Predicate Logic

• Predicate logic, also known as first-order logic, extends propositional logic by introducing quantifiers and predicates.
• A predicate is a function. It takes some variable(s) as arguments; it returns either True or False (but not both) for each combination of the argument values.
• Predicate Logic - It allows us to express more complex statements involving variables, objects, and relationships.
• Predicates - represent properties or relations that can be true or false for specific objects.
• The quantifiers give us the power to express propositions involving entire sets of objects, some of them, enumerate them, etc.
• Universal Quantifier - Indicates that a statement holds true for all elements in a given domain.
• Universal Quantifier - is another way of converting a predicate into a proposition.
• P(x) is true for all x in the universe of discourse. We write ∀xP(x), and say for all x, P(x).
• ∀xP(x) is TRUE if P(x) is true for every single x.
• ∀xP(x) is FALSE if there is an x for which P(x) is false.
• Existential Quantifier - Indicates that there exists at least one element in the domain for which a statement is true.
• ∀ - Universal
• ∃ - Existential
• Variables - Represent unspecified elements in the domain.
• Predicate logic is used in computer science, artificial intelligence, linguistics, and mathematics. It helps formalize reasoning, database queries, and knowledge representation.
• System Specifications - When developing software or hardware systems, engineers and manufacturers need to meet specific requirements and specifications.
• Artifical Intelligence or Automation - Predicate logic is at the core of AI systems. Engineers use it to represent knowledge and make inferences.
• Translating English Sentences - By translating these sentences into logical expressions using predicate logic, we remove ambiguity.
• Circuit Designing - engineers use predicate logic to express circuit behavior, constraints, and functionality.
• Mathematics and Linguistics - Researchers use predicate logic to describe structures precisely.
• Generalized quantifiers extend the standard quantifiers (universal and existential) of modern logic.
• Generalized quantifiers - allow us to express more complex statements involving variables, objects, and relationships.
• Quantifier expressions are variable-binding operators.
• Quantifier Expression - These operators allow us to express statements about sets of objects.
• Generalized quantifiers are entities that generalize the standard quantifiers.
• Universal Instantiation - If we know that a statement holds for all elements in the domain, we can instantly apply it to a specific element.
• Universal Generalization - If we prove a statement for a specific element, we can generalize it to hold for all elements.
• Existential Instantiation - If we know that a statement exists for some element, we can instantiate it to a specific element.
• Existential Generalization - If we prove a statement for a specific element, we can generalize it to exist for some element.