Symbolization of Propositions
Modern logic moves away from the limitations of ordinary language by using symbols. This process, called symbolization, allows us to focus strictly on the logical structure of an argument rather than its verbal content.
Components of Symbolization
- Propositional Constants: Capital letters (A, B, P, Q) represent specific, simple statements.
- Propositional Variables: Small letters (p, q, r) serve as placeholders for any statement.
- Logical Connectives: Symbols that link simple propositions into compound ones.
Logical Constants and Truth-Functions
In modern symbolic logic, the truth-value of a compound proposition is determined entirely by the truth-values of its component parts. The primary logical operators are:
Truth-Table Method for Testing Validity
A truth table is a mechanical procedure that lists all possible combinations of truth-values for the components of an argument. It is used to determine if an argument form is Valid or Invalid.
The Procedure:
- Identify the number of variables (n). The table will have 2n rows.
- Set up columns for each variable and each premise, then for the conclusion.
- Fill in truth values for all possible combinations.
- The Test: An argument is Invalid if there is at least one row where all the premises are True (T) and the conclusion is False (F). If no such row exists, it is Valid.
Shorter Truth-Table Method for Invalidity
The shorter truth-table method is a more efficient "reductio ad absurdum" approach. Instead of drawing the whole table, we try to prove the argument invalid by force.
How to do it:
- Assume Invalidity: Assign F to the conclusion and T to all premises.
- Assign Values: Work backward to assign T or F to the individual variables to satisfy step 1.
- Check for Consistency:
- If you can assign values to all variables consistently without contradiction, the argument is Proven Invalid.
- If you encounter a contradiction (e.g., a variable must be both T and F at once), the argument is Valid.
Exam Focus: Practical Guide
- Exam Tip: In the Shorter Truth-Table, always start with the conclusion if it is an implication (p ¹ q), because there is only ONE way for it to be False (T ¹ F).
- Mnemonic for Implication: Remember "T-F is F". For any other combination (T-T, F-T, F-F), the implication is always True.
- Common Pitfall: Do not confuse "Exclusive Or" with "Inclusive Or." Standard logic uses Inclusive Or (v), where "P or Q" is True even if both are True.
- FAQ: How many rows do I need for 3 variables (P, Q, R)?
Formula: 23 = 8 rows. For 4 variables, you need 16 rows.