Unit 4: Modern Symbolic Logic

Symbolization of Propositions

Modern logic uses symbols to represent statements and their connections, allowing for a more precise analysis of arguments.

Five Basic Truth-Functions

In modern logic, symbols are used to connect simple statements into compound ones. The primary logical operators are:

• Negation (~): "Not" (e.g., ~P means "It is not the case that P").
• Conjunction (•): "And" (e.g., P • Q).
• Disjunction (v): "Or" (e.g., P v Q).
• Implication (¹): "If... then..." (e.g., P ¹ Q).
• Equivalence (º): "If and only if" (e.g., P º Q).

Testing Validity by Truth-Table Method

A truth table is a mechanical decision procedure used to determine the validity of an argument by exploring all possible truth values of its components.

Defining the Operators

The Test for Validity

An argument is Invalid if there is at least one row where all premises are True (T) but the conclusion is False (F). If no such row exists, the argument is Valid.

Shorter Truth-Table Method for Proving Invalidity

The Shorter Truth-Table Method is a more efficient way to prove that an argument is invalid without drawing a full table.

Step-by-Step Procedure

1. Assume Invalidity: Assign False (F) to the conclusion and True (T) to all premises.
2. Back-Solve: Work backward to assign truth values to the individual variables (P, Q, R) based on those assignments.
3. Check for Consistency: If you can successfully assign truth values to all variables without creating a contradiction, the argument is proven Invalid.
4. Contradiction: If you find it impossible to make the premises true while the conclusion is false, the argument is Valid.

Exam Focus: FAQs

• Q: When is a conditional (P ¹ Q) false?
  A: Only when the antecedent (P) is True and the consequent (Q) is False.
• Q: Why use the shorter method?
  A: A full truth table for an argument with 4 variables requires 16 rows. The shorter method saves time by looking for the one specific case that proves invalidity.