Rules of Substitution

Explore how rules of substitution complement inference rules in propositional logic. Understand how logically equivalent sentences can replace each other to prove argument validity, illustrated through examples like De Morgan's laws and modus tollens. This lesson helps you apply these rules to analyze and validate complex arguments effectively.

We'll cover the following...

Valid arguments that defy inference
- Step 1: Assign symbols
- Step 2: Translate to propositional logic syntax
Missing link—substitution rules
- De Morgan’s rule
Back to the grand dinner
- Step 3: Use rules of inference and substitution for validity
Other rules of substitution

Valid arguments that defy inference

There are some arguments in natural language that sound logically valid, but apparently, there’s no way to prove their validity using any combination of inference rules. Here’s an instance:

Imagine there are three friends: Alice, Bob, and Carol. Carol has a rule that says, “If Carol hosts a grand dinner, then Alice and Bob both come to the party.” However, Alice and Bob are known for being unpredictable. One or both of them might not show up. So, the situation is such that either Alice or Bob won’t come to the party. Is it safe to conclude that Carol won’t be hosting the grand dinner?

Let’s translate this to propositional logic.

Step 1: Assign symbols

$\text{A}$ : Alice comes to the party.
$\text{B}$ : Bob comes to the party.
$\text{C}$ : Carol hosts the grand party.

Step 2: Translate to propositional logic syntax

P1: $\text{C} \to (\text{A} ∧ \text{B})$
P2: $\neg A \lor \neg B$ ...

1.Introduction to Logical Thinking

2.Fallacies in Logic

3.Logical Argument

4.What Is Logic?

5.First Theory of Logic: Aristotle’s Logic

6.Second Theory of Logic: Propositional Logic

7.Third Theory of Logic: Predicate Logic

8.Logical Reasoning in Action

9.Wrap Up

Rules of Substitution

Valid arguments that defy inference

Step 1: Assign symbols

Step 2: Translate to propositional logic syntax