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How to represent either/or not both

niaquino
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I hope i'm posting this in the right place. This question could really apply to any section but it came up for me in a logic game (specifically game 4 of Sep 1998).

On the game, one of the rules stipulates: either I or V, but not both, serves on the panel. I immediately figured this would be an easy rule to diagram but I struggled to capture it in one symbol. For example, V :dblline: I only captures the fact that V and I cannot both be selected, but ~I :arrow: V leaves open the possibility that both could be selected.

Is there a single symbol to capture both aspects of the conditional? And if so, is there any value in capturing it with one symbol?
Dave Killoran
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Hi Niaquino,

Yes, this is the right place!

Technically, the way to represent this with a single symbol is: I :dbl: V. Or the contrapositive of that works as well: V :dbl: I.

However, for most students that's probably more confusing than helpful :-D So, I'd probably represent it more like


..... ..... ..... Exactly one of I/V


But the real key is to make sure you show the I/V requirement on the diagram itself, which in this game can be done as:

    ___ ..... ___
    ___ ..... ___
    ___ ..... ___
    I/V ..... I/V
     1 .....     2

The one thing to track here is that you cannot say that if I serves on a panel then V serves on the other panel. But the notation above is telling you that you always have to reserve a space for I or V, so you should be good.

I find that many times with the harder rules, I just need to get the basic idea across in my notation, but then often it's the representation on the diagram itself that makes the rule come alive.

Please let me know if that helps. Thanks!
Dave Killoran
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niaquino
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That is helpful. I think writing it out in words captures the meaning well enough. I feel I get bogged down sometimes trying to concisely notate unusual rules, when simply writing a very brief phrase suffices.
taylorharris24
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What is the contrapositive of this?
Rachael Wilkenfeld
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Hi Taylor,

The contrapositive is formed in this case like any other conditional---flip the terms, and negate the terms.

So if our original statement is "either V or I, but not both" we'd diagram it as shown below:

V :dbl: I

We can read that as "if V, then not I, and if not I, then V."

The contrapositive would switch the a and b, and negate them. It would look like this:

I :dbl: V

We can read this as "if I then not V, and if not V, then I"

Hope that helps!
Rachael