Every symbol in BOOL has two rules: Intro and Elim. You’ve already learned how to use or eliminate a conjunction with &Elim.

Now we’ll learn how to use &Intro to make or “introduce” a conjunction.

Here’s a simple example:

1. P

2. Q

3. P&Q

Premise

Premise

&Intro;1,2

Conjunction always requires two sentences, so the rule &Intro must cite two lines.

The rule allows you to put together the conjuncts in whatever order you want. For example, like this:

1. P

2. Q

3. Q&P

Premise

Premise

&Intro;1,2

When you cite the steps, * use numerical order*. Some logical systems require you to cite the steps in the order that they get conjoined. That would require you to write &Intro;2,1 here, since the Q is conjoined first and it comes from line 2.

Logical systems are tools that can be designed in many ways. **Our system uses numerical order for line citations.**

But remember, logical systems are tools that can be designed in many ways, and our system uses numerical order for line citations.

It’s your turn to practice with the & rules. Here’s a proof that shows the associativity of &. Your job is to write the justification for each line.

You can re-try auto-check answers without starting over.

When you complete longer proofs like that, which automatically check your answers, here’s a helpful tip: if you get one wrong, you don’t have start the whole problem over. You can put the cursor back in that field, delete your answer, and try again.

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