First Order Logic (FOL)

Imagine you have a toolbox for building logical statements. Propositional logic is like a basic set of tools: it lets you connect simple ideas (like “it’s raining” and “the ground is wet”) using “AND,” “OR,” and “NOT.” But the world is much more complex than simple “true” or “false” statements. That’s where First-Order Logic (FOL) comes in – it’s like a supercharged toolbox for representing knowledge and making complex deductions about the world. So in simple words, First Order Logic is a formal system used to express statements about objects and the relationships between them.

Beyond Propositions: It extends Propositional Logic, which deals with simple true/false statements, by introducing concepts like objects, relations between objects, and quantifiers.

Expressions: This allows FOL to express a much wider range of complex facts and relationships that are hard to capture with Propositional Logic alone.
For example, if we say ∀ x ( P(x) → Q(x) ), we’re saying that every object x with property P also has property Q.

The Building Blocks of First Order Logic

  1. Variables: Just like in algebra, variables (usually x, y, z) are placeholders for objects.
  2. Objects or Constants: FOL is all about things in the world. These could be people, places , numbers, or concepts. They are usually denoted by lowercase letters at the beginning of the alphabet, such as a, b, c.
  3. Predicates: Predicates are statements or properties that can be true or false for certain objects. They are represented by uppercase letters followed by parentheses containing variables or constants. These are like labels describing properties or relationships between objects. For example, P(x), Q(x, y).
  4. Quantifiers: These help us express how many things have a certain property.

    ⦿ Universal Quantifier (∀): Denoted by ∀, Means “for all”. It asserts that a statement holds true for all elements in the domain.
    For example, ∀x P(x) means "For all x, P(x) is true."
    ⦿ Existential Quantifier (∃): Denoted by ∃, Means “there exists at least one”. It asserts that a statement holds true for at least one element in the domain.
    For example, ∃x P(x) means "There exists an x such that P(x) is true."
  5. Logical Connectives: First-order logic also uses logical connectives to combine or modify predicates and statements. Common logical connectives include "AND" (∧), "OR" (∨), "NOT" (¬), and "IMPLIES" (→).
  6. Functions: Functions in first-order logic are similar to functions in mathematics. They take objects as arguments and produce another object as a result. Functions are represented by lowercase letters followed by parentheses containing variables or constants.
    For example, f(x), g(x, y).

Examples of statements in First Order Logic:

  1. All humans are mortal:
    • FOL representation: ∀x (Human(x) → Mortal(x))
    • Explanation: This statement says that for every object x, if x is a human, then x is mortal.
  2. Some cats are black:
    • FOL representation: ∃x (Cat(x) ∧ Black(x))
    • Explanation: This statement says that there exists an object x such that x is a cat and x is black.
  3. Every student has a favorite subject:
    • FOL representation: ∀x ∃y (Student(x) → FavoriteSubject(x, y))
    • Explanation: This statement says that for every object x, if x is a student, then there exists an object y such that y is the favorite subject of x.
  4. No dogs can fly:
    • FOL representation: ∀x (Dog(x) → ¬Fly(x))
    • Explanation: This statement says that for every object x, if x is a dog, then x cannot fly.
  5. Some birds can swim:
    • FOL representation: ∃x (Bird(x) ∧ CanSwim(x))
    • Explanation: This statement says that there exists an object x such that x is a bird and x can swim.

These examples demonstrate how First-Order Logic allows us to express relationships and properties of objects in a precise and structured manner. Each statement consists of quantifiers (universal or existential), predicates (like “Human(x)” or “Cat(x)”), and logical connectives (like “→” for implication and “∧” for conjunction), which together define the conditions under which the statements are true.

The Power of Inference: Deducing New Truths

Here’s where FOL gets really cool. Imagine you have these FOL statements:

  1. ∀x (Man(x) → Mortal(x)) (“All men are mortal”)
  2. Man(Socrates) (“Socrates is a man”)

Can you figure out a new piece of knowledge from this? Absolutely! Since all men are mortal, and Socrates is a man, we can deduce:

  1. Mortal(Socrates) (“Socrates is mortal”)

Why Use First Order Logic?

  • Represents complex knowledge: FOL lets us express things propositional logic can’t, like relationships, quantities, and properties of objects.
  • Basis of AI: Knowledge bases and reasoning systems in artificial intelligence often use FOL.
  • Mathematical foundations: FOL has close links to mathematical logic.

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Last Update: 10 May 2024

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