Validity through Truth Tables
Explore how to assess the validity of logical arguments using truth tables. Learn to translate natural language arguments into propositional logic, model possible scenarios, and apply a clear rule to determine if an argument is valid or invalid. This lesson equips you with techniques to automate reasoning using truth tables in logical proofs.
Validity from truth tables
So far, we’ve learned to resolve any sentence’s truth value using truth tables. But the whole point of logic (as well as logical language) is to validate an argument as correct or incorrect reasoning. For instance, the following is an argument in natural language. What would it look like when translated into propositional logic?
The argument above sounds reasonable, and the conclusion follows from the two premises. Let’s look at another argument. Is this argument valid as well?
This one doesn’t seem valid, right?
Note: This example requires knowledge of special linguistics and animals to know that an animal with fur is not necessarily always a cat. Our claim was that propositional logic does away with these constraints.
Question: How can a truth table help us mechanize or automate deciding if an argument is valid or invalid?
Model and truth table
Every row in a truth table depicts a possible world situation or a model of the world. A sentence, and even an argument that uses two symbols, will always have four models possible, no matter the complexity.
Simpler example
Let’s try a simple example for creating a truth table using two symbols (p, q):