Lesson Notes By Weeks and Term v3 - Senior Secondary 2
Logical reasoning ii
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Logical reasoning ii
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: Further Mathematics
Class: Senior Secondary 2
Term: 1st Term
Week: 1
Theme: Pure Mathematics
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
A theorem is a statement that has been proven to be true based on a set of accepted axioms, postulates, definitions, and previously established theorems, using a series of logical deductions. In mathematics, theorems form the foundation of our understanding.
Key Points in Theorem Proving: Axioms/Postulates: These are fundamental statements that are accepted as true without proof. They serve as the starting points for all deductions. E.g., "Through any two distinct points, there is exactly one line." Definitions: Precise meanings assigned to terms or concepts.
`(~W ^ ~F) V P` (using De Morgan's) and then `(~W V P) ^ (~F V P)`. So, two clauses: `(~W V P)` and `(~F V P)`. 3. `R` (already in CNF)
4. Negate goal `P`: `~P` (already in CNF)
Clauses: C1: `~R V W` C2: `~W V P` C3: `~F V P` C4: `R` C5: `~P` (Negated goal)
Step 2: Apply Resolution: Resolve C1 (`~R V W`) and C4 (`R`): Literal `~R` from C1 and `R` from C4 resolve.
Resolvent R1: `W` Resolve R1 (`W`) and C2 (`~W V P`): Literal `W` from R1 and `~W` from C2 resolve.
Resolvent R2: `P` Resolve R2 (`P`) and C5 (`~P`): Literal `P` from R2 and `~P` from C5 resolve. * Resolvent R3: `()` (Empty clause)
Conclusion: Since the empty clause `()` is derived, the assumption `~P` leads to a contradiction.
Therefore, the goal `P` (the farmer can plant maize) is true and follows from the knowledge base. Modelling the World for an intelligent system refers to creating a structured and simplified representation of the relevant aspects of the real environment in which the system operates. This model enables the system to understand its context, predict consequences of actions, and make informed decisions.
Critical Points in Modelling the World:
1. Abstraction: The model should capture essential features while ignoring irrelevant details. For example, modelling a traffic system might focus on vehicle positions and road segments, not the make or colour of cars.
2. Granularity: Deciding the level of detail required. Too much detail can make the model complex and inefficient; too little can make it ineffective.
3. Dynamic Nature: The real world changes, so the model must account for these changes or be updated frequently.
4. Formalism: Using a formal language (like logic) to represent the model ensures unambiguous interpretation by the intelligent system.
5. Completeness and Consistency: Ideally, the model should contain all necessary information and be free from contradictions. Components of a World Model (Logical Agent's Perspective): States: Descriptions of the environment at a particular moment. E.g., `(Weather = Sunny, Road = Dry)`.
Actions: Operations that an agent can perform to change the state. E.g., `Drive(Car, Destination)`.
Percepts: Sensory inputs from the environment. E.g., `(TrafficLight = Green)`.
Goals: Desired states the agent aims to achieve. E.g., `(ArrivedAtWork)`.
Domain Knowledge: Facts and rules about the environment and how actions affect it. E.g., `(If TrafficLight = Red, then cannot Drive)`.
Example: Modelling a Local Nigerian Market An intelligent system designed to help a buyer find the best deals at a local market might model the world using: Predicates: `HasProduct(Vendor, Product)`, `Price(Vendor, Product, Amount, Unit)`, `Location(Vendor, Section)`.
Facts: `HasProduct(MamaNkechi, Yam)`, `Price(MamaNkechi, Yam, ₦2000, Tuber)`, `Location(MamaNkechi, RootSection)`.
Rules: `If HasProduct(V, P) AND Price(V, P, A, U) AND A Q)` converts to `(~P V Q)`.
The Resolution Rule (Propositional): Given two clauses `C1` and `C2`, if `C1` contains a literal `L` and `C2` contains the literal `~L` (the negation of `L`), then a new clause, called the resolvent, can be inferred. The resolvent is formed by combining all literals from `C1` and `C2` except for `L` and `~L`.
Formally: `(A V L)` and `(B V ~L)` implies `(A V B)`. (Here, A and B represent disjunctions of other literals). Refutation Resolution Procedure (Propositional): To prove that a statement `S` follows from a Knowledge Base `KB`:
1. Convert all sentences in `KB` into CNF.
2. Negate the statement `S` to `~S`, and convert `~S` into CNF. Add this to `KB`.
3. Repeatedly apply the resolution rule to pairs of clauses in the `KB` (including the negated query) to produce new resolvents.
4. Add each new resolvent to the `KB`.
5. If the empty clause `()` is derived, it means a contradiction has been found. This implies that the assumption `~S` was false, and therefore `S` must be true. If the empty clause cannot be derived, `S` is not entailed by `KB`.
Worked Example (Propositional Resolution): Scenario: A Nigerian farmer decides whether to plant maize based on weather and soil conditions.
Knowledge Base (KB):
1. If it rains (`R`), the soil is wet (`W`). (`R => W`)
2. If the soil is wet (`W`) or the soil is fertile (`F`), the farmer can plant maize (`P`). (`(W V F) => P`)
3. It rained (`R`).
Goal: Prove that the farmer can plant maize (`P`).
Step 1: Convert KB and negated goal to CNF. 1. `R => W` becomes `~R V W` 2. `(W V F) => P` becomes `~(W V F) V P` which is `(~W ^ ~F) V P` (using De Morgan's) and then `(~W V P) ^ (~F V P)`. So, two clauses: `(~W V P)` and `(~F V P)`. 3. `R` (already in CNF)
4. Negate goal `P`: `~P` (already in CNF)
Clauses: C1: `~R V W` C2: `~W V P` C3: `~F V P` C4: `R` C5: `~P` (Negated goal)
Step 2: Apply Resolution: Resolve C1 (`~R V W`) and C4 (`R`): Literal `~R` from C1 and `R` from C4 resolve.
Resolvent R1: `W` Resolve R1 (`W`) and C2 Recall Predicate Logic: Extends propositional logic by allowing quantification over individuals (objects in the domain), using predicates to describe properties of objects or relationships between them, and variables. This section provides a detailed explanation of the core concepts of Logical Reasoning II, as outlined in the performance objectives.
`Man(Socrates)`, `∀x (Man(x) => Mortal(x))`, `∃y (Student(y))`.
Challenges in FOL Resolution: Variables: Propositions now contain variables, requiring a mechanism to match them.
Quantifiers: Universal (`∀`) and Existential (`∃`) quantifiers must be handled.
Key Concepts for FOL Resolution: Skolemization: A process to eliminate existential quantifiers. If an existential quantifier is not within the scope of a universal quantifier, replace the existentially quantified variable with a unique Skolem constant. E.g., `∃x P(x)` becomes `P(A)`, where `A` is a unique constant. If an existential quantifier is within the scope of a universal quantifier, replace the existentially quantified variable with a Skolem function of the universally quantified variables. E.g., `∀x ∃y Likes(x, y)` becomes `∀x Likes(x, f(x))`, where `f(x)` is a function that returns the `y` that `x` likes.
Unification: The process of finding a substitution (a mapping of variables to terms) that makes two or more atomic expressions (literals without negations) identical.
Example: `P(x, A)` and `P(B, y)` can be unified by the substitution `{x/B, y/A}` to become `P(B, A)`. The goal is to find the most general unifier (MGU), which makes the expressions identical with the minimum possible commitment to specific values.
Generalized Resolution Rule (FOL): Given two clauses `C1 = (L1 V A)` and `C2 = (L2 V B)`, where `L1` and `L2` are literals. If `L1` and `~L2` (or `~L1` and `L2`) can be unified with a most general unifier `θ`, then the resolvent is `(A V B)θ`. The substitution `θ` is applied to all literals in the resolvent.
Refutation Resolution Procedure (FOL): Convert all knowledge base sentences and the negated goal into a set of clauses (Skolemization and dropping universal quantifiers are part of this conversion). Repeatedly select two clauses that contain complementary literals that can be unified. Compute the most general unifier `θ` for the complementary literals. Apply the resolution rule using `θ` to generate a new resolvent clause. Add the new resolvent to the set of clauses. If the empty clause `()` is generated, the goal is proven.
Conceptual Example (FOL Resolution): Knowledge Base: `∀x (Man(x) => Mortal(x))` (All men are mortal) `Man(Socrates)` (Socrates is a man)
Goal: Prove `Mortal(Socrates)` (Socrates is mortal)
Steps: Convert to Clauses: `∀x (Man(x) => Mortal(x))` becomes `∀x (~Man(x) V Mortal(x))`. Drop `∀x`: `(~Man(x) V Mortal(x))` (Clause 1) `Man(Socrates)` (Clause 2)
Negate goal: `~Mortal(Socrates)` (Clause 3)
Apply Resolution and Unification: Resolve Clause 1 (`~Man(x) V Mortal(x)`) and Clause 2 (`Man(Socrates)`). The literals `Man(x)` and `Man(Socrates)` are complementary (one is positive, one is negative) and can be unified with `θ = {x/Socrates}`. Applying `θ` to the resolvent gives: `Mortal(Socrates)`. (Resolvent R1) Resolve R1 (`Mortal(Socrates)`) and Clause 3 (`~Mortal(Socrates)`). The literals `Mortal(Socrates)` and `~Mortal(Socrates)` are complementary.
Resolvent R2: `()` (Empty clause)
Conclusion: The empty clause is derived, so `Mortal(Socrates)` is proven.
Smart Agricultural Advisory Systems: Application: Logical reasoning is central to expert systems that advise Nigerian farmers on crop selection, pest control, and optimal irrigation. These systems model the "world" of agriculture (soil type, weather data, crop diseases, market prices) using logical rules. For example, a rule might be `IF (soil_type = "sandy" AND rainfall threshold AND junction_B_density typical_daily_spend AND transaction_location = "unusual_country") THEN (flag_for_review)`. Logical resolution helps in determining if a transaction matches a fraud pattern or if it's a legitimate transaction.
Integration: This highlights the importance of logic in cybersecurity and protecting consumers' finances, a critical aspect of Nigeria's digital economy. ---