Propositional Logic and Its Applications

 

🧩 1. Propositions

🔍 Definition:

A proposition is a statement that is either true or false, but not both.

🐄 Animal Science Example:
"The cow is ruminant." → ✅ Proposition (It can be verified as true or false)

💹 Economics Example:
"Increasing taxes always decreases consumer spending." → ✅ Proposition (It has a truth

value, though we may debate its accuracy)

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🚫 Not Propositions:

• Questions: “Is the cow healthy?”

• Commands: “Increase the price!”

These cannot be judged as true or false, so they are not propositions.

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🔗 2. Arguments (Valid and Invalid)

🔍 Definition:

An argument is a series of propositions where:

• One or more are premises (assumed to be true)

• One is a conclusion
  The goal: check whether the conclusion logically follows from the premises.

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✅ Valid Argument

If the conclusion logically follows from the premises.

Example (Animal Science):

• Premise 1: All poultry need clean water.

• Premise 2: These birds are poultry.

• ➡️ Conclusion: These birds need clean water.
  → Valid

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❌ Invalid Argument

If the conclusion doesn’t logically follow, even if the premises are true.

Example (Economics):

• Premise 1: Inflation is high.

• Premise 2: High inflation causes reduced purchasing power.

• ➡️ Conclusion: The government caused the inflation.
  → Invalid (The premises don’t justify this conclusion — it adds extra information.)

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🔄 3. Logical Connectives (Operators)

Logical connectives are tools used to combine propositions.

Symbol	Name	Meaning	Example
∧	Conjunction	AND	Cows are ruminants ∧ Cows produce milk
∨	Disjunction	OR	Goat is vaccinated ∨ Sheep is vaccinated
¬	Negation	NOT	¬(The poultry is infected)
→	Implication	IF...THEN	If inflation rises → Savings fall
↔	Biconditional	IF AND ONLY IF	An animal is fertile ↔ It can reproduce

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🧮 4. Truth Tables

A truth table shows all possible truth values of compound propositions.

🔘 Example: Conjunction (AND)

Let

• P = "Cow is ruminant"

• Q = "Cow produces milk"

P	Q	P ∧ Q
T	T	T
T	F	F
F	T	F
F	F	F

👉 AND is only true when both are true.

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🔘 Example: Implication (IF...THEN)

Let

• P = "Feed quality improves"

• Q = "Animal health improves"

P	Q	P → Q
T	T	T
T	F	F
F	T	T
F	F	T

👉 The only false case is when the first is true but the second is false.

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🔁 5. Propositional Equivalences

Some compound propositions are logically equivalent — they always have the same truth value.

⚖️ Common Equivalences:

Name	Equivalence	Meaning
Double Negation	¬(¬P) ≡ P	Not not P is just P
De Morgan's Law	¬(P ∧ Q) ≡ ¬P ∨ ¬Q	Not both → Either not
Contrapositive	P → Q ≡ ¬Q → ¬P	Flip and negate
Distributive Law	P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)	Distribute like algebra

🧪 Example in Animal Science:

Let

• P: The cattle are vaccinated

• Q: The cattle are healthy

Then:
¬(P ∧ Q) ≡ ¬P ∨ ¬Q
→ "The cattle are not both vaccinated and healthy" is the same as "They are not vaccinated OR not healthy."

📈 Example in Economics:

Let

• P: "Subsidy is increased"

• Q: "Production rises"
  Then:
  P → Q ≡ ¬Q → ¬P
  → If increasing subsidies leads to more production, then if production doesn’t rise, subsidies likely weren’t increased.

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🔄 Summary Chart

Concept	Purpose	Example (Animal Science)	Example (Economics)
Propositions	Identify true/false statements	“The goat is healthy.”	“GDP is growing.”
Arguments	Evaluate logical structure	"All hens lay eggs → This is a hen → ?"	“Taxes reduce demand → Taxes rose → ?”
Connectives	Combine/mix statements logically	Cow is healthy ∧ Cow is fed well	Market is stable ∨ Govt is intervening
Truth Tables	Analyze compound statements truthfully	If animal eats → animal grows	If inflation rises → savings fall
Equivalences	Recognize interchangeable forms for simplification	¬(Cow is sick ∧ cow is hungry) ≡ …	¬(Govt acts ∧ economy reacts) ≡ …

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🧠 Why This Matters

Animal Science	Economics
Designing and interpreting research	Building sound economic models
Diagnosing health or behavioural issues	Evaluating cause-effect in markets
Making ethical decisions about interventions	Communicating policy logic
Avoiding flawed reasoning in claims	Avoiding bias and fallacies in predictions

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💬 Final Thought:

“Clear reasoning is the foundation of good science and policy.”

Logical thinking doesn’t just make you smarter — it helps you ask better questions, make informed decisions, and avoid being misled.

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