### Summary

First-Order Logic (FOL) is a formal logical system used in mathematics, philosophy, linguistics, and computer science. In AI, FOL serves as a foundational technique to model knowledge and draw inferences by considering objects, their properties, and relationships between them.

### ELI5

Imagine you are playing with a box of different colored lego blocks. You can express simple things like ‘There is a blue block’ or ‘This block is not red’. Now, if you want to express something like ‘If a block is blue, then it has a circle on it’, you need a special language. And that special language, which lets you talk about things, their traits, and their relationships, is what First-Order Logic is all about.

### In-depth explanation

First Order Logic, also known as Predicate Logic or First Order Predicate Logic (FOPL), is a formal system used in mathematics, linguistics, and computer science. In artificial intelligence, it plays a significant role as a framework for systematically representing logical assertions and drawing inferences based on them.

While propositional logic deals with statements that can be true or false, First Order Logic enhances this by dealing with the inherent structure within propositions. FOL provides the language to represent objects, properties, relations between objects, and quantified statements.

FOL comprises variables, constants, predicates, quantifiers, logical connectives and equality symbols. Variables denote objects within a universe of discourse. Constants symbolize specific objects in the universe. Predicates, also known as relations or properties, generally map to functions in programming. Quantifiers (like ‘For all’ - ∀ and ‘There exists’ - ∃) express properties about every object or some object in the universe. Logical connectives specify the relationships between different predicates or terms within a logical expression.

FOL forms the basis of many logic programming languages like Prolog, which are used for AI applications. Inference rules in FOL, like Modus Ponens and Universal Instantiation, enable automated reasoning. Proving theorems, simplifying expressions, or conveying heuristics are all possible due to these rules, making FOL integral to diverse fields, from AI and cognitive science to automated theorem proving and formal verification.

Predicate, Propositional Logic, Knowledge Representation, Automated Reasoning, Logical Inference, Quantifiers, Universal Quantifier, Existential Quantifier, Formal Verification.