When we first learned the rules of inference, we hadn't yet covered biconditionals. We first learned about biconditionals when learning about the rules of replacement. After learning to do proofs with the rules of inference, we started learning about conditional proofs, and we started proving various rules of replacement. This involved proving biconditionals by first using conditional proof to prove each of the two conditionals they were equivalent to, then conjoining them and using the rule of material equivalence to get the desired biconditional. This works well enough except that the lines can get very long. To streamline the proof of a biconditional and to avoid long lines, I'm introducing a couple rules of inference for biconditionals here. I'll also be using them to prove both forms of the rule of material equivalence. This will give you some more experience with proofs and hammer home what biconditionals and material equivalence are about.

The biconditional is signified by the ≡ symbol, and it means that the expressions on each side share the same truth value. The rules of material equivalence, which we'll cover here, express other details about what a biconditional means. One rule indicates that the two expressions on each side of the biconditional materially imply each other, and the other rule indicates that the two sides of a biconditional are both true or both false. You can watch the video, read the post, or both.

Biconditional Introduction

P ⊃ Q
Q ⊃ P
∴ P ≡ Q

Prove Biconditional Introduction:

1. P ⊃ Q // Premise
2. Q ⊃ P // Premise
Prove: P ≡ Q
3. (P ⊃ Q) & (Q ⊃ P) // 1,2 Conjunction
4. P ≡ Q // 3 Material Equivalence

Biconditional Elimination

P ≡ Q
∴ P ⊃ Q
P ≡ Q
∴ Q ⊃ P

Prove Biconditional Elimination:

1. P ≡ Q // Premise
Prove: P ⊃ Q
2. (P ⊃ Q) & (Q ⊃ P) // 1 Material Equivalence
3. P ⊃ Q // Simplification

Prove this form of Material Equivalence without using it:
(P ≡ Q) ≡ ((P ⊃ Q) & (Q ⊃ P))