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If, Only If, and If and Only If on the LSAT

In a previous blog post, we covered how to interpret basic if/then statements and how to form their contrapositives. In this post, we’ll look at some of the most common alternative forms of if/then statements.

Let’s start with a basic if/then statement and its contrapositive:

    If I am a vegetarian, I don’t eat beef.
    If I do eat beef, I am not a vegetarian.

 
We can also represent these symbolically as:

    If A→ B
    If not B→ not A

 
I recommend getting used to this symbolic notation, as it’s much faster to write out and will help you avoid getting bogged down in the wordiness of the LSAT, particularly in the Logical Reasoning section. Just be careful not to conflate A and not A or B and not B. Remember your triggers!

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Now that we’ve got our standard form written out, let’s look at the first common variation:
 

I don’t eat beef if I’m a vegetarian.

This is a really simple one. We’ve taken the two terms and swapped them, BUT we’ve left the if attached to the A term. This is logically equivalent to our original statement because we moved the if. You can basically just imagine there’s a silent then at the beginning of the sentence, though we don’t put it there in actuality because that wouldn’t make much sense grammatically.


Symbolically, here’s what we’ve got:

    B if A = if A→ B

Now for a slightly trickier one:
 

I’m a vegetarian only if I don’t eat beef.

Here, we’ve kept the terms in their original positions, but we’ve moved the if over to the B term and added the word only. Surprisingly, this is still the logical equivalent of our original statement. Think about it this way: if I can be a vegetarian only if I don’t eat beef and I tell you that I’m a vegetarian, then you can say for certain that I don’t eat beef. Therefore, knowing I’m a vegetarian is sufficient for knowing I don’t eat beef. The A term is still sufficient.

On the flipside, if I tell you that I don’t eat beef, you still can’t be sure I’m a vegetarian because I might eat chicken or fish or pork. The B term is not sufficient, but it is necessary for the A term to be true.

We can represent this symbolically as:

    A only if B = if A→ B

And now for the third variation on the if/then statement:
 

I’m a vegetarian if and only if I don’t eat meat.

Notice that I changed the word beef to meat in this example. Why? Because otherwise the statement would not be logically accurate. Of the three variations we’ve looked at thus far, this one is unique because it is not the logical equivalent of the original if/then statement.

If and only if statements actually create two distinct if/then statements. Let’s represent them symbolically first:

    A if and only if B = if A→ B and if B→ A

In other words, both terms are triggers in an if and only if statement. If I’m a vegetarian, then you know for sure that I don’t eat meat. Likewise, if I don’t eat meat, then you know for sure that I’m a vegetarian. The two terms are thus logical equivalents of each other.

Here’s another example:

    I cycle if and only if it’s not raining.

    If I cycle, it’s definitely not raining. If it’s not raining, I definitely cycle. Therefore, days when I cycle and days when it’s not raining are one and the same. They are logical equivalents.

Furthermore, the contrapositives of both statements are logical equivalents:

    If it’s raining, I don’t cycle. If I don’t cycle, it’s raining.

You can either write out both if/then statements (and their contrapositives) separately, or you can write it as a single statement with a two-way arrow: A ←→ B. This notation means that A and B are always seen together. If you have one, you have the other. If you don’t have one, you can’t have the other.

To recap, here’s how to interpret the most common variations on if/then statements:

    B if A = if A→ B
    A only if B = if A→ B
    A if and only if B = A ←→ B

 
 

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4 Responses to If, Only If, and If and Only If on the LSAT

  1. vasana February 26, 2018 at 3:57 PM #

    How so I solve this- it rains tomorrow if and only if I stay home tomorrow.

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert March 13, 2018 at 12:50 PM #

      This would in theory be expressed and interpreted in the same way as the other “if and only if” statements in this blog article, the ones under the heading “I’m a vegetarian if and only if I don’t eat meat.”

      However, the wording implies a strange, possibly unreal causality. By saying “It rains tomorrow if and only if I stay home tomorrow,” you are suggesting that your decision to stay home controls whether or not it rains. In other words, your actions would control the weather, rather than the weather controlling your actions. The real LSAT wouldn’t have an example like that, indicating a causal relationship that’s not possible in real life. Instead, the LSAT would be more likely to say something like “I will stay home tomorrow if and only if it rains.” In that way,t he weather determines your chosen actions, not the other way around.

  2. Jenna April 1, 2018 at 6:30 PM #

    When I form an if then diagram when do I absolutely use the contrapositive? When both are negated and in all games?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert April 6, 2018 at 1:11 PM #

      Bear in mind that the diagram is something you make for yourself, in order to keep track of what a question is asking, what arrangements a game is implying, and what the answers are. In that sense, you never absolutely have to include contrapositives in your diagrams. Whether or not you include contrapositives is up to you; if you find contrapositives more helpful in the diagram, include them. If diagramming the contrapositives seems to make the diagram too complex or slow you down, then don’t use them.

      With that said, you should always consider contrapositives, at least mentally, anytime you are dealing with if/then scenarios in the Logic Games. Contrapositives are an excellent way to check your interpretation of an if/then scenario and make sure that you’ve interpreted correctly.


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