Truth Tables: Logic and Validity

• Mathematicians use two-valued logic, where every statement is either True or False.
• This concept is called the Law of the Excluded Middle.
• Statements in sentential logic are built from simple statements using logical connectives.
• Truth or falsity depends on the truth or falsity of its components.

Truth Tables

• Used to determine the validity or truth of a compound statement.
• A compound statement consists of one or more simple statements.
• Simple statements are represented by symbols (often letters).
• Each symbol signifies a statement, like “John scored a goal” or “It is raining”.
• Constructing a truth table depends upon the simple statements composing the compound statement.
• A statement can be referred to as a premise or expression, depending on the context.

Steps to Construct a Truth Table:

• Identify the variables present in the logical expression.
• List all possible combinations of true and false values for each variable.
• Evaluate the expression for each combination of truth values and record the result.

Truth Table Example, Two Simple Statements:

• If p=John scored a goal and q=It is raining
• When both John scored a goal and It is raining are both true, both events occurred.

Truth Table Example, Three Simple Statements:

• The number of times T appears consecutively in each column is determined by the number of statements.

• The formula for the number of times T is listed consecutively under statement p is 2 squared.

• The formula for the number of times T is listed consecutively under statement q is 2 to the power of 1.

• The formula for the number of times T is listed consecutively under statement r is 2 to the power of 0.

• The number of times T is listed consecutively in the first statement is half the number of true-false cases.

• The number of true-false cases doubles with the addition of one statement.

• The total number of cases is determined by 2 to the power of the number of statements.

Connectives

• The truth of a compound statement depends on the simple statements and the connective used.
• Connectives are symbols indicating the relationship between simple statements.

Common Connectives:

• Includes negation, conjunction, disjunction, conditional, and biconditional
• When multiple connectives are used, their truth must be determined in a specific order.

Connective Examples:

• p represents the statement "John scored a goal" and q represents "John won the game."

Negation ("Not"):

• Complete negation first.
• The statement is true when the input statement is false.
• ~p represents "John did not score a goal."

Conjunction ("and"):

• Complete conjunction second along with disjunction.
• The statement is true only when both input statements are true; otherwise, it's false.
• p ∧ q represents "John scored a goal and won the game."

Disjunction ("or"):

• Complete disjunction second along with conjunction.
• The statement is false only when both input statements are false; otherwise, it's true.
• p V q represents "John scored a goal or won the game."

Conditional ("if...then"):

• Complete conditional third.
• The statement is false only when the first input statement is true, and the second is false otherwise, it's true.
• p → q represents "If John scored a goal, then won the game."

Biconditional ("if and only if"):

• Complete biconditional last.
• The statement is true when both input statements are true or both are false; otherwise, it's false.
• p ↔ q represents "John scored a goal if and only if he won the game."

Definitions of Terms Related to Truth Tables and Logic:

• Tautology: A logical expression that is always true, regardless of the truth values of its variables.
• Fallacy: An error in reasoning or a faulty argument due to a flaw in the reasoning or premises.
• Contradiction: A logical expression that is always false, regardless of the truth values of its variables.
• Contingency: A logical expression that can be either true or false, depending on the truth values of its variables.

Other Related Terms:

• Proposition: A statement that can be either true or false.
• Truth value: The truth or falsity of a proposition.
• Logical operator: A symbol used to connect propositions to form more complex propositions.
• Truth table: A table that shows the truth values of a logical expression for all possible combinations of truth values of its variables.
• Valid argument: An argument in which the conclusion logically follows from the premises.
• Sound argument: A valid argument with true premises.
• Invalid argument: An argument that is not valid.
• Unsound argument: An argument that is either invalid or has false premises.