PROPOSITIONAL CALCULUS

A proposition is a complete declarative sentence that is either TRUE (truth value T or 1) or FALSE (truth value F or 0), but not both.

These are not propositions!

Connectives and Compound Propositions

PROPOSITIONAL CONNECTIVES

Negation of a Proposition

• The negation of a proposition 𝑃 is denoted by ¬𝑃 and is read as “not 𝑃”.

• The truth table is shown below:

Example 1: Negation of a Proposition

1. 𝑃: It will rain today.

¬𝑃: It will not rain today.

2. 𝑄: Angela is hardworking.

¬𝑄: Angela is not hardworking.

3. 𝑅: You will pass this course.

¬𝑅: You will not pass this course.

Conjunction of Propositions

• The proposition “𝑃 and 𝑄”, denoted by 𝑃 ∧ 𝑄, is called the conjunction of 𝑃 and 𝑄.

• Other keywords: “but”, “nevertheless”

Conjunction of Propositions

• The truth table is shown as follows:

Example 2: Conjunction of a Proposition

1. 𝑃: Sofia is beautiful.

𝑄: Anton is strong.

𝑃 ∧ 𝑄: Sofia is beautiful and Anton is strong.

2. 𝑆: The stock exchange is down.

𝑇: It will continue to decrease.

𝑆 ∧ 𝑇: The stock exchange is down and it will continue to decrease.

Disjunction: Inclusive “or”

• The proposition “𝑃 or 𝑄”, denoted by 𝑃 ∨ 𝑄, is called the disjunction of 𝑃 or 𝑄. This is also referred to as the inclusive “or”.

• Other keywords: “unless”

Disjunction: Inclusive “or”

• The truth table is shown below:

Example 3: Inclusive “or”

1. 𝑃: This lesson is interesting.

𝑄: The lesson is easy.

𝑃 ∨ 𝑄: This lesson is interesting or it is easy.

2. 𝑆: I want to take a diet.

𝑇: The food is irresistible.

𝑆 ∨ 𝑇: I want to take a diet or the food is irresistible.

Disjunction: Exclusive “or”

The disjunction proposition “𝑃 or 𝑄 but not both”, denoted by 𝑃 ⊕ 𝑄, is called the exclusive “or”. The truth table is shown as follows:

Example 4: Exclusive “or”

𝑃: Presidential candidate A wins.

𝑄: Presidential candidate B wins.

𝑃 ⊕ 𝑄 : Either presidential candidate A or B wins.

Implications or Conditionals

• The proposition “If 𝑃, then 𝑄”, denoted by 𝑃 ⇒ 𝑄 is called an implication or a conditional.

• Equivalent propositions: “𝑃 only if 𝑄”, “𝑄 follows from 𝑃”, “𝑃 is a sufficient condition for 𝑄”, “𝑄 whenever 𝑃”

Implications or Conditionals

• The truth table for an implication is shown as follows:

Example 5: Implications

𝑃: It is raining very hard today.

𝑄: Classes are suspended.

𝑃 ⇒ 𝑄: If it is raining very hard today, then classes are suspended.

Related Implication: Converse

• The converse of the proposition “If 𝑃, then 𝑄” is the proposition “If 𝑄, then 𝑃”. In symbols, the converse of 𝑃 ⇒ 𝑄 is 𝑄 ⇒ 𝑃.

• Example: The converse of the proposition 𝑃 ⇒ 𝑄: “If it is raining very hard today, then classes are suspended.” is the proposition 𝑄 ⇒ 𝑃: “If classes are suspended, then it is raining very hard today.”

Related Implication: Contrapositive

• The contrapositive of the proposition “If 𝑃, then 𝑄” is the proposition “If not 𝑄, then not 𝑃”. In symbols, the contrapositive of 𝑃 ⇒ 𝑄 is ¬𝑄 ⇒ ¬𝑃.

• Example: The contrapositive of the proposition 𝑃 ⇒ 𝑄: “If it is raining very hard today, then classes are suspended.” is the proposition ¬𝑄 ⇒ ¬𝑃: “If classes are not suspended, then it is not raining very hard today.”

Related Implication: Inverse

• The inverse of the proposition “If 𝑃, then 𝑄” is the proposition “If not 𝑃, then not 𝑄”. In symbols, the inverse of 𝑃 ⇒ 𝑄 is ¬𝑃 ⇒ ¬𝑄.

• Example: The inverse of the proposition 𝑃 ⇒ 𝑄: “If it is raining very hard today, then classes are suspended.” is the proposition ¬𝑄 ⇒ ¬𝑃: “If it is not raining very hard today, then classes are not suspended.”

Biconditionals

• The proposition “𝑃 if and only if 𝑄”, denoted by 𝑃 ⇔ 𝑄 is called a biconditional.

• Equivalent propositions: “𝑃 is equivalent to 𝑄”, “𝑃 is a necessary and sufficient condition for 𝑄”

Biconditionals

• The truth table for a biconditional is shown as follows:

Example 6: Biconditionals

𝑃: I will pass Matapre.

𝑄: My grade is at least 60.

𝑃 ⇔ 𝑄: I will pass Matapre if and only if my grade is at least 60.

TRUTH TABLE SUMMARY

Truth Tables

• In constructing a truth table, the number of rows is equal to 2𝑛 where 𝑛 is the number of propositional variables.

• For example, if there are 4 propositional variables, then the truth table will consist of 24 = 16.

Assignment of Values

For two propositional variables, we have 4 rows for the truth table and the assignment of values are shown as follows:

Assignment of Values

For three propositional variables, we have 8 rows:

Assignment of values

Types of Propositional Forms

There are three types of propositional forms:

• Tautology

• Contradiction

• Contingency

Tautology

A propositional form that is true under all circumstances is called a tautology.

Example: The proposition 𝑃 ⇒ 𝑄 ⇔ ¬𝑃 ∨ 𝑄

is a tautology. The truth table is shown as follows:

Tautology

Contradiction

A propositional form that is false under all circumstances is called a contradiction.

Example: The proposition ¬ 𝑃 ∧ 𝑄 ⇔ (𝑄 ∧ 𝑃)

is a contradiction. The truth table is shown as follows:

Contradiction

Contingency

A propositional form that is neither a tautology nor a contradiction is called a contingency.

Example: The proposition (𝑃 ∨ 𝑄) ⇒ 𝑅

is a contingency. The truth table is shown as follows:

Contingency

Knowledge Check 1.3

Test 1: Determine if the following propositional forms is a tautology, contradiction or contingency by constructing truth tables for each.

1. 𝑃 ⇔ ¬𝑃

2. 𝑄 ∨ 𝑆 ⇒ ¬𝑃

3. 𝑄 ⇒ 𝑃 ⇔ 𝑃 ∧ 𝑄

4. ¬𝑄 ⇒ 𝑆 ∧ 𝑃 ∨ 𝑆

Test 2: Turn on page 15 and answer number 18.

Logically Equivalent Propositions

Some Logically Equivalent Propositions

Some Logically Equivalent Propositions

Remark: Implications

When is a Mathematical Reasoning Correct?

Rule of Inference: Addition

1. Addition 𝑃

∴ 𝑃 ∨ 𝑄

• In this rule of inference, we can add any propositional variable to another with the use of the logical connective “or”.

Rule of Inference: Simplification

2. Simplification 𝑃 ∧ 𝑄

∴ 𝑃

• Illustration: Give the appropriate conclusion using simplification rule.

(𝑅 ⇒ 𝑆) ∧ ¬𝑇 ∴ _________?

Rule of Inference: Conjunction

3. Conjunction 𝑃 𝑄

∴ 𝑃 ∧ 𝑄

• Illustration: Give the appropriate conclusion using conjunction rule.

𝑆 ⇒ 𝑇 𝑁 ∨ 𝑅

∴ _________?

Rule of Inference: Modus Ponens

4. Modus Ponens 𝑃 ⇒ 𝑄

𝑃 ∴ 𝑄

• Illustration: Give the appropriate conclusion using modus ponens rule.

(𝑀 ∧ 𝑁) ⇒ ¬𝑇 𝑀 ∧ 𝑁

∴ _________?

Rule of Inference: Modus Tollens

5. Modus Tollens 𝑃 ⇒ 𝑄

¬𝑄 ∴ ¬𝑃

• Illustration: Give the appropriate conclusion using modus tollens rule.

𝑀 ⇒ (𝐵 ∧ 𝐶) ¬(𝐵 ∧ 𝐶)

∴ _________?

Rule of Inference: Disjunctive Syllogism

6. Disjunctive Syllogism 𝑃 ∨ 𝑄

¬𝑃 ∴ 𝑄

• Illustration: Give the appropriate conclusion using disjunctive syllogism rule.

𝑍 ∨ (𝑋 ⇒ 𝑌) ¬𝑍

∴ _________?

Rule of Inference: Hypothetical Syllogism

7. Hypothetical Syllogism 𝑃 ⇒ 𝑄 𝑄 ⇒ 𝑅

∴ 𝑃 ⇒ 𝑅

• Illustration: Give the appropriate conclusion using hypothetical syllogism rule.

(𝑋 ∧ 𝑁) ⇒ ¬𝑇 ¬𝑇 ⇒ (𝐴 ∨ 𝐵)

∴ _________?

Some Applications

Some Applications

Knowledge Check 1.4

State the rule of inference by which the conclusion follows from its premise/s.

1. 𝐴 ⇒ (𝐵 ⇔ ¬𝐶)

𝐵 ⇔ ¬𝐶 ⇒ 𝐷

∴ 𝐴 ⇒ 𝐷

2. 𝑇 ∨ 𝑈 ∨ 𝑊

[𝑇 ∨ 𝑈 ∨ 𝑊 ] ∨ 𝑃

Knowledge Check 1.4

3. 𝐸 ⇒ ¬𝐹 ⇒ ¬𝐺 ∨ ¬𝐻

𝐸 ⇒ ¬𝐹

∴ ¬𝐺 ∨ ¬𝐻

4. 𝐼 ⇒ 𝐽 ∨ 𝐾 ⇒ 𝐿

¬ 𝐼 ⇒ 𝐽

∴ 𝐾 ⇒ 𝐿

Knowledge Check 1.4

5. 𝑀 ⇒ ¬𝑁

¬𝑁 ⇒ 𝑄

∴ (𝑀 ⇒ ¬𝑁) ∧ (¬𝑁 ⇒ 𝑄)

6. ¬𝑃 ⇔ 𝑄 ∨ 𝑅 ∨ 𝑆

¬ ¬𝑃 ⇔ 𝑄

∴ 𝑅 ∨ 𝑆