Elementary Logic: Connectives, Quantifiers, and Truth Tables

Introduction to Propositions

• A proposition is a declarative sentence or statement that is either true or false.

• Examples of simple propositions include:

  • Bente pesos kilo ng bigas.

  • Mas safe na sa Pinas.

  • Di ako magnanakaw.

Logical Connectives and Compound Propositions

• Connectives are tools used to join two or more propositions to create compound propositions.

• Each connective has a specific presentation and symbol depending on the type of statement it creates:

• AND (Conjunction)

  • Symbol: $\wedge$

  • Function: Joins statements to form a conjunction.

  • Examples:

    • "My father is a contractor and we are rich."

    • "I am handsome and I am funny."

• OR (Disjunction)

  • Symbol: $\vee$

  • Function: Joins statements to form a disjunction.

  • Examples:

    • "You made my day I am a communist or."

    • "It is raining or it is sunny."

• IF…THEN (Conditional)

  • Symbol: $\rightarrow$

  • Function: Joins statements to form a conditional statement.

  • Examples:

    • "If I am guilty then put me to prison."

    • "If I eat pizza then I am happy."

• IF AND ONLY IF (Biconditional)

  • Symbol: $\leftrightarrow$

  • Function: Joins statements to form a biconditional statement.

  • Examples:

    • "I will stop crying you will bring him home if and only if."

    • "I will join the game if and only if I team up with my crush."

Symbolic Representation and Variables

• Simple statements are represented by the variables $p, q, r, \text{ and } s$.

• Symbolic representation allows for the abbreviation of compound statements:

  • Statement: "The highest mountain in the Philippines is Mt. Apo, and the second highest mountain is Mt. Pulag."

  • Symbolic form: $p \wedge q$.

• Example Case:

  • Let $p =$ "We will go to Camiguin Island."

  • Let $q =$ "We will go to Mactan Island."

  • The compound statement "We will go to Camiguin Island or we will go to Mactan Island" is written as $p \vee q$.

Quantifiers in Logic

• Quantifiers are words that indicate the quantity of elements that satisfy a condition. Common examples include "all," "there exists," and "none."

• Universal Quantifiers: These words either deny the existence of something or stress that every element satisfies a condition.

  • Words like "none" and "no" are used to deny existence.

  • Words like "all" and "every" stress that every element is included.

• Examples of quantified statements:

  • "All women are right."

  • "No men is decent."

Negation of Statements

• Negation is the act of denying or contradicting a statement, idea, or action.

• The symbol for negation is $\sim$.

• Negative markers include words like "no" or "not," and prefixes or suffixes like "-less," which make a statement the opposite of its original meaning.

• Negations must ensure that there is at least one thing that disproves the original proposition.

• Negation Examples:

  • Statement ($p$): "All women are right."

  • Negation 1 ($\sim p$): "Not all women are right."

  • Negation 2 ($\sim p$): "Some women are wrong."

Analysis of Compound Propositions

• Compound propositions can be dismantled into their base parts: propositions, connectives, and quantifiers.

• Analysis of "All my classmates are absent and I am present":

  • First Proposition: "All my classmates are absent."

  • Second Proposition: "I am present."

  • Connective: "and" (conjunction).

  • Quantifier: "All."

Truth Tables and Structural Logic

• A truth table is a systematic way to show the truth value of a compound statement for all possible truth combinations of its constituent simple statements.

• The truth value of a compound proposition is determined by its parts, which may be true and false simultaneously.

• Row Calculation for Truth Tables:

  • For any number $n$ of propositions, the number of rows in the truth table is calculated as $2^n$.

  • A simple proposition (where $n=1$) has $2^1 = 2$ rows (one for True, one for False).

Rules for Determining Truth Values

• Negation ($\sim p$): Flips the truth value. If $p$ is True, $\sim p$ is False. If $p$ is False, $\sim p$ is True.

• Conjunction ($p \wedge q$): True only when both parts are True.

• Disjunction ($p \vee q$): True if at least one part is True.

• Conditional ($p \rightarrow q$): False only when the first part (antecedent) is True and the second part (consequent) is False.

• Biconditional ($p \leftrightarrow q$): True when both parts have the same truth value (both True or both False).

Examples of Truth Table Construction

• Example 1: Conjunction ($p \wedge q$)

  • $p =$ "I like Math."

  • $q =$ "I like Science."

  • Table:

    • Row 1: $p=T, q=T \rightarrow (p \wedge q)=T$

    • Row 2: $p=T, q=F \rightarrow (p \wedge q)=F$

    • Row 3: $p=F, q=T \rightarrow (p \wedge q)=F$

    • Row 4: $p=F, q=F \rightarrow (p \wedge q)=F$

• Example 2: Conditional with Negation ($\sim p \rightarrow q$)

  • Statement: "If I don’t love you then you are free."

  • $p =$ "I love you."

  • $q =$ "You are free."

  • Table:

    • $p=T, \sim p=F, q=T \rightarrow (\sim p \rightarrow q)=T$

    • $p=T, \sim p=F, q=F \rightarrow (\sim p \rightarrow q)=T$

    • $p=F, \sim p=T, q=T \rightarrow (\sim p \rightarrow q)=T$

    • $p=F, \sim p=T, q=F \rightarrow (\sim p \rightarrow q)=F$

• Example 3: Disjunction with internal negation ($\sim p \vee p$)

  • Table:

    • $p=T, \sim p=F \rightarrow (\sim p \vee p)=T$

    • $p=F, \sim p=T \rightarrow (\sim p \vee p)=T$

• Example 4: Conjunction of two negations ($\sim p \wedge \sim q$)

  • Table:

    • $p=T, \sim p=F, q=T, \sim q=F \rightarrow (\sim p \wedge \sim q)=F$

    • $p=T, \sim p=F, q=F, \sim q=T \rightarrow (\sim p \wedge \sim q)=F$

    • $p=F, \sim p=T, q=T, \sim q=F \rightarrow (\sim p \wedge \sim q)=F$

    • $p=F, \sim p=T, q=F, \sim q=T \rightarrow (\sim p \wedge \sim q)=T$

Tautologies

• A tautology is a statement that is true by necessity or by virtue of its logical form.

• A logical tautology is a statement that is always true because it includes all logical possibilities.

• The formula $(\sim p \vee p)$ is an example of a tautology, as shown in Example 3 above, where every result in the final column is True.