Introduction to Logic

Pioneers of Logic

• Augustus De Morgan and George Boole contributed to the advancement of symbolic logic as a mathematical discipline.

Logic Statements

• A declarative sentence that can be classified as either true or false.
• A statement cannot be a question, an instruction, or an opinion.
• Examples: "Every triangle has three sides", "The price of a Samsung Galaxy tablet was Php 11, 000 on December 24, 2014".
• Statements are usually labeled with lowercase letters (p, q, r, s, etc.).
• True statements are denoted by T, and false statements are denoted by F.

Compound Statements

• Formed by connecting two or more statements or by negating a single statement.
• Connectives used to form compound statements: Not, And, Or, If-then.

Negation

• The negation of a simple statement is the opposite of the original statement.
• Usually done by putting "not" into the statement.

Conjunction

• Two simple statements are combined with the word "and".

Disjunction

• A logical connective typically notated as "V" and read aloud as "or".
• Examples: "Today is Friday and it is raining", "It is not raining and I am going to a movie".

Truth Tables

• A breakdown of a logic function by listing all possible values the function can attain.
• A table typically contains several rows and columns, with the top row representing the logical variables and combinations.

Negation, Double Negation, Conjunction, and Disjunction Truth Tables

• Each table lists all possible values for the respective logical operation.

Equivalent Statements

• Theorem that states "If a statement is true, then it is not the case that the statement is not true".

Tautologies

• A statement represented by an entire column of T values in a truth table.
• Such a proposition is always true, no matter what.

Conditional and Biconditional Statements

• One of the most important ways to combine two statements is by the conditional-consequence linkage, also known as the "if-then" form.
• The parts of the conditional: "if p, then q" can be identified by name:
  • p is called the premise, hypothesis, or the antecedent.
  • q is called the conclusion or the consequent.