of a single proposition symbol, which can be true or false.
- Complex sentences: These are constructed from simpler sentences using logical connectives:
- not (¬): negates a sentence
- and (∧): conjunction of two sentences
- or (∨): disjunction of two sentences
- implies (⇒): conditional statement
- if and only if (⇔): biconditional statement
Semantics of Propositional Logic
- Models: These are possible worlds, each of which assigns true or false to every proposition symbol.
- Truth: The truth value of a sentence is determined by the truth values of its components and the connectives used to combine them.
- Truth tables: These tables specify the truth value of a complex sentence for each possible combination of truth values for its components.
Inference in Propositional Logic
- Entailment: A sentence α entails another sentence β if β is true in all worlds where α is true.
- Inference rules: These rules allow new sentences to be derived from existing ones, such as Modus Ponens and And-Elimination.
- Resolution: A complete inference procedure that can determine if a sentence is entailed by a knowledge base.
Applications of Propositional Logic
- Automated reasoning: Used in theorem provers and other systems for formal logic.
- Expert systems: Used to represent knowledge in rule-based systems.
- Planning: Used to represent states, actions, and goals in automated planning systems.
Conclusion
- Propositional logic is a powerful tool for representing and reasoning about knowledge.
- It is the foundation of many AI applications and is an important topic for anyone interested in the field.
Predicate logic
Introduction
- Predicate logic is a more expressive language than propositional logic, allowing for the representation of objects, relations, and functions.
- It is used extensively in AI for knowledge representation, reasoning, and planning.
Syntax
- Constants: Symbols that represent objects (e.g., John, Mary, 1, 2).
- Variables: Symbols that stand for any object (e.g., x, y, z).
- Predicates: Symbols that represent relations or properties (e.g., Brother, >, Red).
- Functions: Symbols that represent functions (e.g., FatherOf, Plus).
- Terms: Expressions that refer to objects (e.g., John, FatherOf(John), x).
- Atoms: Simple sentences that state facts (e.g., Brother(John, Mary), > (2, 1)).
- Quantifiers:
- Universal quantifier (∀): “For all” (e.g., ∀x Person(x)).
- Existential quantifier (∃): “There exists” (e.g., ∃x King(x)).
Semantics
- Interpretation: Specifies which objects, relations, and functions are referred to by the symbols.
- Model: A possible world with objects, relations, and functions.
- Truth: A sentence is true in a model if it is satisfied by the interpretation in that model.
Examples
- “All men are mortal”: ∀x Man(x) ⇒ Mortal(x)
- “There is a king”: ∃x King(x)
- “John is the father of Mary”: FatherOf(John, Mary)
Applications
- Knowledge representation: Representing facts and rules about the world.
- Reasoning: Deriving new knowledge from existing knowledge.
- Planning: Describing states, actions, and goals for planning problems.
Conclusion
- Predicate logic is a powerful tool for representing and reasoning about complex knowledge.
- It is a fundamental part of AI and is used in many applications.
Comparison of Propositional and Predicate Logic
| Feature | Propositional Logic | Predicate Logic |
| Syntax | Propositions, logical connectives (AND, OR, NOT, implies, if and only if) | Constants, variables, predicates, functions, quantifiers (for all, there exists) |
| Semantics | Truth tables, interpretations | Models, interpretations |
| Expressive Power | Limited, can only represent facts | More expressive, can represent objects, properties, relations |
| Examples | “The sky is blue”, “It is raining” | “All men are mortal”, “There exists a cat that is black” |
| Applications | Simple knowledge representation, rule-based systems | Complex knowledge representation, natural language processing, database systems |
References:
- Russell, S., and Norvig, P. Artificial Intelligence: A Modern Approach, 4th Edition, 2020, Pearson.
- Rich, E., Knight, K., & Nair, S. B. Artificial Intelligence. McGraw-Hill International.
- Nilsson, N. J. Artificial Intelligence: A New Synthesis. Morgan Kaufmann.
Note: This content was generated with the assistance of Google’s Gemini AI.