logic

Logic

Introduction to Statements, Negations, and Quantified Statements

Objectives

  • Identify English sentences that are statements.
  • Express statements using symbols.
  • Form the negation of a statement.
  • Express negations using symbols.
  • Translate a negation represented by symbols into English.
  • Express quantified statements in two ways.
  • Write negations of quantified statements.

Definitions and Concepts

Sentences, Statements, and Scientific Hypotheses

  • Simple Sentences: In everyday conversation, we use simple sentences which contain one verb. Examples include:

    • (a) I am tired.
    • (b) Please open the door.
    • (c) I traveled to Mexico by plane.
    • (d) The earth is round.
    • (e) You are absolutely right.
    • (f) Calgary receives more precipitation during summer than winter.
    • (g) Please show your driving license.
    • (h) Have you been driving to school?
    • (i) Stop pushing the door!

  • Simple Statements: Not all simple sentences are statements; only those that can be validated as true or false are considered simple statements. Examples include:

    • (1) It is really sunny outside.
    • (2) My microphone is muted.
    • (3) It is cold outside.
    • (4) It is 14th of January.
    • (5) The earth is round.
    • (6) Commands and questions, like "Please open the door!" or "Have you been driving to school?", cannot be true or false.

  • Scientific Statements (Hypotheses): A statement becomes a scientific statement (hypothesis) if its truth can be verified through experiments, regardless of the outcome. Key points include:

    • Experimental verification must yield numerical or quantitative outcomes.
    • Statements that cannot be verified remain simple statements only.

Symbolic Logic

Using Symbols to Represent Statements

  • Lowercase Letters: In symbolic logic, lowercase letters such as p, q, r, and s represent statements. For example:

    • (p) London is the capital of England.
    • (q) The earth is rotating about its axis.

  • Equivalent Statements: Two statements are equivalent if they both have the same truth value (both T or both F). Examples include:

    • (a) This airplane is heavy. (Equivalent: Today is NOT light.)
    • (b) My computer does not work. (Equivalent: My computer crashed or My computer is broken.)

Negating Simple Statements

  • Definition of Negation: The negation of a statement modifies it so that it changes to the opposite truth value. Examples:

    • The negation of a true statement (e.g., "The earth is round") becomes false, hence "The earth is NOT round."
    • The negation of a false statement (e.g., "Today is cold") becomes true, hence "Today is NOT cold."

  • Methods: Negating can be achieved by adding "NOT" or removing "NOT" from the verb. Examples of negated statements:

    • a) Earth is round. => The earth is NOT round.
    • b) Today is cold. => Today is NOT cold.
    • c) My computer does not work. => My computer does work.
    • Symbolically: Negation is indicated by the symbol ~ before a statement. Example:
    • Let p and q represent the following statements:
      • p: The surface of the earth is static.
      • q: Today is not Monday.
      • Negated versions:
      • a. The surface of the earth is NOT static → ~p
      • b. Today is Monday → ~q

Quantified and Non-quantified Simple Statements

  • Categories: Simple statements are categorized into two types: Quantified and Non-quantified.
  • Quantifiers: Simple statements containing terms like all, some, none are called simple quantified statements. Examples include:
    • All poets are writers.
    • Some people do not wear masks.
    • None of drivers is at fault in this accident.

Negating Quantified Statements

  • Example: Negate the following statement: p: All MRU students wear masks.

  • Possible responses:

    • ~p: Some students do NOT wear masks.
    • ~p: ALL MRU STUDENTS DO NOT WEAR MASKS.
    • ~p: NO MRU STUDENTS WEAR MASKS.
    • ~p: NOT ALL MRU STUDENTS WEAR MASKS.

  • Using Venn Diagrams: Negating quantified statements can be visually represented through Venn diagrams showing the relationship between various sets.

Compound Statements

Types of Compound Statements

  1. Conjunction (AND Statements): Connected using "and" represented as p ∧ q.
  2. Disjunction (OR Statements): Connected using "or" represented as p ∨ q, where it may mean exclusive or inclusive.
  3. Conditional Statements (If-Then): Represented as p → q, where p is the antecedent and q is the consequent.
  4. Biconditional Statements (If and Only If): Represented as p ↔ q, indicating the statements have the same truth value.

Truth Tables

Negation, Conjunction, and Disjunction Truth Tables

  • Truth Table Construction: Truth values are determined by evaluating the truth of statements.
  • Examples:
    • Truth table for conjunction (p ∧ q): Only true when both p and q are true.
    • Truth table for disjunction (p ∨ q): Only false when both p and q are false.

Constructing Truth Tables for Compound Statements

  • Processes to Construct: A systematic approach includes listing out all possible truth values for simple statements and calculating results for compound statements.
  • Examples Include:
    • (p → q)
    • [p ∨ (q ∧ r)]

Logical Equivalences

Definitions of Symbolic Logic

  1. Negation: ~p (not): the opposite truth value of p.
  2. Conjunction: p ∧ q (and): true only if both p and q are true.
  3. Disjunction: p ∨ q (or): false only when both are false.
  4. Conditional: p → q (if-then): false only if p is true and q is false.
  5. Biconditional: p ↔ q (if and only if): true when both have the same truth value.

Arguments

General Format

  • Components of Arguments:
    • Premises (P1, P2, etc.) must lead logically to a conclusion (C).
    • Validity: This can be determined through truth tables where the argument is valid if the conclusion follows necessarily from the premises.
Example of Valid Argument:
  • (Premise 1) If Mr. Scott is still with us, then the power will come on.
  • (Premise 2) The power comes on.
  • (Conclusion) Mr. Scott is still with us.

Validity Check Steps

  1. Represent each statement symbolically.
  2. Combine premises using logical operators to form the antecedent.
  3. Evaluate using truth tables.
Conclusion
  • Ensure to follow the logical flows and clarify definitions of all statements involved in the argument.

This structured approach provides a thorough basis for understanding logic, including the manipulation of statements through negation and logical equivalences, culminating in complex arguments that maintain validity through established logical principles.