Comprehensive Study Guide to Logic and Statements

Fundamentals of Logic and Statements

• Definition of Logic: Logic is the study of methods and principles used to distinguish correct from incorrect arguments and reasoning.
• Statement: A statement is a declarative sentence for which it is meaningful to say it is either true ($T$) or false ($F$).
  • Statements are typically represented by lowercase letters such as $p$, $q$, and $r$.
• Logical Value (LV): The truth or falsity of a statement is referred to as its logical value (or truth value).
  • Example 1: Statement $p$: $2 + 2 = 4$. Logical Value: $T$.
  • Example 2: Statement $q$: Every prime number is odd. Logical Value: $F$ (Note: the number 2 is a prime number and is even).
  • Example 3: Statement $r$: There is a real number $x$ such that $x^{2} + 1 = 0$. Logical Value: $F$ (The solutions for $x^{2} = -1$ are imaginary numbers, not real numbers).

Categorical Statements

• Description: A categorical statement expresses a relationship between two categories or sets: the subject set ($S$) and the predicate set ($P$).
• The Four Types of Categorical Statements:
  1. All S are P: This indicates that every member of the subject set is also a member of the predicate set.
  2. No S are P: This indicates that the subject set and the predicate set have no members in common; they are disjoint.
  3. Some S are P: This indicates that there is at least one member of set $S$ that is also a member of set $P$.
  4. Some S are not P: This indicates that there is at least one member of set $S$ that is not a member of set $P$.

Negation of Statements

• Definition of Negation: The negation of a statement is any statement that is false if and only if the original statement is true. A statement and its negation always have opposite truth values.
• Symbolism: The symbol $\sim$ is used for negation. The expression $\sim p$ is read as "not $p$."
  • If $p$ is true, then $\sim p$ is false.
  • If $p$ is false, then $\sim p$ is true.
• Examples of Negation Truth Values:
  • Statement $p$: $2 + 2 = 4$ (LV: $T$). Negation $\sim p$: $2 + 2 \neq 4$ (LV: $F$).
  • Statement $q$: Every prime number is odd (LV: $F$). Negation $\sim q$: It is not true that every prime number is odd (LV: $T$).
  • Statement $r$: There is a real number $x$ such that $x^{2} + 1 = 0$ (LV: $F$). Negation $\sim r$: There is no real number $x$ such that $x^{2} + 1 = 0$ (LV: $T$).
• Negation of Quantified Statements:
  • Negation of All: If a statement claims "All objects of a certain type have a particular property," its negation asserts that "at least one of those objects does not have the property."
    • Example: $p$: All roses are red. $\sim p$: There is at least one rose that is not red.
  • Negation of Some: If a statement claims "Some object of a certain type has a particular property," its negation asserts that "none of those objects has the property."
    • Example: $p$: Some roses are red. $\sim p$: No roses are red.

Logical Operations and Notation

• Summary Table of Notations:
  • Negation: Meaning: "not". Symbol: $\sim$.
  • Conjunction: Meaning: "and". Symbols: $\bullet$ or $\wedge$.
  • Disjunction: Meaning: "or". Symbol: $\vee$.
  • Conditional: Meaning: "if…then". Symbol: $\rightarrow$.
  • Biconditional: Meaning: "if and only if". Symbol: $\leftrightarrow$.
• Specific Negation Rules:
  • Statement: $A$ or $B$. Negation: not $A$ and not $B$.
  • Statement: $A$ and $B$. Negation: Not $A$ or not $B$.
  • Statement: If $A$, then $B$. Negation: $A$ and not $B$.

Conditional Statements

• Definition: A conditional statement is a sentence of the form "if $p$, then $q$," where $p$ and $q$ are given statements.
• Symbolic Form: $p \rightarrow q$ (read as "$p$ implies $q$").
• Example:
  • $p$: $x$ is divisible by 4.
  • $q$: $x$ is even.
  • $p \rightarrow q$: If $x$ is divisible by 4, then $x$ is even.

Variations of Conditional Statements

Given a conditional statement $p \rightarrow q$, the following variations can be formed:

• Converse: $q \rightarrow p$ (If $q$, then $p$).
• Inverse: $\sim p \rightarrow \sim q$ (If not $p$, then not $q$).
• Contrapositive: $\sim q \rightarrow \sim p$ (If not $q$, then not $p$).
• Example 1: Divisibility:
  • Conditional: If $x$ is divisible by 4, then $x$ is even.
  • Converse: If $x$ is even, then $x$ is divisible by 4.
  • Inverse: If $x$ is not divisible by 4, then $x$ is not even.
  • Contrapositive: If $x$ is not even, then $x$ is not divisible by 4.
• Example 2: Geometry:
  • Conditional: If a quadrilateral is a rectangle, then it has two pairs of parallel sides.
  • Converse: If a quadrilateral has two pairs of parallel sides, then it is a rectangle.
  • Inverse: If a quadrilateral is not a rectangle, then it does not have two pairs of parallel sides.
  • Contrapositive: If a quadrilateral does not have two pairs of parallel sides, then it is not a rectangle.

Logical Equivalence and Biconditionals

• Logical Equivalence:
  • A conditional statement is logically equivalent to its contrapositive. Either both statements are true, or both are false.
  • The converse and inverse of a conditional statement are logically equivalent to each other. Either both are true, or both are false.
  • Truth Value Case Study:
    • Conditional ($p \rightarrow q$): If $x$ is divisible by 4, then $x$ is even. (Logical Value: $T$).
    • Contrapositive ($\sim q \rightarrow \sim p$): If $x$ is not even, then $x$ is not divisible by 4. (Logical Value: $T$).
    • Converse ($q \rightarrow p$): If $x$ is even, then $x$ is divisible by 4. (Logical Value: $F$, e.g., $x=6$).
    • Inverse ($\sim p \rightarrow \sim q$): If $x$ is not divisible by 4, then $x$ is not even. (Logical Value: $F$, e.g., $x=6$).
• Biconditional Statements:
  • Definition: A biconditional is a combination of a conditional and its converse. It is expressed as "$p$ if and only if $q$."
  • Symbolic Form: $p \leftrightarrow q$.
  • Truth Values:
    • A biconditional statement is true when both its conditional ($p \rightarrow q$) and converse ($q \rightarrow p$) are true.
    • A biconditional statement is false when either its conditional or its converse (or both) are false.
  • Example:
    • Conditional: If $x$ is divisible by 4, then $x$ is even. ($T$).
    • Converse: If $x$ is even, then $x$ is divisible by 4. ($F$).
    • Biconditional: $x$ is divisible by 4 if and only if $x$ is even. (Logical Value: $F$).