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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.
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”
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”
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”
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”
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 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:
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:
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:
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:
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.
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)
∴ _________?
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