Limitations of Propositional Logic and Introduction to Predicate Logic
Explore the limitations of propositional logic and the need for predicate logic. This guide explains why propositional logic is insufficient for expressing complex statements involving quantifiers and introduces predicate logic as a more expressive system.
Limitations of Propositional Logic and Introduction to Predicate Logic
Limitations of Propositional Logic
Propositional logic uses simple statements (propositions) that are either true or false. However, it has limitations:
- Can't Handle Quantifiers: It can't express statements about "all" or "some" things. For example, you can't directly represent "All dogs are mammals" in propositional logic.
- Can't Describe Properties: It can't describe properties of objects or relationships between them. For example, "x is a prime number" can't be expressed.
- Limited Inferencing: It struggles with arguments that require reasoning about individual objects or groups of objects.
Examples Illustrating the Limitations
Example 1:
- "All equipment in the chemistry lab is functioning properly."
- "The chemistry lab in my college is functioning properly."
Propositional logic alone can't deduce whether the college's chemistry lab is functioning properly based on the general statement.
Example 2:
- "Harry is playing."
- "If Harry is playing, then she will not watch a movie."
We can easily conclude that Harry won't watch a movie, but this is simple deductive reasoning.
Example 3:
- "A virus infiltrated computer system A."
- "A virus infiltrated computer system B."
Propositional logic can't directly infer that a virus infiltrated the entire city network.
Predicate Logic
Predicate logic solves many of propositional logic's limitations. It allows us to make statements about properties of objects and relationships between them. A statement in predicate logic has two parts:
- Subject: The object or variable the statement is about (e.g., "x").
- Predicate: The property or relationship being described (e.g., "is red").
We write this as P(x), where P is the predicate and x is the subject.
Examples of Predicates
One-Variable Predicates
Example 1: P(x) = x > 3. Then P(4) is true (4 > 3), and P(2) is false (2 is not > 3).
Example 2: P(x) = "A virus infiltrated computer system x". If systems CS20 and Business were infiltrated, P(CS20) and P(Business) are true, but P(CS10) is false (assuming CS10 wasn't affected).
Two-Variable Predicates
Example 1: Q(a, b) = a = b + 6. Q(3, 6) is false (3 ≠ 6 + 6), and Q(6, 0) is true (6 = 0 + 6).
Example 2: Q(a, b) = a = b - 5. Q(7, 4) is false (7 ≠ 4 -5), and Q(0, 5) is true (0 = 5 - 5).
n-ary Predicates
Predicates can have any number of variables. We write this as P(x1, x2, ..., xn), representing the value of P for the n-tuple (x1, x2, ..., xn).
Conclusion
Predicate logic is a powerful extension of propositional logic. It provides the tools to make statements about specific objects and their properties, significantly enhancing our ability to represent and reason about complex situations.