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: ...