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: $¬\text{A} ∨ ¬\text{B}$

• ∴ $\neg \text{C}$

Now, let’s try out all the known rules of inference. But can we prove that P1 and P2 logically lead to the conclusion? Are we missing something important in trying to avoid using a truth table?

Missing link—substitution rules

Does it mean there’s more to logic than the simple rules of inferences? Thankfully, some logicians devised using logically equivalent sentences to substitute others. For instance:
$\text{A} \stackrel{T}{\equiv} \neg(\neg(\text{A}))$

The operator $\stackrel{T}{\equiv}$ denotes a tautological equivalence between the logical sentences on both sides. The rule above is known as double negation, and it helps substitute one side with another because both always have the same truth values.

Let’s see another rule of substitution that suggests that Carol doesn’t have to host the grand dinnerThis points to the argument at the start of the lesson..

De Morgan’s rule

Imagine yourself saying the following to a guy named De Morgan:

I can study maths and I can study logic, but I will not do both.

• $\text{A}$= I can study maths.

• $\text{B}$= I can study logic.

Augustus De Morgan, a celebrated British mathematician, would translate this sentence in symbolic logic as follows:

$\neg(\text{A} \land \text{B})$

Then, through the following truth table, he would find another symbolic logic expression (last column) equivalent to the abovementioned expression (second-last column).