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Dave Britten · (This post was last modified: 05-30-2015 06:33 PM by Dave Britten.)

This is from Professor Stewart's Cabinet of Mathematical Curiosities, a rather good book so far. Should make for good beach/camping reading

The logic on this one particular puzzle isn't adding up for me, though.

Quote:No cat that wears a heron suit is unsociable.
No cat without a tail will play with a gorilla.
Cats with whiskers always wear heron suits.
No sociable cat has blunt claws.
No cats have tails unless they have whiskers.
Therefore:
No cat with blunt claws will play with a gorilla.
Is the deduction logically correct?

I came up with yes, it is. But then I checked the answer in the back of the book. (Spoilers below.)

Quote:The deduction is incorrect. Consider a cat with blunt claws that plays with a gorilla, does not wear a heron suit, has a tail, has no whiskers, and is unsociable. The first five statements are all true, but the sixth is not.

Wait, huh?

Quote:No cats have tails unless they have whiskers.

Quote:...has a tail, has no whiskers...

Am I parsing that wrong, or do those two qualities contradict rule five? To me, that rule reads "tail -> whiskers".

Gerald H · (This post was last modified: 05-30-2015 07:11 PM by Gerald H.)

Yes, you're parsing wrong.

"Unless" is "If not" so

No cats have tails unless they have whiskers

is

Only if cats have whiskers then they have tails

or if you like

Whiskers == tails.

Dave Britten · 05-30-2015, 08:01 PM

(05-30-2015 07:02 PM)Gerald H Wrote: Yes, you're parsing wrong.

"Unless" is "If not" so

No cats have tails unless they have whiskers

is

Only if cats have whiskers then they have tails

or if you like

Whiskers == tails.

Well then that's an even more direct contradiction to say "tail iff whiskers", since he says...

Quote:...has a tail, has no whiskers...

...and says that meshes with all of the first five statements.

I was interpreting it as "no cats have tails, unless of course they have whiskers, in which case it's okay to have a tail", i.e. tail -> whiskers, or conversely, !whiskers -> !tail.

Gerald H · 05-31-2015, 07:19 AM

Really here you're dealing with a natural language, prone to equivocation & varying in interpretation between speakers, & trying to transform it into a uniquely determined logical form.

A futile task, doomed to failure, the value of the quest questionable.

The silence of other members does not mean they find your or my form correct unless they contradict me.

Thomas Radtke · 05-31-2015, 10:35 AM

I read this:

- cat is unsociable as it has blunt claws
- cat doesn't wear heron suit as it is unsociable
- cat has no whiskers as it doesn't wear a heron suit
- cat has-right now-no tail as it has no whiskers
- cat-right now-doesn't play with the gorilla as it has no tail

I'm apparently wrong and I don't understand your discussion. Please explain.

Bill (Smithville NJ) · 05-31-2015, 12:21 PM

Hi Dave,

This brings back memories of my senior year in college where I took a Philosophy course in Logic. It was sort of half Philosophy and half mathematics. I can remember taking similar word logic problems and converting them to 1 and 0 in logic tables and then solving for whether the conclusion was correct based on the mathematics of the logic table.

Unfortunately, I do not really remember how we set up the logic tables from the word problem. Been too long ago.

Bill
Smithville, NJ

Dave Britten · 05-31-2015, 05:21 PM

(05-31-2015 12:21 PM)Bill (Smithville NJ) Wrote: Hi Dave,

This brings back memories of my senior year in college where I took a Philosophy course in Logic. It was sort of half Philosophy and half mathematics. I can remember taking similar word logic problems and converting them to 1 and 0 in logic tables and then solving for whether the conclusion was correct based on the mathematics of the logic table.

Unfortunately, I do not really remember how we set up the logic tables from the word problem. Been too long ago.

Bill
Smithville, NJ

That's basically what I did, recalling the Boolean logic we did in discrete math back in college.

You can express the rules thusly (using my interpretation for rule 5):

Quote:heron suit -> sociable
not(tail) -> not(play with a gorilla)
whiskers -> heron suit
sociable -> not(blunt claws)
tail -> whiskers

blunt claws -> not(play with a gorilla)

Then it's a matter of making transitive associations from "blunt claws" to "not(play with a gorilla)", using contrapositives where necessary (which is nearly every step).

Quote:blunt claws -> not(sociable)
not(sociable) -> not(heron suit)
not(heron suit) -> not(whiskers)
not(whiskers) -> not(tail)
not(tail) -> not(play with a gorilla)

But he's claiming "tail and not(whiskers)" is valid, which would contradict how I'm interpreting rule 5 (tail -> whiskers).

Reading it again (and again, and again, and again), it seems like it might be possible to interpret it as "The situation in which a cat has a tail unless it has whiskers does not arise", or alternatively, "Having whiskers does not necessitate the lack of a tail".

If that's how it's meant to be understood, then that rule actually doesn't reveal any relationship between tail and whiskers, and thus the deduction would be incorrect.

Time to go dig out that old discrete math textbook and see if I've forgotten some important detail about Boolean logic.

Thomas Radtke · 06-01-2015, 05:12 AM

Maybe the author made a mistake here.

EdS2 · 06-01-2015, 08:45 AM

(05-30-2015 06:32 PM)Dave Britten Wrote:
Quote:No cats have tails unless they have whiskers.

Native English speaker here - from the UK, as is Ian Stewart. Looks like I agree with your analysis:

Quote:No cats have tails unless they have whiskers.

(I agree with the earlier poster that "unless" means "if not" - but it is one of the easiest logical words to misapprehend.)
Let me rewrite a few times:

Quote:All cats are without tails unless they have whiskers.

Quote:if a cat is without whiskers, it will be without a tail.

Quote:no whiskers => no tail.

Quote:tail => whiskers

Quote:tail and no whiskers never happens

which invalidates Stewart's counterexample.

According to http://www.cs.nott.ac.uk/~vxc/g51mcs/ch01_logic.pdf

Quote:No cats have tails unless they have whiskers.

is equivalent to

Quote:(Not tailed) OR whiskered

which we can transform to

Quote:Not (tailed AND not whiskered)

which is the same interpretation, I think.

That pdf concludes that the deduction is correct, and gives a derivation.

Dave Britten · 06-01-2015, 11:03 AM

(06-01-2015 05:12 AM)Thomas Radtke Wrote: Maybe the author made a mistake here.

Yeah, I had figured that was a very real possibility, but wanted more input before making that assumption. After all, Prof Stewart has no doubt studied math much more than I have. This one's probably just a simple transcription/editing error, and proposition 5 was meant to be worded differently. At least it gave me an excuse to review my math 225 textbook from... 12 years ago? Egad.