Formal logic diagramming help Forum

[risktaker]

"Unless public transportation becomes much more popular, traffic will become more congested and pollution from cars will get worse."

TMCnot and PWnot----->PTMP
PTMPnot------>TMC or PW
This is what i thought the disgram should be like. The answer shows the contra positive of this as:
PTMPnot----->TMCnot and PWnot
Why does "and" not switch to "or" when the contrapositive is taken?
Any help greatly appreciated.

[theZeigs]

Weird, the post that is currently right below yours has a similar thing...I just wrote a long thought-process there:

http://top-law-schools.com/forums/viewt ... 6&t=103815

To summarize, "unless" means "if not" so your FIRST if/then should be

PTMPnot --> TMC and PCW

and the contrapositive

TMCnot OR PCWnot --> PTMP.

In English: If public transportation is NOT more popular, then traffic will be more congested AND pollution form cars will get worse. The only way to prevent traffic from being congested is for public transportation to be more popular. The only way to prevent pollution from cars from being worse is for public transportation to be more popular. Thus, if traffic is not more congested, then public transportation must have become more popular. Similarly, if pollution from cars is not worse, then public transportation is more popular.

[theZeigs]

*MODS* - Am I in violation of any trademark/copyright issue here because of the LRB? Thanks.

Ahhh...so I just got tripped up on a similar type of question:

"Unless they find an eyewitness and put the defendant on the stand, they will lose the case."

Here, if you use my (uncorrected) method in the previous post, you diagram

-FEW & -DOS --> LC
therefore
-LC --> FEW or DOS

BUT


this is not the correct way of doing it! The reason is because the contrapositive is now not correct. (here: if I didn't lose the case, then either I found an eyewitness or I put the defendant on the stand. But really, I had to do BOTH according to the language of the prompt in order to not lose the case!) The key here is if you do it my way, negation (i.e. "if no") causes any "AND" that's part of the "IF NOT" clause to become "OR" and any "OR" must become "AND".

This is why the LRB suggests doing something called the "Unless Equation" (TM). To paraphrase, basically, anything modified by "unless" (or anything that means "if no...") becomes the necessary condition.The rest is then negated and is the sufficient. Unfortunately, in the LRB, they don't mention that when you negate something, like in my method, ANDs become ORs. They only tell you to negate "the rest" and make the "unless" the sufficient.

For my example: "Unless they find an eyewitness and put the defendant on the stand, they will lose the case."

-LC --> FEW and DOS
therefore
-FEW or -DOS --> LC

For your example: "Unless public transportation becomes much more popular, traffic will become more congested and pollution from cars will get worse."

-TMC or -PCW --> PTMP
therefore
-PTMP --> TMC AND PCW

So you can use either method you like, but just remember that if you say that "unless" is "if no" you are effectively negating whatever that clauses is, and must then reverse any AND/OR that are present.

This has been helpful for me to think through this, I can't believe the LRB people didn't put this in the book. Although I haven't finished it yet, so we'll see

[blhblahblah]

"Unless public transportation becomes much more popular, traffic will become more congested and pollution from cars will get worse."

You both are confused. For the sake of posterity, the credited diagram is:

~Congested AND ~Pollution worse --> Public transportation popular

contrapositive:

~Public transporation popular --> Congested OR Pollution worse

/end thread.

[blhblahblah]

Also, this isn't formal logic. It's conditional logic.

Formal logic deals strictly with categories and quantities. Conditional logic deals with sufficiency and necessity.

[theZeigs]

I have to admit, this is not my strength.

I am quoting directly from the LRB when I discuss the "Unless equation." (page 126 in my book, which I got in 2008)

This is why the LRB suggests doing something called the "Unless Equation" (TM). To paraphrase, basically, anything modified by "unless" (or anything that means "if no...") becomes the necessary condition.The rest is then negated and is the sufficient.

So: "Unless public transportation becomes much more popular, traffic will become more congested and pollution from cars will get worse."

Put the "Unless" as necesssary ( ... ---> PTMP) and negate the rest and put as sufficient (-TMC and -PCW --> PTMP) but I think when you negate you SHOULD make the AND into OR.

Obviously, I think that the LRB didn't go far enough in explaining that when a clause has an AND/OR in it and must be negated, the AND becomes OR and the OR becomes AND. If I am correct and LRB is correct otherwise, then we should get the same answer as risktaker's book, but....

The answer shows the contra positive of this as:
PTMPnot----->TMCnot and PWnot

I think that he must have typed the wrong answer here. I think he meant:

PTMPnot ----> TMC and PW

Risktaker, can you advise?

[bluejayk]

Also, this isn't formal logic. It's conditional logic.

Formal logic deals strictly with categories and quantities. Conditional logic deals with sufficiency and necessity.

No, different systems of formal logic vary, but they often include conditional logic. People on this board are typically referring to sentential logic, one type of formal logic. That's just translating natural language into logical sentences using the operators "and", "or", "not", "if/then".

[risktaker]

Okay. This is quoting from LSAT 180 from Kaplan TM. This was their way of diagramming it, which is different from the other poster.
"If public transport isn't popular, then we'll have more traffic and pollution." TM
I have a feeling that with unless statements when there's "and" involved, the contrapositive retains the "and". The same also goes for "or" i think.

So it would be PTP~------>More traffic and more pollution
Contrapositive: More traffic~ and more pollution~------->PTP

Any thoughts on this would be appreciated.

[tomwatts]

"Unless public transportation becomes much more popular, traffic will become more congested and pollution from cars will get worse."

My thought process is this:
"Unless" means "if not." So it becomes "If public transportation doesn't become much more popular, traffic will become more congested and pollution from cars will get worse."

Now I represent that (~ is negation):
~public transport more popular -> traffic more congested AND pollution worse
~traffic more congested OR ~pollution worse -> public transit more popular

As a double-check, you could turn this back into words. If traffic doesn't become more congested or pollution doesn't get worse, then we know that public transit got more popular, because unless public transit gets more popular, traffic will become more congested and pollution will get worse.

As for why "and" becomes "or" and vice-versa when negating, I think of it this way. I could say, "If I get good grades and a good score on the LSAT, I'll get into law school." Then if I come to you and say, "Man, life sucks. I didn't get into law school," you couldn't conclude that I BOTH got terrible grades AND got a bad score on the LSAT; all you could say was that I screwed up at least one of those two things, either by failing to get good grades or by failing to get a good score (or both). So the symbols look like this:
good grades AND good score -> get in
~get in -> ~good grades OR ~good score

What was originally creating the conditional (an "if," an "unless") is irrelevant. The rules are the same regardless.

[risktaker]

Yeah, that makes sense. Thank you.

[chewdak]

To get this concept, it helps me to break complex statements into simple ones.
A --> B and C is equivalent to two statements: A --> B and A --> C
contrapositive of either will suffice to negate A:
~B --> ~A or ~C --> ~A, and can be shortened to ~B or ~C --> ~A

[theZeigs]

To get this concept, it helps me to break complex statements into simple ones.
A --> B and C is equivalent to two statements: A --> B and A --> C
contrapositive of either will suffice to negate A:
~B --> ~A or ~C --> ~A, and can be shortened to ~B or ~C --> ~A

This is a nice, concise summary. +1

And I was right! You had mis-entered what the book said! Man, I lost sleep last night because of that. I even was talking about it with my lawyer friend, who replied "I care about real problems." HAHAH

[risktaker]

Oh man. Yeah, sorry about that.

[theZeigs]

Oh man. Yeah, sorry about that.

No worries.