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### Nonmonotonic Logic

Ahmed Salman Malik

Overview

- Monotonic Logic
- Nonmonotonic Logic
- Usage and Applications
- Comparison with other forms of logic
- Related Topics
- Summary

Monotonic Logic

- Standard type of logic
- If proven true, will be true forever
- Facts provided can’t be modified

- Doesn’t always fit in real life.
- Sidra is in Doha and Doha is in Qatar, so Sidra is in Qatar.
- Sidra can always take a trip to United States

Nonmonotonic Logic

- New facts can be added
- Current facts can be modified
- Conclusion can change
- If A → B before new fact
- Conclusion might change after new fact

Example

- Consider the following example
- All balls bounce
- Football is a ball
- Does football bounce

- Of course?
- What about a football with no air filled in?

- Conclusions change with new facts

Real life usage

- Used in artificial intelligent systems
- For its adaptability
- Adding, removing and modifying facts
- To reach appropriate conclusions for appropriate scenarios

Application

- Consider an example that can’t be handled by monotonic logic
- Birds can fly

- Seems logical, right?

Exceptions?

- What about exceptions?
- Ostrich, Penguins

Exceptions

- Bird(x) → Flies(x)
- How to handle exceptions?
- Bird(x) Ù Abnormal(x) → Flies(x)

- Through nonmonotonic logic, we handle exceptions

Handling Exceptions

- We know
- Ostrich(x) → Abnormal(x)
- Ostrich is not a normal bird
We conclude

- Ostrich(x) → Bird(x) Ù Flies(x)

- We make all exceptions this way

Specifying Defaults

- Monotonic logic has formal systems to handle defaults
- Defaults: Known facts and rules

- Nonmonotoniclogic uses incomplete and uncertain information to form patterns for decision making
- Abduction: Interpretation of the rules and facts.

Reasoning

- Default logic
- The predicate logic used as set of inferences

- Modal Operator - consistent with known facts
- x,y: A(x,y) Ù B(x,y)
C (x,y)

- Here B(x,y) is the Modal Operator

- x,y: A(x,y) Ù B(x,y)

Abduction

- For a given fact
- A(x) → B(x) and
- A(x) is sufficient for B(x)
- Although, A(x) is not required for B(x)

Comparison with Probability

- Nomonontonic logic isn’t compatible with probability
- The uncertainty and addition of facts disturb the probability model.

Comparison with Classical Logic

- Always results in a conclusion
- Classical logic might loop forever with incomplete information
- Would return wrong answer, instead of none.

Related Topics

- Modal Logic
- Modelling reasoning about knowledge, actions or time

- Epistemic Logic
- Uses modal logic to reason about knowledge

- Deontic Logic
- The representation of normative knowledge

Summary

- Adapts with addition, removal and modification of new facts.
- Handling exceptions
- Used in artificial intelligence for decision making purposes
- Helpful where predicate or classic logic falls short

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