Proofs by Natural Deduction

This lesson brings together what we’ve learned about the rules of inference and substitution to prove the validity of arguments. Using these rules to prove validity (instead of proving it through truth tables) is also termed proof by natural deduction.

Entailment

Usually, we write an argument in the following form:

Premise 1,

Premise 2,

Premise 3,

∴ Conclusion.

Here we consider the following:

• We read the comma-separated premises to mean Premise 1 $∧$ Premise 2 $∧$ Premise 3 (ANDed).

• We read that the ANDed set of premises entails (implying/deducing) the conclusion.

• We denote entailment by $⊢$ such that: (Premise 1 $∧$ Premise 2 $∧$ Premise 3) $⊢$ Conclusion.

Simply put, if we create the same symbolic arrangement as that in the argument’s given conclusion by applying the rules of inference and/or substitution, we can prove the argument’s validity through entailment. Let’s apply this to an sample scenario.

First example

Suppose there’s a science competition that a student named Ian can participate in if he both studies $(\text{S})$ and is interested $(\text{I})$. However, it’s known that Ian didn’t participate in the competition $(¬\text{C})$, but it’s also known that he’s still interested $(\text{I})$. From these circumstances, we can conclude that Ian didn’t study $(¬\text{S})$.

We’ve already identified the unique symbols used within the argument. Here’s how we translate the natural language argument to propositional logic syntax: