Formal Logic help!! Forum

[kky215]

I am having difficulty in making FL diagram involving either or. How do you make FL diagram and its contrapositive with the 3 cases listed below? Thanks in advance

1. Either A or B or both
Ex) if either A or B or both are present, then so is C.

2. Either A or B but not both
Ex) if either A or B but not both are present, then C is.

3. If it is not the case that both A and B are present, then C is.

[Cerebro]

1. Either A or B or both
Ex) if either A or B or both are present, then so is C.

A or B --> C

If you prefer function syntax:
OR(A,B) --> C

2. Either A or B but not both
Ex) if either A or B but not both are present, then C is.

you can do:
(A or B) and ~(A and B) ---> C

or more compact using xor (exclusive or) -- NOTE: Haven't seen this mentioned in PS material, not sure if other books use this or not, but I do:

A xor B ---> C

If you prefer function syntax:
XOR(A,B) ---> C

3. If it is not the case that both A and B are present, then C is.

~A and ~B ---> C

or

~(A and B) ---> C

function syntax (for completeness):

~AND(A,B) ---> C

[katjust]

1.

Contrapositive of the "Ex"

~C --> ~(A or B)

2.

Contrapositive of the "Ex"

~C --> (A & B) and ~(A or B)

3. Contrapositive

~C --> (A & B)

[Joeshan520]

This is the easy stuff. Wait till you have to diagram stuff like "Neither Ross nor Ann is a vegan or vegetarian"

[kky215]

~ross and ~ann --> vegan or vegetarian
~vegan and ~vegetarian --> ross or ann

No?

This is the easy stuff. Wait till you have to diagram stuff like "Neither Ross nor Ann is a vegan or vegetarian"

[gaud]

http://plato.stanford.edu/entries/necessary-sufficient/

[kky215]

Thanks for clearing this up.

1. Either A or B or both
Ex) if either A or B or both are present, then so is C.

A or B --> C

If you prefer function syntax:
OR(A,B) --> C

2. Either A or B but not both
Ex) if either A or B but not both are present, then C is.

you can do:
(A or B) and ~(A and B) ---> C

or more compact using xor (exclusive or) -- NOTE: Haven't seen this mentioned in PS material, not sure if other books use this or not, but I do:

A xor B ---> C

If you prefer function syntax:
XOR(A,B) ---> C

3. If it is not the case that both A and B are present, then C is.

~A and ~B ---> C

or

~(A and B) ---> C

function syntax (for completeness):

~AND(A,B) ---> C

[Cerebro]

I don't know if it helps you or not, but I've sometimes found it helpful to construct truth tables using Excel (or you can draw them out manually using a pencil and piece of paper). The one below is a very simple example based on the two variables, A and B, that were discussed in the OP. You should be able to understand what these notations mean automatically when working with these expressions, but if you are confused about what ~(A xor B) means, for example, the visualization in a table like this might be useful to help your understanding.

[TylerJonesMPLS]

1. Either A or B or both
Ex) if either A or B or both are present, then so is C.

The LSAT only uses Inclusive Or, which can also be expressed as And/Or. When you see A or B on the LSAT, you can assume it means And/Or.

So, either A or B or both = A or B

1. Ex: (A or B) > C
Sometimes it’s useful to separate them:
(A > C) and (B > C)

The contrapositive is ~C > ~(A or B) which can be simplified to:
~C > ~ A & ~ B

By DeMorgan’s Laws.

By the way, either A or B (using inclusive Or = And/Or) is logially equivalent to ~A > B and also logically equivalent to Unless.

2. Either A or B but not both.

[(A or B) & ~(A & B)] this could also be expressed as:

[(~A > B) & (A > ~B)].
In English this would be expressed as:
“Not Neither A Nor B, AND Not Both A and B”.

2. Ex:
[(~A > B) & (A > ~B)] > C


Contrapositve is~C > ~[(~A > B) & (A > ~B)] which can be simplified to:

~C > [~(~A > B) or ~(A > ~B)] which can be simplified to:

~C >[(~A & ~B) or (A & B)] Again, by DeMorgan’s Laws

In English, the contrapositive would be expressed as “Neither Or Both”.



3. If it is not the case that both A and B are present, then C is.
~(A & B) > C which can be simplified to:
~A or ~B > C
Contrapositive: ~C > (A & B)

[TylerJonesMPLS]

~ross and ~ann --> vegan or vegetarian
~vegan and ~vegetarian --> ross or ann

No?

This is the easy stuff. Wait till you have to diagram stuff like "Neither Ross nor Ann is a vegan or vegetarian"

Maybe more like:

~r(Vegan) & ~r(Vegetarian)
~a(Vegan) & ~a(Vegetarian)

[TylerJonesMPLS]

1. Either A or B or both
Ex) if either A or B or both are present, then so is C.

A or B --> C

If you prefer function syntax:
OR(A,B) --> C

2. Either A or B but not both
Ex) if either A or B but not both are present, then C is.

you can do:
(A or B) and ~(A and B) ---> C

or more compact using xor (exclusive or) -- NOTE: Haven't seen this mentioned in PS material, not sure if other books use this or not, but I do:

A xor B ---> C

If you prefer function syntax:
XOR(A,B) ---> C

3. If it is not the case that both A and B are present, then C is.

~A and ~B ---> C

or

~(A and B) ---> C

function syntax (for completeness):

~AND(A,B) ---> C

I don't think that ~A & ~B is the logical equivalent of ~(A & B). You distribute a negation across a parenthesis by negating the terms and changing the "&" to "or" (or vice versa). (DeMorgan's Laws)
So, ~(A & B) is equivalent to ~A or ~B .

As an example using Apples and Bagels,
~(A & B) could mean any of three things: Apples but no Bagels, Bagels but no Apples, no Apples and no Bagels.
All we know is that there can't be both Apples and Bagels.

~A & ~B can only mean one thing: no Apples and no Bagels.

[VUSisterRayVU]

~ross and ~ann --> vegan or vegetarian
~vegan and ~vegetarian --> ross or ann

No?

This is the easy stuff. Wait till you have to diagram stuff like "Neither Ross nor Ann is a vegan or vegetarian"

Maybe more like:

~r(Vegan) & ~r(Vegetarian)
~a(Vegan) & ~a(Vegetarian)

Yeah, I don't know that I would really diagram that? Ross isn't vegan or vegetarian. Ann isn't vegan or vegetarian. Pretty cut and dry?

[Cerebro]

I don't think that ~A & ~B is the logical equivalent of ~(A & B). You distribute a negation across a parenthesis by negating the terms and changing the "&" to "or" (or vice versa). (DeMorgan's Laws)
So, ~(A & B) is equivalent to ~A or ~B .

As an example using Apples and Bagels,
~(A & B) could mean any of three things: Apples but no Bagels, Bagels but no Apples, no Apples and no Bagels.
All we know is that there can't be both Apples and Bagels.

~A & ~B can only mean one thing: no Apples and no Bagels.

Thank you, Tyler Jones. I stand corrected!

[TylerJonesMPLS]

Thanks, Celebro, for being so nice about it. I was really tired and in a hurry when I wrote that, and later I was afraid I had been impolite. Cheers!