First Order Predicate Logic

This topic will explain

• What is Predicate Logic
• Syntax of Predicate logic
• Semantics of Predicate Logic
• Examples of Predicate Logic

In predicate logic, also known as first-order logic, statements are expressed using predicates, variables, quantifiers, and logical connectives. The syntax and semantics of predicate logic define the rules for constructing well-formed formulas and the meanings of those formulas.

Here's an explanation of the syntax and semantics of predicate logic:

Syntax:
1. Variables: Variables represent unspecified individuals or objects and are typically denoted by lowercase letters such as x, y, or z.
2. Constants: Constants represent specific individuals or objects and are typically denoted by uppercase letters such as a, b, or c.
3. Predicates: Predicates represent properties or relations and are typically denoted by uppercase letters or symbols. For example, "P(x)" represents a unary predicate that applies to the variable x, and "Q(x, y)" represents a binary predicate that applies to variables x and y.
4. Quantifiers: Quantifiers are used to express the scope of variables in a formula. The two main quantifiers in predicate logic are the universal quantifier (∀) and the existential quantifier (∃).
5. Connectives: Logical connectives such as conjunction (∧), disjunction (∨), implication (→), and negation (¬) are used to combine or modify statements.

The syntax of predicate logic specifies how these elements can be combined to form well-formed formulas. It includes rules for parentheses placement, quantifier scoping, and the proper use of variables, constants, predicates, and connectives.

Semantics:
The semantics of predicate logic define the meanings and interpretations of the syntax. It specifies how the logical connectives, quantifiers, variables, and predicates are interpreted.

1. Interpretations: An interpretation assigns meanings to the constants, predicates, and functions in a logic system. It provides a domain of discourse or universe of discourse, which consists of objects that the variables can refer to.

2. Predicates: Predicates are interpreted as properties or relations over the objects in the domain of discourse. For example, if P(x) represents "x is a prime number," the interpretation of P(x) determines which objects in the domain satisfy the predicate.

3. Quantifiers: The universal quantifier (∀) asserts that a statement holds for all objects in the domain, while the existential quantifier (∃) asserts that a statement holds for at least one object in the domain.

4. Connectives: The logical connectives such as conjunction (∧), disjunction (∨), implication (→), and negation (¬) have defined truth tables that specify the truth values of compound statements based on the truth values of their constituent parts.

By combining the syntax and semantics of predicate logic, we can construct well-formed formulas and determine their truth values based on the interpretations of predicates, quantifiers, and connectives. This allows us to reason and make logical deductions using predicate logic.

Examples:

Certainly! Here are a few examples of statements expressed using predicate logic:

1. Example with Unary Predicate:
Let's say we have a unary predicate "Even(x)" which represents "x is an even number."
- Statement: "For all x, x is an even number."
- Predicate Logic Representation: ∀x Even(x)

2. Example with Binary Predicate:
Consider a binary predicate "Likes(x, y)" which represents "x likes y."
- Statement: "There exists someone who likes everyone."
- Predicate Logic Representation: ∃x ∀y Likes(x, y)

3. Example with Quantifiers and Connectives:
Let's say we have a unary predicate "Human(x)" representing "x is a human" and a binary predicate "Parent(x, y)" representing "x is a parent of y."
- Statement: "Every human has a parent who is also a human."
- Predicate Logic Representation: ∀x (Human(x) → ∃y (Human(y) ∧ Parent(y, x)))

4. Example with Negation:
Consider a unary predicate "Positive(x)" representing "x is a positive number."
- Statement: "There does not exist a positive number."
- Predicate Logic Representation: ¬∃x Positive(x)

These examples demonstrate how predicate logic can express statements involving predicates, variables, quantifiers, and connectives. It provides a formal and precise representation of relationships, properties, and logical statements in various domains.